Let X_1, X_2, ..., X_n, and Y_1, Y_2, ..., Y_m be independent random variables,
ID: 3227135 • Letter: L
Question
Let X_1, X_2, ..., X_n, and Y_1, Y_2, ..., Y_m be independent random variables, with the X' s being a random sample from a N (mu_x, sigma^2_x) distribution, and the Y' s being a random sample from a N (mu_y, sigma^2_Y) distribution. Suppose that all four parameters, mu_x, sigma^2_X, mu_Y, sigma^2_Y, are unknown, but also suppose that it is known that sigma^2_X = 5 sigma^2_Y. Use this information to develop a t - ratio that has n + m - 2 degrees of freedom, and use that to find a formula for a 100 (1 - alpha) % confidence interval for mu_X - mu_Y.Explanation / Answer
Point estimate for 100(1-alpha)% CI=
(MuX-MuY)
pooled std deviation Sp= sqrt(((n-1)Sx^2-(m-1)Sy^2)/(n+m-2))
{now using given information Sx^2 = 5Sy^2
Sp= sqrt(((n-1)Sx^2-5(m-1)Sx^2)/(n+m-2)) = Sx* sqrt((n-5m+4)/(n+m-2))
pooled std error= Sp*sqrt(1/n+1/m) = Sx* sqrt(((n-5m+4)/(n+m-2))*(1/n+1/m))
thus,
100(1-alpha)% CI:
(MuX-MuY) +- t * Std error=
=(MuX-MuY) +- t*Sx* sqrt(((n-5m+4)/(n+m-2))*(1/n+1/m))
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