{"id":5437,"date":"2022-08-22T23:23:27","date_gmt":"2022-08-22T23:23:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=5437"},"modified":"2022-08-22T23:23:48","modified_gmt":"2022-08-22T23:23:48","slug":"12d-inclass","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/12d-inclass\/","title":{"raw":"12D InClass","rendered":"12D InClass"},"content":{"raw":"<div class=\"textbox key-takeaways\">\r\n<h3>Question 1<\/h3>\r\n1) Do you think that the font used in printed instructions influences how difficult people think it will be to follow those instructions? Would you rather read a set of instructions printed in Arial font or Mistral font (the fonts are shown below)?<\/div>\r\nIn previous lessons, you constructed confidence interval estimates for a population\u00a0 proportion, a difference in proportions, and a population mean. The form of those confidence intervals was:\r\n\r\nestimate \u00b1 margin of error\r\n\r\nWhen you are interested in estimating a difference in population means using data from independent samples, the confidence interval has the same form. The estimate used to construct the interval is the difference in sample means, [latex]\\bar{x}_{1}-\\bar{x}_{2}[\/latex], and the margin of error\u00a0is calculated using the standard error for a difference in sample means and a critical value from the t Distribution. You will use technology to calculate the margin of error, so we won\u2019t worry about the formula for standard error or margin of error at this point (you\u00a0 will see them in In-Class Activity 13.C).\r\n\r\nCompared to the formula for proportions, the margin of error here is calculated a little differently\u2014instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate t Distribution. This is not surprising if you think back to your work with the standardized t-statistic in In-Class Activity 12.B.\r\n\r\nIn the preview assignment, you used data from a study that investigated the effect of using a cell phone while driving on reaction time.[footnote]Strayer, D. L., &amp; Johnston, W. A. (2001, November 1). Driven to distraction: Dual-task studies of\u00a0 simulated driving and conversing on a cellular telephone. Psychological Science, 12(6), 462\u2013466.\u00a0 https:\/\/doi.org\/10.1111\/1467-9280.00386[\/footnote] In this study, 64 students were randomly assigned to one of two groups. Students in both groups were asked to drive in a driving simulator and to press a brake button as quickly as possible when they saw a red light. Response times (in milliseconds) were measured. Students in one group used their cell phones while driving in the simulator and students in the other group did not use their cell phones. This dataset is built into the data analysis tool, and you will use these data to estimate the difference between the mean reaction time for people who use their cell phones while driving and the mean reaction time for people who do not use their cell phones while driving.\r\n\r\nRecall from your previous work that when you want to use a confidence interval, you follow a process that includes deciding what confidence interval method might be used, checking the assumptions for the chosen method to make sure it is appropriate to use,\u00a0 calculating the confidence interval, and interpreting the interval in context.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\n2) In the preview assignment, the two-sample t confidence interval was selected, and\u00a0 you verified that the necessary assumptions were reasonable. Now, we will look at\u00a0 calculating the confidence interval and interpreting it in context.\r\n\r\na) Go to the DCMP Comparing Two Population Means tool at\u00a0 https:\/\/dcmathpathways.shinyapps.io\/2sample_mean\/. You will use this tool\u00a0 to calculate confidence intervals for a difference in population means.\r\n\r\nUnder the Confidence and Significance Tests tab:\r\n<ul>\r\n \t<li>Select the Reaction Times 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 95% confidence level.<\/li>\r\n<\/ul>\r\nBased on the output, what is the 95% confidence interval for the difference in mean\u00a0 reaction times between people who use their cell phones while driving and people\u00a0 who do not use their cell phones while driving?\r\n\r\nb) Think about how you have previously interpreted confidence intervals for a\u00a0 difference. How would you interpret the confidence interval you just\u00a0 calculated?\r\n\r\nc) What does the confidence interval tell you about the effect of using a cell\u00a0 phone while driving on reaction time?