4.57.* The standard error formula y = / n treats the population size as infinitely large relative to the sample size n. The formula

4.57.* The standard error formula σ¯y = σ/
√
n treats the population size as infinitely large relative to the sample size n. The formula for σ¯y for a finite population size denoted by N is

The term
(N − n)/(N − 1) is called the finite population correction.

(a) When n = 300 students are selected from a college student body of size N = 30, 000, show that σ¯y = 0.995σ/
√
n. (In practice, n is usually small relative to N, so the correction has little influence.)

(b) If n = N (i.e., we sample the entire population), show that σ¯y = 0. In other words, no sampling error occurs, because ¯y = μ.

(c) For n = 1, explain why the sampling distribution of ¯y and its standard error are identical to the population distribution and its standard deviation.

Oy= N-n VN-1 n