{"id":5428,"date":"2022-08-22T22:11:34","date_gmt":"2022-08-22T22:11:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=5428"},"modified":"2022-08-22T22:14:38","modified_gmt":"2022-08-22T22:14:38","slug":"12c-inclass","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/12c-inclass\/","title":{"raw":"12C InClass","rendered":"12C InClass"},"content":{"raw":"There are many situations where you might\u00a0be interested in estimating a population\u00a0mean.\u00a0For example, you might be interested in\u00a0collecting data from a random sample of\u00a0students who graduated from a two-year\u00a0college in 2020 to learn about student loans.\u00a0If you asked each student in the sample the\u00a0amount of their student loan debt, you could\u00a0then use the data to estimate the mean\u00a0student loan debt for two-year college graduates.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 1<\/h3>\r\n1) What are examples of other situations where you might want to estimate a population mean?\r\n\r\n<\/div>\r\nIn Lesson 10, you constructed confidence interval estimates for a population proportion and a difference in proportions when certain assumptions\/conditions were met. The form of those confidence intervals was\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/26121803\/Picture32-300x200.jpg\" alt=\"Several coins and some cash, as well as a paper that reads \u201cStudent Debt.\u201d There is a jagged vertical arrow.\" width=\"300\" height=\"200\" \/> Credit: iStock\/Darren415[\/caption]\r\n<p style=\"text-align: center;\">estimate \u00b1 margin of error<\/p>\r\nwhere the margin of error was calculated by multiplying the standard error of the estimate by a z-critical value corresponding to the desired confidence level.\r\n\r\nWhen the population parameter that you are interested in estimating is a population mean, the confidence interval has the same form. The estimate used to construct the\u00a0interval is the sample mean, [latex]\\bar{x}[\/latex], and the standard error used is the standard error of the sample mean, [latex]\\frac{s}{\\sqrt{n}}[\/latex].\r\n\r\nThe margin of error is calculated a little differently\u2014instead of multiplying the value of\u00a0 the standard error by a value from the normal distribution, it is multiplied by a value from\u00a0 the appropriate t Distribution. This is not surprising if you think back to your work with\u00a0 the standardized t-statistic in In-Class Activity 12.B.\r\n\r\nThe formula for a confidence interval for a population mean is\r\n\r\n[latex]\\bar{x}\\pm(t-critical\\;value)\\frac{s}{\\sqrt{n}}[\/latex]\r\n\r\nThe t-critical value (t-score in the data analysis tool) in the confidence interval will\u00a0 depend on the sample size (degrees of freedom for the t Distribution [latex]=n-1[\/latex]) and the confidence level. This interval is often called a <strong>one-sample t interval<\/strong>.\r\n\r\nWhile you will use technology to do the calculations, you can see from this confidence interval formula that you are just taking the sample mean and forming an interval around\u00a0 it by subtracting and adding the margin of error to get an interval of plausible values for\u00a0 the population mean.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 2<\/h3>\r\n2) The General Social Survey (GSS) collects data from a representative sample of adults in the United States on a number of attitudes and behaviors:\u00a0 https:\/\/gss.norc.org\/About-The-GSS.\r\n\r\nOne of the questions asked as part of the survey is how many hours are spent watching TV on a typical day. The dataset we will be using consists of responses\u00a0 from a sample of 1,555 adults from the 2018 survey.\r\n\r\na) Describe the population mean that these data could be used to estimate. Be\u00a0 sure your description includes the population of interest.\r\n\r\nb) Go to the DCMP Inference for a Population Mean tool at\r\n\r\nhttps:\/\/dcmathpathways.shinyapps.io\/Inference_mean\/. You will use this tool to calculate confidence intervals for a population mean. Under the\r\n\r\nConfidence and Significance Tests tab:\r\n<ul>\r\n \t<li>Select the TV Hours dataset.<\/li>\r\n \t<li>For Type of Inference, select \u201cConfidence Interval.