A drug store chain provides an app to its customers to track their shopping habits. One statistic the app tracks is the amount of money the customer saves by purchasing sale items. The company's sales

team pulls data from the previous year for a random sample of 50 customers. They find that the

mean amount of money saved by these customers in the previous year is $154 with a standard

deviation of $26.

(a) Construct a 99% confidence interval for the true mean amount of money saved by all customers

in the previous year by purchasing sale items.

(b) The sales team would like to repeat this study with the goal of obtaining a smaller margin of

error. Propose two changes that would decrease the margin of error. What are potential

drawbacks if those changes are implemented?

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.