Question

Suppose a small finite population consists of only N = 7 numbers:

20 22 25 27 30 33 34

I. Find population mean and standard deviation
II. Draw all possible samples of size 2 without replacement
III. Construct sampling distribution
IV. Find sample mean and standard deviation
V. Verify/justify your results.

Answer 1

I. Find population mean and standard deviation

$\mu = \frac{20+ 22+ 25+ 27 +30 +33 +34}{7} =27.28 \\ \sigma = \sqrt{\frac{1}{n}[(20-27.28)^2 +(22-27.28)^2 +...+(33-27.28)^2 +(34-27.28)^2]} = 5.34$

II. Draw all possible samples of size 2 without replacement

Number of samples $= \frac{7!}{2!(7-2)!} = \frac{6 \times 7}{2} = 21$

III. Construct sampling distribution

IV. Find the sample mean and standard deviation

$Sample\; mean = \frac{21+22.5+...+32+33.5}{21} = 27.28 \\ SD = \sqrt{\frac{1}{n} [(21-27.28)^+(22.5-27.28)^2 +...+(32-27.28)^2 +(33.5-27.28)^2]} = 3.27$

V. Verify/justify your results.

Population mean and sample mean are equal, but population standard deviation is greater than sample standard deviation.