Question 1<\/h3>\r\nA water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?\r\n\r\n<\/div>\r\n\r\nQuestion 2<\/h3>\r\nIt was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.\r\n\r\n \t- What is the population parameter of interest in this situation?<\/li>\r\n \t
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\r\n \t
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\r\n<\/ol>\r\n<\/div>\r\n\r\n
Question 3<\/h3>\r\nIn the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.\r\n\r\n<\/div>\r\n\r\nQuestion 4<\/h3>\r\nThe FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[footnote]Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/[\/footnote]\r\n\r\n \t- What is the sample proportion?<\/li>\r\n \t
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\r\n<\/ol>\r\n<\/div>\r\nHow likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.\r\n\r\nSince the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?\r\n\r\nn this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:\r\n\r\n[latex] P(\\hat{p}\\geq 0.2) = 0.000002 [\/latex]\r\n\r\n
Question 5<\/h3>\r\nWhat can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nWhat are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?\r\n\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nConsider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?\r\n\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nIn the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/19161639\/11A-InClass-1-225x300.jpg\")
<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 2<\/h3>\r\nIt was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.\r\n\r\n \t- What is the population parameter of interest in this situation?<\/li>\r\n \t
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\r\n \t
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\r\n<\/ol>\r\n<\/div>\r\n\r\n
Question 3<\/h3>\r\nIn the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.\r\n\r\n<\/div>\r\n\r\nQuestion 4<\/h3>\r\nThe FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[footnote]Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/[\/footnote]\r\n\r\n \t- What is the sample proportion?<\/li>\r\n \t
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\r\n<\/ol>\r\n<\/div>\r\nHow likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.\r\n\r\nSince the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?\r\n\r\nn this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:\r\n\r\n[latex] P(\\hat{p}\\geq 0.2) = 0.000002 [\/latex]\r\n\r\n
Question 5<\/h3>\r\nWhat can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nWhat are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?\r\n\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nConsider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?\r\n\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nIn the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 3<\/h3>\r\nIn the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.\r\n\r\n<\/div>\r\n\r\nQuestion 4<\/h3>\r\nThe FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[footnote]Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/[\/footnote]\r\n\r\n \t- What is the sample proportion?<\/li>\r\n \t
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\r\n<\/ol>\r\n<\/div>\r\nHow likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.\r\n\r\nSince the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?\r\n\r\nn this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:\r\n\r\n[latex] P(\\hat{p}\\geq 0.2) = 0.000002 [\/latex]\r\n\r\n
Question 5<\/h3>\r\nWhat can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nWhat are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?\r\n\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nConsider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?\r\n\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nIn the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 4<\/h3>\r\nThe FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[footnote]Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/[\/footnote]\r\n\r\n \t- What is the sample proportion?<\/li>\r\n \t
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\r\n<\/ol>\r\n<\/div>\r\nHow likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.\r\n\r\nSince the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?\r\n\r\nn this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:\r\n\r\n[latex] P(\\hat{p}\\geq 0.2) = 0.000002 [\/latex]\r\n\r\n
Question 5<\/h3>\r\nWhat can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nWhat are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?\r\n\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nConsider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?\r\n\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nIn the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 5<\/h3>\r\nWhat can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nWhat are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?\r\n\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nConsider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?\r\n\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nIn the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 6<\/h3>\r\nWhat are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?\r\n\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nConsider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?\r\n\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nIn the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 7<\/h3>\r\nConsider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?\r\n\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nIn the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 8<\/h3>\r\nIn the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 9<\/h3>\r\nNotice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.\r\n\r\nIn your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n<\/p>\n
Credit: iStock\/kedsanee<\/p>\n
\nQuestion 1<\/h3>\n
A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n
\n- What is the population parameter of interest in this situation?<\/li>\n
- The compliance threshold for water safety is 10%. Let\u2019s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation\u00a0 as a hypothesis test, what is the null hypothesis?<\/li>\n
