{"id":5572,"date":"2022-09-27T10:09:52","date_gmt":"2022-09-27T10:09:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5572"},"modified":"2022-10-18T07:53:39","modified_gmt":"2022-10-18T07:53:39","slug":"18b-coreq","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/18b-coreq\/","title":{"raw":"18B Coreq","rendered":"18B Coreq"},"content":{"raw":"
\r\n
The previous in-class activity used a bootstrap confidence interval to estimate a population mean. In the next in-class activity, we will see how bootstrap confidence intervals can be used to estimate other population parameters. To support the preview assignment and the in-class activity, this corequisite support activity revisits two measures of center (the mean and the median) and reviews how confidence intervals for a difference in means are interpreted.<\/div>\r\n
Two Measures of Center\u2014The Mean and the Median<\/div>\r\n
Every year, bullfrogs compete in a jumping contest at the Calaveras County Jumping Frog Jubilee (a contest inspired by a short story by Mark Twain). One year, researchers recorded the jump distances of frogs entered in the contest.[footnote]Astley, H. C., Abbott, E. M., Azizi, E., Marsh, R. L., & Roberts, T. J. (2013). Chasing maximal performance: A cautionary tale from the celebrated jumping frogs of Calaveras County.The Journal of Experimental Biology, 216(21), 3947\u20133953.[\/footnote] The followingare the jump distances (in meters) for a sample of 15 bullfrogs.<\/div>\r\n
\r\n\r\n\r\n\r\n\r\n
0.1<\/td>\r\n0.4<\/td>\r\n0.6<\/td>\r\n0.8<\/td>\r\n1.3<\/td>\r\n1.5<\/td>\r\n1.6<\/td>\r\n1.7<\/td>\r\n<\/tr>\r\n
1.8<\/td>\r\n1.8<\/td>\r\n1.9<\/td>\r\n1.9<\/td>\r\n1.9<\/td>\r\n2.0<\/td>\r\n2.2<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe jump distances have been arranged in order from the shortest distance to the longest distance.<\/div>\r\n
For Questions 1\u20135, you can make the dotplot and calculate the mean and median by hand, or you can use the app at https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/.<\/div>\r\n
\r\n
\r\n

Question 1<\/h3>\r\n1) Construct a dotplotof the 15 jump distances.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n

Question 2<\/h3>\r\n2) Describe the shape of the distribution of jump distances.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n

Question 3<\/h3>\r\n3) What are the values of the mean and median for this dataset?\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n

Question 4<\/h3>\r\n4) What characteristic of the data distribution explains why the median is greater than the mean for this dataset?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
\r\n

Question 5<\/h3>\r\n5) Which do you think is a better choice for describing a typical value for this dataset\u2013the mean or the median?\r\n\r\n<\/div>\r\n<\/div>\r\n
Interpreting Confidence Intervals for a Difference in Means<\/div>\r\n
Recall that a confidence interval for a difference in population means is interpreted as an interval of plausible values for the difference in means. For example, suppose that each student in a random sample of 50 first-year students at a college and each student in a random sample of 50 second-year students from the college are asked how many hours of sleep they get on a typical weekday night. The data from these two samples are used to construct a confidence interval for the difference \ud835\udf07\ud835\udc39\u2212\ud835\udf07\ud835\udc46, where \ud835\udf07\ud835\udc39 is the mean number of sleep hours for first-year students and \ud835\udf07\ud835\udc46 is the mean number of sleep hours for second-year students. Because the samples are random samples and the sample sizes are both greater than 30, it would be appropriate to use a two-sample t confidence interval. If the 95% confidence interval was (0.4, 1.0), we would note the following:<\/div>\r\n
\u2022Plausible values for \ud835\udf07\ud835\udc39\u2212\ud835\udf07\ud835\udc46 are between 0.4 and 1.0.<\/div>\r\n
\u2022All of the plausible values are positive, which corresponds to \ud835\udf07\ud835\udc39 being greater than \ud835\udf07\ud835\udc46.<\/div>\r\n
\u2022We are 95% confident that the mean number of hours of sleep for first-year students is greater than the mean number of hours of sleep for second-year students by somewhere between 0.4 and1.0 hours.<\/div>\r\n
\r\n
\r\n

Question 6<\/h3>\r\n6) Suppose that the 95% confidence interval for \ud835\udf07\ud835\udc39\u2212\ud835\udf07\ud835\udc46was (\u22121.2, \u22120.5). Interpret this interval.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n

