{"id":5261,"date":"2022-08-19T01:14:31","date_gmt":"2022-08-19T01:14:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5261"},"modified":"2022-08-19T01:14:31","modified_gmt":"2022-08-19T01:14:31","slug":"10b-preview","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/10b-preview\/","title":{"raw":"10B Preview","rendered":"10B Preview"},"content":{"raw":"Preparing for the next class\r\n\r\nIn the next in-class activity, you will need to identify the components of a confidence\u00a0 interval from a summary provided by the DCMP Inference for a Population Proportion\u00a0 tool. You will also need to construct confidence intervals with technology and interpret\u00a0 their meaning.\r\n\r\nIn In-Class Activity 8.B, we explored data from a national survey of college students\u00a0 conducted by the American College Health Association. One of the questions posed to\u00a0 the participants was, \u201cOn how many of the last 7 days did you take a nap?\u201d Out of the\u00a0 38,440 participants, 13,888 stated they had not taken a nap in the last 7 days.\r\n\r\nThe image below shows the results of calculating a confidence interval to estimate the\u00a0 population proportion of college students who have not had a nap in the last 7 days.\r\n\r\n\"\"\r\n
\r\n

Question 1<\/h3>\r\nGiven the previous calculations shown, which of the following is the point-estimate\u00a0 for the proportion of college students who have not had a nap in the last 7 days?\r\n
    \r\n \t
  1. 0.3613<\/li>\r\n \t
  2. 0.0025<\/li>\r\n \t
  3. 0.3565<\/li>\r\n \t
  4. 0.3565<\/li>\r\n<\/ol>\r\nHint: The point-estimate is the proportion calculated from our sample.\r\n\r\n<\/div>\r\n
    \r\n

    Question 2<\/h3>\r\nWhich of the following confidence levels was used to calculate the confidence\u00a0 interval presented in the previous image?\r\n
      \r\n \t
    1. 90%<\/li>\r\n \t
    2. 95%<\/li>\r\n \t
    3. 99%<\/li>\r\n \t
    4. It is impossible to know.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Question 3<\/h3>\r\nGiven the calculations shown, identify the lower bound and upper bound of the\u00a0 calculated confidence interval.\r\n
      \r\n\r\n\r\n\r\n\r\n
      Lower bound<\/td>\r\nUpper bound<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n \r\n\r\nAccurately interpreting a confidence interval is just as important as ensuring our\u00a0 calculations are correct. One common but incorrect interpretation is that the confidence\u00a0 level is the probability (expressed as a percentage) that the population proportion is\u00a0 contained within the bounds of our confidence interval.\r\n\r\nFor example, using the confidence interval calculated previously, a student might\u00a0 incorrectly state:\r\n\r\n\u201cThere is a 95% chance that the population proportion of college students who have not\u00a0 had a nap in the last 7 days is between 0.3565 and 0.3661, or 35.65% and 36.61%.\u201d\r\n\r\nRather than measuring the likelihood that a single confidence interval contains the\u00a0 population proportion, the confidence level instead tells us the percentage of all confidence intervals that we\u2019d expect to contain the population proportion, if we were to\u00a0 repeatedly take random samples and construct confidence intervals around our point estimates.\r\n\r\nLet\u2019s explore this idea a bit more.\r\n\r\n<\/div>\r\n
      \r\n

      Question 4<\/h3>\r\nSuppose we knew that the population proportion of college students who did not\u00a0 take a nap in the last week was 0.40, or 40%. Of course, in reality we wouldn\u2019t know\u00a0 the population proportion,\u2014which is why we\u2019re creating a confidence interval to\u00a0 begin with\u2014but let\u2019s assume that we did.\r\n\r\nNow suppose we took 10 random samples of 100 college students and constructed\u00a0 a 95% confidence interval for each one. We can run this simulation using the DCMP\u00a0 Explore Coverage and Confidence Intervals tool available at\r\n\r\nhttps:\/\/dcmathpathways.shinyapps.io\/ExploreCoverage\/.\r\n\r\nThe following picture illustrates a simulated result of this situation using the tool (i.e.,\u00a0 10 confidence intervals constructed from our 10 random samples).\r\n\r\n\"\"\r\n
        \r\n \t
      1. The red lines in the previous picture represent the confidence intervals that\u00a0 did not contain the population proportion 0.40. Out of the 10 confidence\u00a0 intervals from this simulated result, how many of them did not contain the\u00a0 population proportion 0.40?\r\nHint: The red lines are those that did not contain the population proportion\u2014how\u00a0 many red lines are in the previous picture?<\/li>\r\n \t
      2. The green lines in the previous picture represent the confidence intervals\u00a0 that did contain the population proportion that we assumed (0.40). Out of the\u00a0 10 confidence intervals from this simulated result, how many of them did\u00a0 contain the population proportion 0.40?\r\n\r\nHint: The green lines are those that actually did contain the population proportion\u2014 how many green lines are in the previous picture?<\/li>\r\n \t
      3. Given your answers to Parts a and b, what proportion of the 10 confidence\u00a0 intervals did contain the population proportion 0.40? What is this value as a percentage?\r\n\r\nHint: In Part b, you calculated the number of intervals containing the proportion 0.40.\u00a0 Divide this value by 10 to calculate the proportion.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
        \r\n