- The residents who lead the charge in the FWS were convinced there was a\u00a0 higher proportion of homes in the city with contaminated water than is\u00a0 allowable under federal guidelines. What was their alternative hypothesis?<\/li>\n<\/ol>\n<\/div>\n\n
Question 3<\/h3>\n
In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n\n- What is the sample proportion?<\/li>\n
- The DEQ claimed that the city was compliant with federal guidelines and only\u00a0 10% of residences actually had contaminated water. According to the DEQ\u00a0 (and our corresponding null hypothesis), what was the assumed population\u00a0 proportion of residences with contaminated water?<\/li>\n<\/ol>\n<\/div>\n
How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n
Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n
n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n
[latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n
\nQuestion 5<\/h3>\n
What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n
\nQuestion 7<\/h3>\n
Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n
\nQuestion 8<\/h3>\n
In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n
In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n
Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n
<\/p>\n
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li>
- Barry-Jester, A. M. (2016, January 26). What went wrong in Flint.<\/em> FiveThirtyEight. https:\/\/fivethirtyeight.com\/features\/what-went-wrong-in-flint-water-crisis-michigan\/ ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5309","chapter","type-chapter","status-publish","hentry"],"part":5305,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions"}],"predecessor-version":[{"id":5311,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/revisions\/5311"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5305"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 10<\/h3>\r\nBased on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.\r\n\r\nSummarize how the FWS provided evidence that contributed to these decisions.\r\n\r\n<\/div>\r\n ","rendered":"
As we\u2019ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.<\/p>\n
In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint\u2019s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]<\/sup><\/a><\/p>\n Credit: iStock\/kedsanee<\/p>\n A water sample is \u201ccontaminated\u201d if it contains lead levels of 15 ppb or above. What\u00a0 percentage of residences in the FWS\u2019s sample returning contaminated water\u00a0 samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?<\/p>\n<\/div>\n It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not\u00a0 compliant with federal water safety regulations. In other words, it was their\u00a0 responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.<\/p>\n In the FWS\u2019s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,\u00a0 strong evidence against the null hypothesis that the city was compliant and strong\u00a0 evidence that there was a higher proportion of homes with lead-contaminated water\u00a0 than had been believed). Explain.<\/p>\n<\/div>\n The FWS took a sample of 271 residences in Flint spread throughout the city, and\u00a0 they found contaminated water samples in 20% of their sample.[2]<\/sup><\/a><\/p>\n How likely would it have been for the FWS to obtain a sample with a proportion of\u00a0 contaminated residences as high as 0.20 if the city was actually compliant? We can use\u00a0 the sampling distribution of the sample proportion and the normal distribution to\u00a0 determine this probability.<\/p>\n Since the normal distribution is continuous, it does not make sense to consider the\u00a0 probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we\u2019re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other\u00a0 words, how likely are we to have a sample proportion fall in that high range of 20% or\u00a0 more if the true population proportion is only 10%?<\/p>\n n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:<\/p>\n [latex]P(\\hat{p}\\geq 0.2) = 0.000002[\/latex]<\/p>\n What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?<\/p>\n<\/div>\n What are the two possible explanations for the high proportion of residences with\u00a0 contaminated water that the FWS found?<\/p>\n<\/div>\n Consider the large proportion of residences with contaminated water found by the\u00a0 FWS. What would you conclude about the water in Flint based on that sample?<\/p>\n<\/div>\n In the context of the hypothesis test, would you reject or fail to reject the null\u00a0 hypothesis?<\/p>\n<\/div>\n Notice that in conducting this hypothesis test, we began by assuming the null\u00a0 hypothesis was true (i.e., it was already the assumption that Flint was compliant with\u00a0 federal regulations). The FWS then collected evidence in support of the alternative\u2014 that Flint was not compliant. The possibilities are then that the FWS obtained\u00a0 enough evidence to reject the null hypothesis or that it did not obtain enough\u00a0 evidence to reject the null hypothesis. As we saw in the preview assignment, the\u00a0 only possible conclusions of a hypothesis test are to reject the null hypothesis or to\u00a0 fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in\u00a0 which one \u201caccepts\u201d the null hypothesis.<\/p>\n In your own words, describe the only possible conclusions of this hypothesis test,\u00a0 and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.<\/p>\n<\/div>\n Based on the results of the FWS, as well as a study done by a pediatrician on the\u00a0 blood lead levels of children in Flint, the city switched the water supply back to the\u00a0 previous source, and the governor declared a state of emergency a few months\u00a0 later.<\/p>\n Summarize how the FWS provided evidence that contributed to these decisions.<\/p>\n<\/div>\n <\/p>\n<\/p>\nQuestion 1<\/h3>\n
Question 2<\/h3>\n
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Question 3<\/h3>\n
Question 4<\/h3>\n
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Question 5<\/h3>\n
Question 6<\/h3>\n
Question 7<\/h3>\n
Question 8<\/h3>\n
Question 9<\/h3>\n
Question 10<\/h3>\n