. You will be comparing the sampling distributions for two different estimators of a, the population standard deviation. When trying to estimate the standard deviation of a population (a) from a sample of size n we could use: .X-}_(2 .X-X'2 SI: 21(1 ) 01' 32: 22(1 ) nl n (51 is the sample standard deviation). The graphs below give the sampling distributions produced by these estimators when drawing a sample of size 8 from a normal population with mean ,u = 3 and standard deviation 0 = 5. What do you notice about the mean of the standard deviations produced using the .31 estimator compared to the 32 estimator compared to the true population stande deviation? Why do we prefer to use the 31 formulation when we have a sample of data and are interested in estimating the population standard deviation? (You should use the resulting histograms to help you answer the question and use the word bias.) \f2. Retail stores experience their heaviest volume of transactions that include returns on December 26th and December 27th each year. The distribution for the Number of Items Returned (X) by Macys custOmers who do a return transaction on those days last year is given in the table below. It has mean: ,u = 2.61 and variance 02 z 1.80. Number of Items Returned Probability 1 0.25 0.28 0.2 0.17 0.08 0.02 (mandamus (a) Is this population distribution left skewed, symmetric, or right skewed? How do you know? (b) In what percent of returns did retail workers see three or more items in the return