\r\n\r\nHint: Both endpoints of the interval are positive. What does this tell you about\u00a0 the mean reaction times?\r\n\r\nd) Which of the following is a correct interpretation of the confidence level of\u00a0 95%?\r\n\r\n(a) The probability that the actual value of the difference in population means between 12.3 and 90.9 milliseconds is 0.95.\r\n\r\n(b) 95% of people using their cell phones while driving have a reaction time between 12.3 and 90.9 milliseconds.\r\n\r\n(c) If this method was used to construct a confidence interval for the\u00a0 difference in population means for many different pairs of samples from\u00a0 the population, about 95% of the intervals would contain the actual\u00a0 difference in population means.\r\n\r\n(d) The mean difference in mean reaction times is guaranteed to be in the\u00a0 interval from 12.3 to 90.9 milliseconds.\r\n\r\ne) If you were to use this sample to calculate a 90% confidence interval rather\u00a0 than a 95% confidence interval, how would the width of the two intervals\u00a0 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\u00a0 interval.\r\n\r\n(c) The 95% confidence interval will be narrower than the 90% confidence\u00a0 interval.\r\n\r\n<\/div>\r\nAs was the case with previous inference methods, there are a few\u00a0assumptions\/conditions that you should check before using the two-sample t confidence interval. These were introduced in the preview assignment and are included here as a\u00a0 reminder:\r\n<ol>\r\n \t<li>The samples are independent.<\/li>\r\n \t<li>Each sample is a random sample from the corresponding population of interest or\u00a0 it is reasonable to regard the sample as if it were a random sample. It is\u00a0 reasonable to regard the sample as a random sample if it was selected in a way\u00a0 that should result in the sample being representative of the population. If the data\u00a0 are from an experiment, you just need to check that there was random\u00a0 assignment to experimental groups\u2014this substitutes for the random sample\u00a0 condition and also results in independent samples.<\/li>\r\n \t<li>For each population, the distribution of the variable that was measured is\u00a0 approximately normal, or the sample size for the sample from that population is\u00a0 large. Usually, a sample of size 30 or more is considered to be \u201clarge.\u201d If a sample\u00a0 size is less than 30, you should look at a plot of the data from that sample (a\u00a0 dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make\u00a0 sure that the distribution looks approximately symmetric and that there are no\u00a0 outliers.<\/li>\r\n<\/ol>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 3<\/h3>\r\n3) Researchers at the University of Michigan[footnote]Song, H., &amp; Schwarz, N. (2008, October 1). If it's hard to read, it's hard to do: Processing fluency affects\u00a0 effort prediction and motivation. Psychological Science, 19(10), 986\u2013988. https:\/\/doi.org\/10.1111\/j.1467- 9280.2008.02189.x\u00a0[\/footnote] investigated whether the font used in\u00a0 printed instructions influences how difficult people think it will be to complete a task.\u00a0 Participants were randomly assigned to one of two groups. Those in one group read\u00a0 instructions on how to make a Japanese sushi roll that were printed in an easy-to read font (Arial). Participants in the second group read the same instructions, but they were printed in a hard-to-read font (Mistral). All participants were then asked to\u00a0 say how long they thought it would take to prepare the sushi roll according to the\u00a0 given instructions.\r\n\r\nThe mean estimated time to prepare the sushi roll for the sample who read the\u00a0 instructions in Arial font was 22.71 minutes and the sample standard deviation was\u00a0 13.76 minutes. The mean estimated time to prepare the sushi roll for the other\u00a0 sample who read the instructions in Mistral font was 36.15 minutes and the sample\u00a0 standard deviation was 15.30 minutes. For the purposes of this example, suppose\u00a0 that the two sample sizes were both 34.\r\n\r\na) Is the two-sample t interval an appropriate way to estimate the difference in\u00a0 mean estimated times to prepare the sushi roll between those who read the\u00a0 instructions in the easy-to-read font and those who read the instructions in\u00a0 the hard-to-read font?\r\n\r\nHint: Are the assumptions for the two-sample t confidence interval\u00a0 reasonably met?