\u201d<\/li>\r\n \t<li>Use the slider for the confidence level to select a 90% confidence level.<\/li>\r\n<\/ul>\r\nBased on the output, what is the 90% confidence interval for the mean number of hours spent watching TV on a typical day for adults in the United States?\r\n\r\nc) Think about how you have previously interpreted confidence intervals for proportions. How would you interpret the confidence interval you just calculated?\r\n\r\nd) Which of the following is a correct interpretation of the confidence level of 90%?\r\n\r\n(a) The probability that the actual value of the population mean is between 2.820 and 3.057 hours is 0.90.\r\n\r\n(b) 90% of U.S. adults watch between 2.820 and 3.057 hours of TV on a typical day.\r\n\r\n(c) If this method was used to construct a confidence interval for the mean for many different samples from the population, about 90% of the intervals would contain the actual population mean.\r\n\r\n(d) The mean number of hours that U.S. adults spend watching TV on a typical day is guaranteed to be in the interval from 2.820 to 3.057.\r\n\r\ne) If you were to use this sample to calculate a 95% confidence interval rather than a 90% confidence interval, how would the width of the two intervals compare?\r\n\r\n(a) The width of the two intervals would be the same.\r\n\r\n(b) The 95% confidence interval will be wider than the 90% confidence interval.\r\n\r\n(c) The 95% confidence interval will be narrower than the 90% confidence interval.\r\n\r\nf) Use the tool to compute the 95% confidence interval. Is it consistent with your answer to Part E? Explain.\r\n\r\n<\/div>\r\nAs was the case with previous inference methods, there are a few assumptions\/conditions that you should check before using the one-sample t interval. Two important ones are:\r\n<ul>\r\n \t<li>The sample is a random sample from the population of interest or it is reasonable\u00a0 to regard the sample as if it were a random sample. It is reasonable to regard the\u00a0 sample as a random sample if it was selected in a way that should result in a\u00a0 sample that is representative of the population.<\/li>\r\n \t<li>The population distribution of the variable that was measured is approximately\u00a0 normal, or the sample size is large. Usually, a sample of size 30 or more is\u00a0 considered to be \u201clarge.\u201d If the sample size is less than 30, you should look at a\u00a0 plot of the data (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make sure that the distribution looks approximately symmetric and\u00a0 that there are no outliers.<\/li>\r\n<\/ul>\r\nFor the TV hours example, the sample size is large and the sample was selected to be\u00a0 representative of adults in the United States, so the one-sample t confidence interval is\u00a0 an appropriate way to estimate the population mean.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\n3) Researchers in New York carried out a study to investigate how many calories are consumed when people eat lunch at fast-food restaurants.[footnote]Dumanovsky, T., Nonas, C. A., Huang, C. Y., Silver, L. D., &amp; Bassett, M. T. (2009, July). What people\u00a0 buy from fast-food restaurants: Caloric content and menu item selection, New York City 2007. Obesity 17(7), 1369\u20131374. https:\/\/onlinelibrary.wiley.com\/doi\/full\/10.1038\/oby.2009.90[\/footnote] They asked people\u00a0 eating lunch at different locations of McDonald\u2019s, Burger King, and Wendy\u2019s if they\u00a0 would give them their receipts after they had ordered, and then they used the\u00a0 receipts to see what had been ordered to determine the number of calories in the\u00a0 meals. A total of 3,857 meals were analyzed in the study, and the researchers\u00a0 believed that this sample was representative of lunch meals eaten at fast-food\u00a0 restaurants.\r\n\r\nThe mean calorie content for the sample was 857 calories and the sample standard\u00a0 deviation was 677 calories.\r\n\r\na) Is the one-sample t interval an appropriate way to estimate the mean calorie\u00a0 content of fast-food lunches? Recall that in the preview assignment, you\u00a0 decided that the population distribution of the calorie content for fast food\u00a0 lunches was not normal. Are the assumptions for the one-sample t interval\u00a0 reasonable?