The mean of a sample of size n has standard deviation n, where ???? is the population

The mean of a sample of size n has standard deviation σ∕√n, where ???? is the population standard deviation. When sampling without replacement, a more accurate expression can be obtained by multiplying by a correction factor. Specifically, if the sample size is more than 5% of the population size, it is better to compute the standard deviation of the sample mean as

where N is the population size and n is the sample size. The factor

is called the finite population correction factor.

a. One hundred students took an exam. The standard deviation of the 100 scores was 10.

Twenty exams were chosen at random as part of a class assessment. Use the finite population correction to compute the standard deviation of the mean of the 20 exams.

b. In general, is the standard deviation computed with the correction smaller or larger than the standard deviation computed without it?

c. Use the finite population correction to show that if all 100 exams are sampled, the standard deviation of the sample mean is 0.

Explain why this is so.

Transcribed Image Text:

N-n N-1