\r\n\r\nb) Use the data analysis tool to calculate a 90% confidence interval for the\u00a0 difference in mean estimated times to prepare the sushi roll between those\u00a0 who read the instructions in the easy-to-read font and those who read the\u00a0 instructions in the hard-to-read font.\r\n\r\nGo to the DCMP Comparing Two Population Means tool at\r\n\r\nhttps:\/\/dcmathpathways.shinyapps.io\/2sample_mean\/.\r\n\r\nUnder the Confidence and Significance Tests tab:\r\n<ul>\r\n \t<li>Select \u201cSummary Statistics\u201d from the drop-down menu under \u201cEnter\u00a0 Data.\u201d<\/li>\r\n \t<li>Type in \u201cEstimated Time to Complete\u201d for the name of the variable, and type in \u201cEasy to Read\u201d for the Group 1 label and \u201cHard to Read\u201d for the\u00a0 Group 2 label.<\/li>\r\n \t<li>Enter the sample sizes and the sample means and standard deviations for this 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 90% confidence level.<\/li>\r\n<\/ul>\r\nc) Interpret the confidence interval from Part B.\r\n\r\nd) What does the confidence interval tell you about whether the font used in the printed instructions influences how difficult people think it will be to complete a task?\r\n\r\nHint: Both endpoints of the interval are negative. What does this tell you about the difference in mean estimated times to complete the task?\r\n\r\ne) What recommendation would you make to a company who provides printed instructions with their products based on your analysis of the data from this\u00a0 experiment?\r\n\r\n<\/div>","rendered":"<div class=\"textbox key-takeaways\">\n<h3>Question 1<\/h3>\n<p>1) Do you think that the font used in printed instructions influences how difficult people think it will be to follow those instructions? Would you rather read a set of instructions printed in Arial font or Mistral font (the fonts are shown below)?<\/p><\/div>\n<p>In previous lessons, you constructed confidence interval estimates for a population\u00a0 proportion, a difference in proportions, and a population mean. The form of those confidence intervals was:<\/p>\n<p>estimate \u00b1 margin of error<\/p>\n<p>When you are interested in estimating a difference in population means using data from independent samples, the confidence interval has the same form. The estimate used to construct the interval is the difference in sample means, [latex]\\bar{x}_{1}-\\bar{x}_{2}[\/latex], and the margin of error\u00a0is calculated using the standard error for a difference in sample means and a critical value from the t Distribution. You will use technology to calculate the margin of error, so we won\u2019t worry about the formula for standard error or margin of error at this point (you\u00a0 will see them in In-Class Activity 13.C).<\/p>\n<p>Compared to the formula for proportions, the margin of error here is calculated a little differently\u2014instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate t Distribution. This is not surprising if you think back to your work with the standardized t-statistic in In-Class Activity 12.B.<\/p>\n<p>In the preview assignment, you used data from a study that investigated the effect of using a cell phone while driving on reaction time.<a class=\"footnote\" title=\"Strayer, D. L., &amp; Johnston, W. A. (2001, November 1). Driven to distraction: Dual-task studies of\u00a0 simulated driving and conversing on a cellular telephone. Psychological Science, 12(6), 462\u2013466.\u00a0 https:\/\/doi.org\/10.1111\/1467-9280.00386\" id=\"return-footnote-5437-1\" href=\"#footnote-5437-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> In this study, 64 students were randomly assigned to one of two groups. Students in both groups were asked to drive in a driving simulator and to press a brake button as quickly as possible when they saw a red light. Response times (in milliseconds) were measured. Students in one group used their cell phones while driving in the simulator and students in the other group did not use their cell phones. This dataset is built into the data analysis tool, and you will use these data to estimate the difference between the mean reaction time for people who use their cell phones while driving and the mean reaction time for people who do not use their cell phones while driving.<\/p>\n<p>Recall from your previous work that when you want to use a confidence interval, you follow a process that includes deciding what confidence interval method might be used, checking the assumptions for the chosen method to make sure it is appropriate to use,\u00a0 calculating the confidence interval, and interpreting the interval in context.