\r\n\r\nb) Use the tool to calculate a 95% confidence interval for the mean calorie\u00a0 content of fast-food lunches. Go to the DCMP Inference for a Population\u00a0 Mean tool at https:\/\/dcmathpathways.shinyapps.io\/Inference_mean\/. Under the Confidence and Significance Tests tab:\r\n<ul>\r\n \t<li>Select \u201cSummary Statistics\u201d from the drop-down menu under \u201cEnter Data.\u201d<\/li>\r\n \t<li>Type in \u201cCalorie Content\u201d for the name of the variable, and then enter the\u00a0 sample size and the sample mean and standard deviation for this\u00a0 example.<\/li>\r\n \t<li>For Type of Inference, select \u201cConfidence Interval.\u201d<\/li>\r\n \t<li>Use the slider for the confidence level to select a 95% confidence level.<\/li>\r\n<\/ul>\r\nc) Interpret the confidence interval from Part B.\r\n\r\nd) An article on the website Global News included the following statement:\r\n\r\n\u201cAlthough every person\u2019s daily caloric intake is individual, based on their\u00a0 personal goals and needs, nutrition experts estimate that average daily\u00a0 consumption at each meal should be broken down as follows: 300 to 400\u00a0 calories for breakfast and 500 to 700 calories each for lunch and dinner.\u00a0 Snacks shouldn\u2019t exceed 200 calories.\u201d\r\n\r\nDo you think that your confidence interval from Part B provides evidence that the mean calorie content of fast-food lunches does not meet the\u00a0 recommendations in the given quote? Explain.\r\n\r\n<\/div>","rendered":"<p>There are many situations where you might\u00a0be interested in estimating a population\u00a0mean.\u00a0For example, you might be interested in\u00a0collecting data from a random sample of\u00a0students who graduated from a two-year\u00a0college in 2020 to learn about student loans.\u00a0If you asked each student in the sample the\u00a0amount of their student loan debt, you could\u00a0then use the data to estimate the mean\u00a0student loan debt for two-year college graduates.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 1<\/h3>\n<p>1) What are examples of other situations where you might want to estimate a population mean?<\/p>\n<\/div>\n<p>In Lesson 10, you constructed confidence interval estimates for a population proportion and a difference in proportions when certain assumptions\/conditions were met. The form of those confidence intervals was<\/p>\n<div style=\"width: 310px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/26121803\/Picture32-300x200.jpg\" alt=\"Several coins and some cash, as well as a paper that reads \u201cStudent Debt.\u201d There is a jagged vertical arrow.\" width=\"300\" height=\"200\" \/><\/p>\n<p class=\"wp-caption-text\">Credit: iStock\/Darren415<\/p>\n<\/div>\n<p style=\"text-align: center;\">estimate \u00b1 margin of error<\/p>\n<p>where the margin of error was calculated by multiplying the standard error of the estimate by a z-critical value corresponding to the desired confidence level.<\/p>\n<p>When the population parameter that you are interested in estimating is a population mean, the confidence interval has the same form. The estimate used to construct the\u00a0interval is the sample mean, [latex]\\bar{x}[\/latex], and the standard error used is the standard error of the sample mean, [latex]\\frac{s}{\\sqrt{n}}[\/latex].<\/p>\n<p>The margin of error is calculated a little differently\u2014instead of multiplying the value of\u00a0 the standard error by a value from the normal distribution, it is multiplied by a value from\u00a0 the appropriate t Distribution. This is not surprising if you think back to your work with\u00a0 the standardized t-statistic in In-Class Activity 12.B.<\/p>\n<p>The formula for a confidence interval for a population mean is<\/p>\n<p>[latex]\\bar{x}\\pm(t-critical\\;value)\\frac{s}{\\sqrt{n}}[\/latex]<\/p>\n<p>The t-critical value (t-score in the data analysis tool) in the confidence interval will\u00a0 depend on the sample size (degrees of freedom for the t Distribution [latex]=n-1[\/latex]) and the confidence level. This interval is often called a <strong>one-sample t interval<\/strong>.