Question

A population consists of N=7 numbers 1, 2, 2, 3, 4, 5, and XX. Draw all possible samples 

of size n=3 without replacement from this population and find the sample proportion of 

odd numbers in the samples. Construct the sampling distribution of sample proportion 

and verify 𝜇𝑝̂ = 𝑃 and 𝜎

2

𝑝̂ =

𝑝𝑞

𝑛

.

𝑁−𝑛

𝑁−1

where 𝑞 = 1 − 𝑝

Answer 1

Let XX= 5

Population: 1, 2, 2, 3, 4, 5, 5

N=7

Number of odd numbers =4

Proportion of odd numbers: $p =\frac{4}{7}$

n=3

Number of samples $K= \frac{7!}{3!(7-3)!}= \frac{5 \times 6 \times 7}{2 \times 3}= 35$

Sampling distribution of sample proportion 𝑝̂

Mean of sample proportion:

$\mu_{\hat{p}}= \frac{\sum \hat{p}}{K}=\frac{20}{35}= \frac{4}{7}$

Variance of sample proportion:

$\sigma_{\hat{p}}= \frac{\sum \hat{p}^2}{K}- \mu^2_{\hat{p}} \\ = \frac{120/9}{35} - (\frac{4}{7})^2 \\ = \frac{120}{315}- \frac{16}{49} \\ = \frac{8}{147}$

Variance of sample proportion:

$\sigma_{\hat{p}}= \frac{8}{147}$