Question

A random sample is selected from a population with mean and standard deviation . Determine the mean and standard deviation of the sampling distribution of for each of the following sample sizes:

a. n = 9
b. n = 15
c. n = 36
d. n = 50
e. n = 100
f. n = 400

Answer 1

a. Mean = 100, S.D. = 3.333

b. Mean = 100, S.D. = 2.582

c. Mean = 100, S.D. = 1.667

d. Mean = 100, S.D. = 1.414

e. Mean = 100, S.D. = 1

f. Mean = 100, S.D. = 0.5

Explanation:

The question is incomplete:

Population mean: 100

Population standard deviation: 10.

The mean for any sampling distribution is equal to the population mean.

The standard deviation for the sampling distribution depends on the population standard deviation and the sample size as:

`\sigma_s=(\sigma)/(√(n))`

We can calculate the parameters of the sampling distributions as:

a. n = 9

`\mu_s=\mu=100\\\\ \sigma_s=(\sigma)/(√(n))=(10)/(√(9))=(10)/(3)=3.333`

b. n = 15

`\mu_s=\mu=100\\\\ \sigma_s=(\sigma)/(√(n))=(10)/(√(15))=(10)/(3.873)=2.582`

c. n = 36

`\mu_s=\mu=100\\\\ \sigma_s=(\sigma)/(√(n))=(10)/(√(36))=(10)/(6)=1.667`

d. n = 50

`\mu_s=\mu=100\\\\ \sigma_s=(\sigma)/(√(n))=(10)/(√(50))=(10)/(7.071)=1.414`

e. n = 100

`\mu_s=\mu=100\\\\ \sigma_s=(\sigma)/(√(n))=(10)/(√(100))=(10)/(10)=1`

f. n = 400

`\mu_s=\mu=100\\\\ \sigma_s=(\sigma)/(√(n))=(10)/(√(400))=(10)/(20)=0.5`