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<p>2) In the preview assignment, the two-sample t confidence interval was selected, and\u00a0 you verified that the necessary assumptions were reasonable. Now, we will look at\u00a0 calculating the confidence interval and interpreting it in context.<\/p>\n<p>a) Go to the DCMP Comparing Two Population Means tool at\u00a0 https:\/\/dcmathpathways.shinyapps.io\/2sample_mean\/. You will use this tool\u00a0 to calculate confidence intervals for a difference in population means.<\/p>\n<p>Under the Confidence and Significance Tests tab:<\/p>\n<ul>\n<li>Select the Reaction Times 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 95% confidence level.<\/li>\n<\/ul>\n<p>Based on the output, what is the 95% confidence interval for the difference in mean\u00a0 reaction times between people who use their cell phones while driving and people\u00a0 who do not use their cell phones while driving?<\/p>\n<p>b) Think about how you have previously interpreted confidence intervals for a\u00a0 difference. How would you interpret the confidence interval you just\u00a0 calculated?<\/p>\n<p>c) What does the confidence interval tell you about the effect of using a cell\u00a0 phone while driving on reaction time?<\/p>\n<p>Hint: Both endpoints of the interval are positive. What does this tell you about\u00a0 the mean reaction times?<\/p>\n<p>d) Which of the following is a correct interpretation of the confidence level of\u00a0 95%?<\/p>\n<p>(a) The probability that the actual value of the difference in population means between 12.3 and 90.9 milliseconds is 0.95.<\/p>\n<p>(b) 95% of people using their cell phones while driving have a reaction time between 12.3 and 90.9 milliseconds.<\/p>\n<p>(c) If this method was used to construct a confidence interval for the\u00a0 difference in population means for many different pairs of samples from\u00a0 the population, about 95% of the intervals would contain the actual\u00a0 difference in population means.<\/p>\n<p>(d) The mean difference in mean reaction times is guaranteed to be in the\u00a0 interval from 12.3 to 90.9 milliseconds.<\/p>\n<p>e) If you were to use this sample to calculate a 90% confidence interval rather\u00a0 than a 95% confidence interval, how would the width of the two intervals\u00a0 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\u00a0 interval.<\/p>\n<p>(c) The 95% confidence interval will be narrower than the 90% confidence\u00a0 interval.<\/p>\n<\/div>\n<p>As was the case with previous inference methods, there are a few\u00a0assumptions\/conditions that you should check before using the two-sample t confidence interval. These were introduced in the preview assignment and are included here as a\u00a0 reminder:<\/p>\n<ol>\n<li>The samples are independent.<\/li>\n<li>Each sample is a random sample from the corresponding population of interest or\u00a0 it is reasonable to regard the sample as if it were a random sample. It is\u00a0 reasonable to regard the sample as a random sample if it was selected in a way\u00a0 that should result in the sample being representative of the population. If the data\u00a0 are from an experiment, you just need to check that there was random\u00a0 assignment to experimental groups\u2014this substitutes for the random sample\u00a0 condition and also results in independent samples.<\/li>\n<li>For each population, the distribution of the variable that was measured is\u00a0 approximately normal, or the sample size for the sample from that population is\u00a0 large. Usually, a sample of size 30 or more is considered to be \u201clarge.\u201d If a sample\u00a0 size is less than 30, you should look at a plot of the data from that sample (a\u00a0 dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make\u00a0 sure that the distribution looks approximately symmetric and that there are no\u00a0 outliers.<\/li>\n<\/ol>\n<div class=\"textbox key-takeaways\">\n<h3>Question 3<\/h3>\n<p>3) Researchers at the University of Michigan<a class=\"footnote\" title=\"Song, H., &amp; Schwarz, N. (2008, October 1). If it's hard to read, it's hard to do: Processing fluency affects\u00a0 effort prediction and motivation. Psychological Science, 19(10), 986\u2013988. https:\/\/doi.org\/10.1111\/j.1467- 9280.2008.02189.x\u00a0\" id=\"return-footnote-5437-2\" href=\"#footnote-5437-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a> investigated whether the font used in\u00a0 printed instructions influences how difficult people think it will be to complete a task.