18B Coreq

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.
Two Measures of Centerβ€”The Mean and the Median
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] The followingare the jump distances (in meters) for a sample of 15 bullfrogs.
0.1 0.4 0.6 0.8 1.3 1.5 1.6 1.7
1.8 1.8 1.9 1.9 1.9 2.0 2.2

The jump distances have been arranged in order from the shortest distance to the longest distance.

For Questions 1–5, 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/.

Question 1

1) Construct a dotplotof the 15 jump distances.

Question 2

2) Describe the shape of the distribution of jump distances.

Question 3

3) What are the values of the mean and median for this dataset?

Question 4

4) What characteristic of the data distribution explains why the median is greater than the mean for this dataset?

Question 5

5) Which do you think is a better choice for describing a typical value for this dataset–the mean or the median?

Interpreting Confidence Intervals for a Difference in Means
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 πœ‡πΉβˆ’πœ‡π‘†, where πœ‡πΉ is the mean number of sleep hours for first-year students and πœ‡π‘† 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:
β€’Plausible values for πœ‡πΉβˆ’πœ‡π‘† are between 0.4 and 1.0.
β€’All of the plausible values are positive, which corresponds to πœ‡πΉ being greater than πœ‡π‘†.
β€’We 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.

Question 6

6) Suppose that the 95% confidence interval for πœ‡πΉβˆ’πœ‡π‘†was (βˆ’1.2, βˆ’0.5). Interpret this interval.

Question 7

7) What would you conclude if 0 wasincluded in a confidence interval forπœ‡πΉβˆ’πœ‡π‘†?

  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), 3947–3953. ↵