Let's go back to the dataset TenMileRace from Lab 1. Load this dataset (you will have to load the mosaicData library). We saw in class that we can construct confidence intervals for means of small (n less than or equal to 30) data sets as long as the sample comes from a population that is normally distributed. The net race times follow a normal distribution, so we will construct confidence intervals for the mean net race time based on samples of size 30. Construct a random sample of 30 net race times: > racetimes30=Ten Mile Race $net[sample (1:8636, 30)] The t.test function in R allows us to construct confidence intervals in a single line: > t.test(racetimes30) One Sample t-test data: racetimes 30 t = 35.126, df = 29, p-value mean (Ten Mile Race $net) [1] 5599.065 This value is indeed in the confidence interval. 4. Our theory tells us that 95% of the time, or 19 times out of 20, the true population mean will be contained in the 95% confidence interval. (The other 5% of the time, our sample happened to be "weird" - ie, with a distribution much different from that of the population.) Generate 20 different random samples of size 25 from the population of net race times and construct confidence intervals. How many of them contained the true mean? Note again that this theory only works for small samples if they come from a normally-distributed population. If we compute a 95% confidence interval the same way for a small sample that is very abnormal, the range we get will not be a true 95% confidence interval. That is, it will not be the case that 95% of confidence intervals obtained with our method will contain the true population mean.