Let X any Y equal the number of miles per gallon achieved by compact cars and mi

Question

Let X any Y equal the number of miles per gallon achieved by compact cars and midsized cars, respectively, as reported in fuel economy ratings. Assume that mu X = 24.5, sigma X = 3.8, mu Y = 21.3, and sigma Y = 2.7. Let X- and Y- be the sample means of independent random samples of eight observations of X and Y, respectively. (a) Find the means and variances of X- and Y-. (b) Assuming both X and Y are normally distributed, how is X- - Y- distributed? (c) Find P(X- > Y-).

Explanation / Answer

means of X bar= mean of x (as mean is an unbiased estimator)=24.5, s.d of x bar= (1/(sample size)^.5)*s.d. of x=(1/(8^.5))*3.8=1.3435 same for y bar=21.3, s.d. of y bar=(1/(8^.5))*2.7=.9545(upto 4 decimal place) , X normal so x bar normally distributed so is y bar as they are independendent so x bar -y bar follows normal. the praobability is same as of X>Y X-Y folows normal with mean=3.2 and s.d.=(3.8^2+2.7^2)^.5=4.66 then convert to a standard normal and use normal table to get the value

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