(a) Show that the sample variance s 2 = [Pn i=1(xi − x¯) 2 ]/(n− 1) can also be expressed as s 2 = [Pn i=1 x 2 i − ( Pn i=1 xi) 2n ]/(n − 1). At a medical center, a sample of 36 days showed thefollowing number of cardiograms done each day.
25 31 20 32 20 24 43 22 57 23 35 22 43 26 56 21 19 29 36 32 3332 44 32 52 44 51 45 47 20 31 27 37 30 18 28
(b) (1 point) Find the sample mean ¯x and the sample variance s2 x .
(c) (2 points) Construct a stem and leaf plot for the data andfind the sample median.
(d) (3 points) Construct a 86% confidence interval for thepopulation µ.
(e) (4 points) A researcher wishes to test the claim that theaverage number of cardiograms done each day is equal to or greaterthan 33. Is there evidence to support the claim at α = 0.05? Findthe p-value.
(f) (1 point) Let x1, x2, · · ·, x36 be the data of 36 days ofcardiograms above; let a and b be any nonzero constants. If y1 = ax1 +b, y2 = a x2 +b, · · ·, y36 = a x36 +b, and let ¯y and s 2 y bethe sample mean and the sample variance of the yi ’s, respectively.What is the relationship between ¯x and ¯y? What is therelationship between s 2 x and s 2 y ?
(g) (5 points) Show that s 2 x is an unbiased estimator of thepopulation mean σ 2 .