        Question 5<\/h3>\r\nNow suppose we took 200 random samples of 100 college students and constructed\u00a0 a 95% confidence interval for each one. The picture below illustrates a simulated\u00a0 result of this situation (i.e., 200 confidence intervals constructed from our 200\u00a0 random samples).\r\n\r\n\"\"\r\n
          \r\n \t
        1. The red lines in the previous picture represent the confidence intervals that\u00a0 did not contain the population proportion 0.40. Out of the 200 confidence intervals, how many of them did not contain the population proportion 0.40?\r\nHint: The red lines are those that did not contain the population proportion\u2014how\u00a0 many red lines are in the previous picture? It might be easier to count the red\u00a0 squares to identify the number of confidence intervals that did not contain the\u00a0 population proportion.<\/li>\r\n \t
        2. Given your answer to the previous question and knowing there are 200 total confidence intervals, how many confidence intervals from this simulated result did contain the population proportion?\r\nHint: Subtract the value you calculated in Part a from 200 to identify the remaining\u00a0 number of intervals that did contain the population proportion.<\/li>\r\n \t
        3. Given your previous answers, what proportion of the 200 confidence\u00a0 intervals did contain the population proportion 0.40? What is this value as a\u00a0 percentage?\r\nHint: In Part b, you calculated the number of intervals containing the proportion 0.40. Divide this value by 200 to calculate the proportion.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIf we wanted to, we could simulate taking an even larger number of samples from this\u00a0 population and constructing a confidence interval for each one.\r\n\r\n(If you\u2019d like, feel free to explore this further using the tool at\u00a0\u00a0https:\/\/dcmathpathways.shinyapps.io\/ExploreCoverage<\/a>. Be sure to set the population\u00a0 proportion to 0.40 and the sample size to 200.)\r\n\r\nIn the long run (i.e., if we were to repeatedly take more and more samples from the\u00a0 population and construct a confidence interval for each one), we would expect the\u00a0 proportion of confidence intervals containing the population proportion to be equal to the\u00a0 chosen confidence level.\r\n\r\nPut another way, if we were constructing 95% confidence intervals, then in the long run, we would expect 95% of the intervals to contain the population proportion.\r\n\r\nAn important take-away is that the 95% (or some other chosen level of confidence) is\u00a0 associated with the method used to create the interval and not the likelihood that an\u00a0 individual interval contains the population proportion.\r\n\r\nThis means that anytime we provide a statement to interpret the meaning of a\u00a0 confidence interval, we should be sure to state that the confidence level is a measure\u00a0 of our confidence in the method.\r\n
          \r\n

          Question 6<\/h3>\r\nIn 2010, researchers asked subjects whether they would be willing to pay much\u00a0 higher gas prices to protect the environment. The image below includes a 99%\u00a0 confidence interval, estimating the population proportion of individuals who would be\u00a0 willing to pay higher gas prices to protect the environment.\r\n\r\n\"\"\r\n\r\nWhich of the following is the correct interpretation of the interval calculated above?\r\n
            \r\n \t
          1. There is a 99% chance that the population proportion of individuals willing to pay\u00a0 higher gas prices is between 0.4332 and 0.5029, or 43.32% and 50.29%.<\/li>\r\n \t
          2. We can be 99% confident that the population proportion of individuals willing to\u00a0 pay higher gas prices is between 0.4332 and 0.5029, or 43.32% and 50.29%.<\/li>\r\n<\/ol>\r\nHint: Remember that the confidence level is not measuring the likelihood that an\u00a0 individual interval contains the population proportion.\r\n\r\n<\/div>\r\n
            \r\n

            Question 7<\/h3>\r\nIn the 2019 National College Health Assessment, 12,499 college students out of a\u00a0 total of 38,337 reported that they feel \u201cvery safe\u201d on their campuses at night.[footnote]American College Health Association-National College Health Assessment. (2020). Undergraduate student reference group data report, Fall 2019<\/em>. https:\/\/www.acha.org\/NCHA\/ACHA NCHA_Data\/Publications_and_Reports\/NCHA\/Data\/Reports_ACHA-NCHAIII.aspx[\/footnote]\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"

