4.58.* A general rule states that for independent observations, the variance of
yi is the sum of the variances, which is nσ2 for n observations.

(a) Explain intuitively why
yi would have a larger variance than a single observation y. 

(b) Since the variance of a probability distribution is σ2 = E(y − μ)2, explain why the variance of the sampling distribution of ¯y is

(Hint: The second expression represents a sum with n2 in the denominator, which is a constant that can be put in front of the summation.)

(c) From (b), explain why the standard error equals σ¯y = σ/
√
n.

Transcribed Image Text:

E - n = E 2 - n 1/2 (yi - np) = 11 (no) = 02 n n n