Question 7<\/h3>\r\n7) What would you conclude if 0 wasincluded in a confidence interval for\ud835\udf07\ud835\udc39\u2212\ud835\udf07\ud835\udc46?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"
\n
The previous in-class activity used a bootstrap confidence interval to estimate a population mean. In the next in-class activity, we will see how bootstrap confidence intervals can be used to estimate other population parameters. To support the preview assignment and the in-class activity, this corequisite support activity revisits two measures of center (the mean and the median) and reviews how confidence intervals for a difference in means are interpreted.<\/div>\n
Two Measures of Center\u2014The Mean and the Median<\/div>\n
Every year, bullfrogs compete in a jumping contest at the Calaveras County Jumping Frog Jubilee (a contest inspired by a short story by Mark Twain). One year, researchers recorded the jump distances of frogs entered in the contest.[1]<\/sup><\/a> The followingare the jump distances (in meters) for a sample of 15 bullfrogs.<\/div>\n
\n\n\n\n\n
0.1<\/td>\n0.4<\/td>\n0.6<\/td>\n0.8<\/td>\n1.3<\/td>\n1.5<\/td>\n1.6<\/td>\n1.7<\/td>\n<\/tr>\n
1.8<\/td>\n1.8<\/td>\n1.9<\/td>\n1.9<\/td>\n1.9<\/td>\n2.0<\/td>\n2.2<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The jump distances have been arranged in order from the shortest distance to the longest distance.<\/p><\/div>\n

For Questions 1\u20135, you can make the dotplot and calculate the mean and median by hand, or you can use the app at https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/.<\/div>\n
\n
\n

Question 1<\/h3>\n

1) Construct a dotplotof the 15 jump distances.<\/p>\n<\/div>\n<\/div>\n

\n
\n

Question 2<\/h3>\n

2) Describe the shape of the distribution of jump distances.<\/p>\n<\/div>\n<\/div>\n

\n
\n

Question 3<\/h3>\n

3) What are the values of the mean and median for this dataset?<\/p>\n<\/div>\n<\/div>\n

\n
\n

Question 4<\/h3>\n

4) What characteristic of the data distribution explains why the median is greater than the mean for this dataset?<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n
\n

Question 5<\/h3>\n

5) Which do you think is a better choice for describing a typical value for this dataset\u2013the mean or the median?<\/p>\n<\/div>\n<\/div>\n

Interpreting Confidence Intervals for a Difference in Means<\/div>\n
Recall that a confidence interval for a difference in population means is interpreted as an interval of plausible values for the difference in means. For example, suppose that each student in a random sample of 50 first-year students at a college and each student in a random sample of 50 second-year students from the college are asked how many hours of sleep they get on a typical weekday night. The data from these two samples are used to construct a confidence interval for the difference \ud835\udf07\ud835\udc39\u2212\ud835\udf07\ud835\udc46, where \ud835\udf07\ud835\udc39 is the mean number of sleep hours for first-year students and \ud835\udf07\ud835\udc46 is the mean number of sleep hours for second-year students. Because the samples are random samples and the sample sizes are both greater than 30, it would be appropriate to use a two-sample t confidence interval. If the 95% confidence interval was (0.4, 1.0), we would note the following:<\/div>\n
\u2022Plausible values for \ud835\udf07\ud835\udc39\u2212\ud835\udf07\ud835\udc46 are between 0.4 and 1.0.<\/div>\n
\u2022All of the plausible values are positive, which corresponds to \ud835\udf07\ud835\udc39 being greater than \ud835\udf07\ud835\udc46.<\/div>\n
\u2022We are 95% confident that the mean number of hours of sleep for first-year students is greater than the mean number of hours of sleep for second-year students by somewhere between 0.4 and1.0 hours.<\/div>\n
\n
\n

Question 6<\/h3>\n

6) Suppose that the 95% confidence interval for \ud835\udf07\ud835\udc39\u2212\ud835\udf07\ud835\udc46was (\u22121.2, \u22120.5). Interpret this interval.<\/p>\n<\/div>\n<\/div>\n

\n
\n

Question 7<\/h3>\n

7) What would you conclude if 0 wasincluded in a confidence interval for\ud835\udf07\ud835\udc39\u2212\ud835\udf07\ud835\udc46?<\/p>\n<\/div>\n<\/div>\n<\/div>\n

  1. Astley, H. C., Abbott, E. M., Azizi, E., Marsh, R. L., & Roberts, T. J. (2013). Chasing maximal performance: A cautionary tale from the celebrated jumping frogs of Calaveras County.The Journal of Experimental Biology, 216(21), 3947\u20133953. ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":23592,"menu_order":68,"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-5572","chapter","type-chapter","status-publish","hentry"],"part":5563,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5572","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":2,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5572\/revisions"}],"predecessor-version":[{"id":5669,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5572\/revisions\/5669"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/5563"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5572\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5572"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5572"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5572"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5572"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}