2. The owner of a shoe factory wants to estimate the average length of left feet of adult men in the city. Let Y; be the length (in cm) of the left foot of adult male ; let X, be his height (in cm). A census tells us the average height of adult men in the city is 168cm. The owner conducts a SRS of 100 adult men in the city and observes the following results:  To be clear, s2x and s2y are the observed sample variances of X and Y respectively, while  is the observed sample covariance of X and Y. You can assume that the number of men in the city is very large.

(a) Will a 95% C.L. for the total yield calculated via ratio estimation have a greater or smaller width than an equivalent C.I. calculated via SRS formulae? Justify your answer without calculating either C.I.

(b) Write down the point estimates for the population mean foot length from the SRS estimator; the ratio estimator and the regression estimator,

(c) If we define  then show

(d) It can be shown that the estimated variance of the regression estimator is ^p where is the sample correlation between X and Y. Using this or otherwise, estimate the variances of the ratio estimator and the regression estimator.

(e) Which estimator do you recommend to the owner? Justify your answer.