            Preparing for the next class<\/p>\n

            In the next in-class activity, you will need to identify the components of a confidence\u00a0 interval from a summary provided by the DCMP Inference for a Population Proportion\u00a0 tool. You will also need to construct confidence intervals with technology and interpret\u00a0 their meaning.<\/p>\n

            In In-Class Activity 8.B, we explored data from a national survey of college students\u00a0 conducted by the American College Health Association. One of the questions posed to\u00a0 the participants was, \u201cOn how many of the last 7 days did you take a nap?\u201d Out of the\u00a0 38,440 participants, 13,888 stated they had not taken a nap in the last 7 days.<\/p>\n

            The image below shows the results of calculating a confidence interval to estimate the\u00a0 population proportion of college students who have not had a nap in the last 7 days.<\/p>\n

            \"\"<\/p>\n

            \n

            Question 1<\/h3>\n

            Given the previous calculations shown, which of the following is the point-estimate\u00a0 for the proportion of college students who have not had a nap in the last 7 days?<\/p>\n

              \n
            1. 0.3613<\/li>\n
            2. 0.0025<\/li>\n
            3. 0.3565<\/li>\n
            4. 0.3565<\/li>\n<\/ol>\n

              Hint: The point-estimate is the proportion calculated from our sample.<\/p>\n<\/div>\n

              \n

              Question 2<\/h3>\n

              Which of the following confidence levels was used to calculate the confidence\u00a0 interval presented in the previous image?<\/p>\n

                \n
              1. 90%<\/li>\n
              2. 95%<\/li>\n
              3. 99%<\/li>\n
              4. It is impossible to know.<\/li>\n<\/ol>\n<\/div>\n
                \n

                Question 3<\/h3>\n

                Given the calculations shown, identify the lower bound and upper bound of the\u00a0 calculated confidence interval.<\/p>\n

                \n\n\n\n\n
                Lower bound<\/td>\nUpper bound<\/td>\n<\/tr>\n
                <\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                 <\/p>\n

                Accurately interpreting a confidence interval is just as important as ensuring our\u00a0 calculations are correct. One common but incorrect interpretation is that the confidence\u00a0 level is the probability (expressed as a percentage) that the population proportion is\u00a0 contained within the bounds of our confidence interval.<\/p>\n

                For example, using the confidence interval calculated previously, a student might\u00a0 incorrectly state:<\/p>\n

                \u201cThere is a 95% chance that the population proportion of college students who have not\u00a0 had a nap in the last 7 days is between 0.3565 and 0.3661, or 35.65% and 36.61%.\u201d<\/p>\n

                Rather than measuring the likelihood that a single confidence interval contains the\u00a0 population proportion, the confidence level instead tells us the percentage of all confidence intervals that we\u2019d expect to contain the population proportion, if we were to\u00a0 repeatedly take random samples and construct confidence intervals around our point estimates.<\/p>\n

                Let\u2019s explore this idea a bit more.<\/p>\n<\/div>\n

                \n

                Question 4<\/h3>\n

                Suppose we knew that the population proportion of college students who did not\u00a0 take a nap in the last week was 0.40, or 40%. Of course, in reality we wouldn\u2019t know\u00a0 the population proportion,\u2014which is why we\u2019re creating a confidence interval to\u00a0 begin with\u2014but let\u2019s assume that we did.<\/p>\n

                Now suppose we took 10 random samples of 100 college students and constructed\u00a0 a 95% confidence interval for each one. We can run this simulation using the DCMP\u00a0 Explore Coverage and Confidence Intervals tool available at<\/p>\n

                https:\/\/dcmathpathways.shinyapps.io\/ExploreCoverage\/.<\/p>\n

                The following picture illustrates a simulated result of this situation using the tool (i.e.,\u00a0 10 confidence intervals constructed from our 10 random samples).<\/p>\n

                \"\"<\/p>\n

                  \n
                1. The red lines in the previous picture represent the confidence intervals that\u00a0 did not contain the population proportion 0.40. Out of the 10 confidence\u00a0 intervals from this simulated result, how many of them did not contain the\u00a0 population proportion 0.40?
                  \nHint: The red lines are those that did not contain the population proportion\u2014how\u00a0 many red lines are in the previous picture?<\/li>\n
                2. The green lines in the previous picture represent the confidence intervals\u00a0 that did contain the population proportion that we assumed (0.40). Out of the\u00a0 10 confidence intervals from this simulated result, how many of them did\u00a0 contain the population proportion 0.40?\n

                  Hint: The green lines are those that actually did contain the population proportion\u2014 how many green lines are in the previous picture?<\/li>\n

                3. Given your answers to Parts a and b, what proportion of the 10 confidence\u00a0 intervals did contain the population proportion 0.40? What is this value as a percentage?\n

