Question

20 babies born in one week in a local hospital had the following weights (in pounds): 9.6, 8.8, 5.1, 7.7, 6.1, 8.9, 8, 9.2, 5.7, 9.1, 8.5, 7.3, 9.3, 9.6, 5.2, 9.9, 7.6, 9, 7.2, 8.5 (a) Create a QQ plot and histogram of the weights. Do you think it is reasonable to assume that the population distribution is normal? Explain your answer. (b) Regardless of your answer to (a), use R to perform the bootstrap with 3000 resamplings to create a 98% CI for µ. (Show your R code and its output - you can copy and paste the code given in lecture or discussion.) (c) Now construct a 98% CI for µ by hand using the t-tables, and compare it to your bootstrap-t confidence interval. (e) Compute the power of the test if the true population mean is µA = 15. (f) Using s = 4.88 as our best guess of σ, approximately what sample size would be required to achieve a power of 0.8 if the true population mean is µA = 15? Give your answer as the smallest whole number that meets the criterion.

Answer 1

a

While the QQ plot does suggest that the values might be coming from a normal distribution, the histogram on its own is pretty inconclusive.

The bootstrap method returns a

.

The formula for the confidence interval is

For and , looking up the t-distribution table yields .

The mean and standard deviation can be calculated as

Finally, the can be calculated as

The confidence interval calculated using a t-score is similar to the one obtained through bootstrapping, albeit a bit wider.