\u00a0 Participants were randomly assigned to one of two groups. Those in one group read\u00a0 instructions on how to make a Japanese sushi roll that were printed in an easy-to read font (Arial). Participants in the second group read the same instructions, but they were printed in a hard-to-read font (Mistral). All participants were then asked to\u00a0 say how long they thought it would take to prepare the sushi roll according to the\u00a0 given instructions.<\/p>\n<p>The mean estimated time to prepare the sushi roll for the sample who read the\u00a0 instructions in Arial font was 22.71 minutes and the sample standard deviation was\u00a0 13.76 minutes. The mean estimated time to prepare the sushi roll for the other\u00a0 sample who read the instructions in Mistral font was 36.15 minutes and the sample\u00a0 standard deviation was 15.30 minutes. For the purposes of this example, suppose\u00a0 that the two sample sizes were both 34.<\/p>\n<p>a) Is the two-sample t interval an appropriate way to estimate the difference in\u00a0 mean estimated times to prepare the sushi roll between those who read the\u00a0 instructions in the easy-to-read font and those who read the instructions in\u00a0 the hard-to-read font?<\/p>\n<p>Hint: Are the assumptions for the two-sample t confidence interval\u00a0 reasonably met?<\/p>\n<p>b) Use the data analysis tool to calculate a 90% confidence interval for the\u00a0 difference in mean estimated times to prepare the sushi roll between those\u00a0 who read the instructions in the easy-to-read font and those who read the\u00a0 instructions in the hard-to-read font.<\/p>\n<p>Go to the DCMP Comparing Two Population Means tool at<\/p>\n<p>https:\/\/dcmathpathways.shinyapps.io\/2sample_mean\/.<\/p>\n<p>Under the Confidence and Significance Tests tab:<\/p>\n<ul>\n<li>Select \u201cSummary Statistics\u201d from the drop-down menu under \u201cEnter\u00a0 Data.\u201d<\/li>\n<li>Type in \u201cEstimated Time to Complete\u201d for the name of the variable, and type in \u201cEasy to Read\u201d for the Group 1 label and \u201cHard to Read\u201d for the\u00a0 Group 2 label.<\/li>\n<li>Enter the sample sizes and the sample means and standard deviations for this 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 90% confidence level.<\/li>\n<\/ul>\n<p>c) Interpret the confidence interval from Part B.<\/p>\n<p>d) What does the confidence interval tell you about whether the font used in the printed instructions influences how difficult people think it will be to complete a task?<\/p>\n<p>Hint: Both endpoints of the interval are negative. What does this tell you about the difference in mean estimated times to complete the task?<\/p>\n<p>e) What recommendation would you make to a company who provides printed instructions with their products based on your analysis of the data from this\u00a0 experiment?<\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5437-1\">Strayer, D. L., &amp; Johnston, W. A. (2001, November 1). Driven to distraction: Dual-task studies of\u00a0 simulated driving and conversing on a cellular telephone. Psychological Science, 12(6), 462\u2013466.\u00a0 https:\/\/doi.org\/10.1111\/1467-9280.00386 <a href=\"#return-footnote-5437-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-5437-2\">Song, H., &amp; Schwarz, N. (2008, October 1). If it's hard to read, it's hard to do: Processing fluency affects\u00a0 effort prediction and motivation. Psychological Science, 19(10), 986\u2013988. https:\/\/doi.org\/10.1111\/j.1467- 9280.2008.02189.x\u00a0 <a href=\"#return-footnote-5437-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"menu_order":11,"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-5437","chapter","type-chapter","status-publish","hentry"],"part":5315,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5437","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":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5437\/revisions"}],"predecessor-version":[{"id":5438,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5437\/revisions\/5438"}],"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\/5437\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5437"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5437"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5437"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5437"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}