Your Packaging Company Fills Various Types of Flour into Bags ${ } ^ { \sigma } { } _ { Y }$

Essay

Your packaging company fills various types of flour into bags. Recently there have been complaints from one chain of stores: a customer returned one opened 5 pound bag which weighed significantly less than the label indicated. You view the weight of the bag as a random variable which is normally distributed with a mean of 5 pounds, and, after studying the machine specifications, a standard deviation of 0.05 pounds.
(a)You take a sample of 20 bags and weigh them. Sketch below what the average pattern of individual weights might look like. Let the horizontal axis indicate the sampled bag number (1, 2, …, 20). On the vertical axis, mark the expected value of the weight under the null hypothesis, and two (? 1.96)standard deviations above and below the expected value. Draw a line through the graph for E(Y)+ 2 ${ } ^ { \sigma } { } _ { Y }$ , E(Y), and E(Y)- 2 ${ } ^ { \sigma } Y$ How many of the bags in a sample of 20 will you expect to weigh either less than 4.9 pounds or more than 5.1 pounds?
(b)You sample 25 bags of flour and calculate the average weight. What is the distribution of the average weight of these 25 bags? Repeating the same exercise 20 times, sketch what the distribution of the average weights would look like in a graph similar to the one you drew in (b), where you have adjusted the standard error of $\bar { Y }$ accordingly.
(c)For each of the twenty observations in (c)a 95% confidence interval is constructed. Draw these confidence intervals, using the same graph as in (c). How many of these 20 confidence intervals would you expect to weigh 5 pounds under the null hypothesis?