                  Hint: In Part b, you calculated the number of intervals containing the proportion 0.40.\u00a0 Divide this value by 10 to calculate the proportion.<\/li>\n<\/ol>\n<\/div>\n

                  \n

                  Question 5<\/h3>\n

                  Now suppose we took 200 random samples of 100 college students and constructed\u00a0 a 95% confidence interval for each one. The picture below illustrates a simulated\u00a0 result of this situation (i.e., 200 confidence intervals constructed from our 200\u00a0 random samples).<\/p>\n

                  \"\"<\/p>\n

                    \n
                  1. The red lines in the previous picture represent the confidence intervals that\u00a0 did not contain the population proportion 0.40. Out of the 200 confidence intervals, how many of them did not contain the population proportion 0.40?
                    \nHint: The red lines are those that did not contain the population proportion\u2014how\u00a0 many red lines are in the previous picture? It might be easier to count the red\u00a0 squares to identify the number of confidence intervals that did not contain the\u00a0 population proportion.<\/li>\n
                  2. Given your answer to the previous question and knowing there are 200 total confidence intervals, how many confidence intervals from this simulated result did contain the population proportion?
                    \nHint: Subtract the value you calculated in Part a from 200 to identify the remaining\u00a0 number of intervals that did contain the population proportion.<\/li>\n
                  3. Given your previous answers, what proportion of the 200 confidence\u00a0 intervals did contain the population proportion 0.40? What is this value as a\u00a0 percentage?
                    \nHint: In Part b, you calculated the number of intervals containing the proportion 0.40. Divide this value by 200 to calculate the proportion.<\/li>\n<\/ol>\n<\/div>\n

                    If we wanted to, we could simulate taking an even larger number of samples from this\u00a0 population and constructing a confidence interval for each one.<\/p>\n

                    (If you\u2019d like, feel free to explore this further using the tool at\u00a0\u00a0https:\/\/dcmathpathways.shinyapps.io\/ExploreCoverage<\/a>. Be sure to set the population\u00a0 proportion to 0.40 and the sample size to 200.)<\/p>\n

                    In the long run (i.e., if we were to repeatedly take more and more samples from the\u00a0 population and construct a confidence interval for each one), we would expect the\u00a0 proportion of confidence intervals containing the population proportion to be equal to the\u00a0 chosen confidence level.<\/p>\n

                    Put another way, if we were constructing 95% confidence intervals, then in the long run, we would expect 95% of the intervals to contain the population proportion.<\/p>\n

                    An important take-away is that the 95% (or some other chosen level of confidence) is\u00a0 associated with the method used to create the interval and not the likelihood that an\u00a0 individual interval contains the population proportion.<\/p>\n

                    This means that anytime we provide a statement to interpret the meaning of a\u00a0 confidence interval, we should be sure to state that the confidence level is a measure\u00a0 of our confidence in the method.<\/p>\n

                    \n

                    Question 6<\/h3>\n

                    In 2010, researchers asked subjects whether they would be willing to pay much\u00a0 higher gas prices to protect the environment. The image below includes a 99%\u00a0 confidence interval, estimating the population proportion of individuals who would be\u00a0 willing to pay higher gas prices to protect the environment.<\/p>\n

                    \"\"<\/p>\n

                    Which of the following is the correct interpretation of the interval calculated above?<\/p>\n

                      \n
                    1. There is a 99% chance that the population proportion of individuals willing to pay\u00a0 higher gas prices is between 0.4332 and 0.5029, or 43.32% and 50.29%.<\/li>\n
                    2. We can be 99% confident that the population proportion of individuals willing to\u00a0 pay higher gas prices is between 0.4332 and 0.5029, or 43.32% and 50.29%.<\/li>\n<\/ol>\n

                      Hint: Remember that the confidence level is not measuring the likelihood that an\u00a0 individual interval contains the population proportion.<\/p>\n<\/div>\n

                      \n

                      Question 7<\/h3>\n

                      In the 2019 National College Health Assessment, 12,499 college students out of a\u00a0 total of 38,337 reported that they feel \u201cvery safe\u201d on their campuses at night.[1]<\/sup><\/a><\/p>\n<\/div>\n

                       <\/p>\n

                       <\/p>\n

                      1. American College Health Association-National College Health Assessment. (2020). Undergraduate student reference group data report, Fall 2019<\/em>. https:\/\/www.acha.org\/NCHA\/ACHA NCHA_Data\/Publications_and_Reports\/NCHA\/Data\/Reports_ACHA-NCHAIII.aspx ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":574340,"menu_order":6,"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-5261","chapter","type-chapter","status-publish","hentry"],"part":5220,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5261","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\/5261\/revisions"}],"predecessor-version":[{"id":5266,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5261\/revisions\/5266"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5220"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5261\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5261"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5261"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5261"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}