1. Student researchers took a random sample of 100 students and calculated the ratio of backpack weight to body weight for each student. They found a sample mean ratio of 0.07713, and a sample standard deviation of 0.03664.
(a) Determine a 90%, 95%, and 99% confidence interval for the population mean µ.
(b) Do these intervals suggest that the average student carries less than 10% of his or her body weight in his or her backpack? Explain.
2. Suppose X1, . . . , Xn are i.i.d. Normal random variables with unknown mean µ and unknown variance σ2.
(a) Describe how to find c1 and c2 such that
(You will not be able to get values for c1 and c2 until you have a value for n)
(b) Use your answer from part (a) to determine a formula for a 95% confidence interval for σ2.
(c) Determine the width of the 95% t-interval for µ, as a function of the sample standard deviation Sn-1, when n = 5, when n = 10, and when n = 20.
(d) Determine the width of the 95% chi-square-interval for σ2, as a function of the sample standard deviation Sn-1, when n = 5, when n = 10, and when n = 20.
3. Consider again the setup of problem 1.
(a) Determine a 95% confidence interval for the population standard deviation σ.
(b) Recalculate the 95% confidence interval for the mean by (incorrectly) applying the z-interval formula as if σ = 0.03664. Compare your answer to the corresponding 95% t-interval from problem 1.(a).
(c) Now consider what would happen if we had incorrectly used z-intervals instead of t-intervals for µ if our results had been from smaller sample sizes. Calculate 95% z-intervals and t-intervals for µ for sample sizes of n = 5, n = 10, and n = 20.
Briefly compare and summarize your results.