Question

Solve the problem below.

Consider all samples of size of 5 from this population:

2, 5, 7, 9, 10, 11, 12

a. Compute the population mean.

b. Compute the population variance.

c. Compute the population standard deviation.

d. Compute the mean of the sampling distribution of the sample means and

compare it the mean of the population.

e. Compute the variance of the sampling distribution of the sample means.

f. Compute the standard deviation of the sampling distribution of the sample

means.

Answer 1

a. The population mean

$\mu = \frac{2+5+7+9+10+11+12}{7}= 8$

b. The population variance

$s^2 = \frac{1}{n-1} \sum (x_i - \mu)^2 \\ s^2 = \frac{1}{7-1}((2-8)^2 + (5-8)^2 + (7-8)^2 + (9-8)^2 + (10 -8)^2 + (11 -8)^ + (12 -8)^2) \\ = \frac{1}{6}( 36 + 9 + 1 + 1+ 4 + 9 + 16 ) \\ = 12.66$

c. The population standard deviation

$s = \sqrt{24.61} = 3.55$

d. The number of all possible sample = ⁷C₅=21 and the corresponding sample mean = sum of each five values/5

The mean of the sampling distribution of the sample means = 8

The mean of the sampling distribution of the sample means is equal to compare it the mean of the population.

e. The variance of the sampling distribution of the sample means

$s^2 = \frac{1}{21-1} \times 15.2 = 0.76$

f. The standard deviation of the sampling distribution of the sample means

$s = \sqrt{0.76} = 0.87$