<\/p>\n<p>While you will use technology to do the calculations, you can see from this confidence interval formula that you are just taking the sample mean and forming an interval around\u00a0 it by subtracting and adding the margin of error to get an interval of plausible values for\u00a0 the population mean.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 2<\/h3>\n<p>2) The General Social Survey (GSS) collects data from a representative sample of adults in the United States on a number of attitudes and behaviors:\u00a0 https:\/\/gss.norc.org\/About-The-GSS.<\/p>\n<p>One of the questions asked as part of the survey is how many hours are spent watching TV on a typical day. The dataset we will be using consists of responses\u00a0 from a sample of 1,555 adults from the 2018 survey.<\/p>\n<p>a) Describe the population mean that these data could be used to estimate. Be\u00a0 sure your description includes the population of interest.<\/p>\n<p>b) Go to the DCMP Inference for a Population Mean tool at<\/p>\n<p>https:\/\/dcmathpathways.shinyapps.io\/Inference_mean\/. You will use this tool to calculate confidence intervals for a population mean. Under the<\/p>\n<p>Confidence and Significance Tests tab:<\/p>\n<ul>\n<li>Select the TV Hours dataset.<\/li>\n<li>For Type of Inference, select \u201cConfidence Interval.\u201d<\/li>\n<li>Use the slider for the confidence level to select a 90% confidence level.<\/li>\n<\/ul>\n<p>Based on the output, what is the 90% confidence interval for the mean number of hours spent watching TV on a typical day for adults in the United States?<\/p>\n<p>c) Think about how you have previously interpreted confidence intervals for proportions. How would you interpret the confidence interval you just calculated?<\/p>\n<p>d) Which of the following is a correct interpretation of the confidence level of 90%?<\/p>\n<p>(a) The probability that the actual value of the population mean is between 2.820 and 3.057 hours is 0.90.<\/p>\n<p>(b) 90% of U.S. adults watch between 2.820 and 3.057 hours of TV on a typical day.<\/p>\n<p>(c) If this method was used to construct a confidence interval for the mean for many different samples from the population, about 90% of the intervals would contain the actual population mean.<\/p>\n<p>(d) The mean number of hours that U.S. adults spend watching TV on a typical day is guaranteed to be in the interval from 2.820 to 3.057.<\/p>\n<p>e) If you were to use this sample to calculate a 95% confidence interval rather than a 90% confidence interval, how would the width of the two intervals compare?<\/p>\n<p>(a) The width of the two intervals would be the same.<\/p>\n<p>(b) The 95% confidence interval will be wider than the 90% confidence interval.<\/p>\n<p>(c) The 95% confidence interval will be narrower than the 90% confidence interval.<\/p>\n<p>f) Use the tool to compute the 95% confidence interval. Is it consistent with your answer to Part E? Explain.<\/p>\n<\/div>\n<p>As was the case with previous inference methods, there are a few assumptions\/conditions that you should check before using the one-sample t interval. Two important ones are:<\/p>\n<ul>\n<li>The sample is a random sample from the population of interest or it is reasonable\u00a0 to regard the sample as if it were a random sample. It is reasonable to regard the\u00a0 sample as a random sample if it was selected in a way that should result in a\u00a0 sample that is representative of the population.<\/li>\n<li>The population distribution of the variable that was measured is approximately\u00a0 normal, or the sample size is large. Usually, a sample of size 30 or more is\u00a0 considered to be \u201clarge.\u201d If the sample size is less than 30, you should look at a\u00a0 plot of the data (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make sure that the distribution looks approximately symmetric and\u00a0 that there are no outliers.<\/li>\n<\/ul>\n<p>For the TV hours example, the sample size is large and the sample was selected to be\u00a0 representative of adults in the United States, so the one-sample t confidence interval is\u00a0 an appropriate way to estimate the population mean.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>3) Researchers in New York carried out a study to investigate how many calories are consumed when people eat lunch at fast-food restaurants.<a class=\"footnote\" title=\"Dumanovsky, T., Nonas, C. A., Huang, C. Y., Silver, L. D., &amp; Bassett, M. T. (2009, July). What people\u00a0 buy from fast-food restaurants: Caloric content and menu item selection, New York City 2007. Obesity 17(7), 1369\u20131374. https:\/\/onlinelibrary.wiley.com\/doi\/full\/10.1038\/oby.2009.90\" id=\"return-footnote-5428-1\" href=\"#footnote-5428-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> They asked people\u00a0 eating lunch at different locations of McDonald\u2019s, Burger King, and Wendy\u2019s if they\u00a0 would give them their receipts after they had ordered, and then they used the\u00a0 receipts to see what had been ordered to determine the number of calories in the\u00a0 meals. A total of 3,857 meals were analyzed in the study, and the researchers\u00a0 believed that this sample was representative of lunch meals eaten at fast-food\u00a0 restaurants.<\/p>\n<p>The mean calorie content for the sample was 857 calories and the sample standard\u00a0 deviation was 677 calories.<\/p>\n<p>a) Is the one-sample t interval an appropriate way to estimate the mean calorie\u00a0 content of fast-food lunches? Recall that in the preview assignment, you\u00a0 decided that the population distribution of the calorie content for fast food\u00a0 lunches was not normal. Are the assumptions for the one-sample t interval\u00a0 reasonable?<\/p>\n<p>b) Use the tool to calculate a 95% confidence interval for the mean calorie\u00a0 content of fast-food lunches. Go to the DCMP Inference for a Population\u00a0 Mean tool at https:\/\/dcmathpathways.shinyapps.io\/Inference_mean\/. Under the Confidence and Significance Tests tab:<\/p>\n<ul>\n<li>Select \u201cSummary Statistics\u201d from the drop-down menu under \u201cEnter Data.\u201d<\/li>\n<li>Type in \u201cCalorie Content\u201d for the name of the variable, and then enter the\u00a0 sample size and the sample mean and standard deviation for this\u00a0 example.<\/li>\n<li>For Type of Inference, select \u201cConfidence Interval.\u201d<\/li>\n<li>Use the slider for the confidence level to select a 95% confidence level.<\/li>\n<\/ul>\n<p>c) Interpret the confidence interval from Part B.<\/p>\n<p>d) An article on the website Global News included the following statement:<\/p>\n<p>\u201cAlthough every person\u2019s daily caloric intake is individual, based on their\u00a0 personal goals and needs, nutrition experts estimate that average daily\u00a0 consumption at each meal should be broken down as follows: 300 to 400\u00a0 calories for breakfast and 500 to 700 calories each for lunch and dinner.\u00a0 Snacks shouldn\u2019t exceed 200 calories.\u201d<\/p>\n<p>Do you think that your confidence interval from Part B provides evidence that the mean calorie content of fast-food lunches does not meet the\u00a0 recommendations in the given quote? Explain.<\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5428-1\">Dumanovsky, T., Nonas, C. A., Huang, C. Y., Silver, L. D., &amp; Bassett, M. T. (2009, July). What people\u00a0 buy from fast-food restaurants: Caloric content and menu item selection, New York City 2007. Obesity 17(7), 1369\u20131374. https:\/\/onlinelibrary.wiley.com\/doi\/full\/10.1038\/oby.2009.90 <a href=\"#return-footnote-5428-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"menu_order":8,"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-5428","chapter","type-chapter","status-publish","hentry"],"part":5315,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5428","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\/23592"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5428\/revisions"}],"predecessor-version":[{"id":5432,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5428\/revisions\/5432"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5315"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5428\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5428"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5428"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5428"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5428"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}