Answer to Question #199398 in Statistics and Probability for JAFFAR HUSSAIN

Question #199398

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 πœ‡π‘Μ‚ = 𝑃 and 𝜎

2

𝑝̂ =

π‘π‘ž

𝑛

.

π‘βˆ’π‘›

π‘βˆ’1

where π‘ž = 1 βˆ’ 𝑝


1
Expert's answer
2021-06-01T03:33:02-0400

Let XX= 5

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

N=7

Number of odd numbers =4

Proportion of odd numbers: p=47p =\frac{4}{7}

n=3

Number of samples K=7!3!(7βˆ’3)!=5Γ—6Γ—72Γ—3=35K= \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:

ΞΌp^=βˆ‘p^K=2035=47\mu_{\hat{p}}= \frac{\sum \hat{p}}{K}=\frac{20}{35}= \frac{4}{7}

Variance of sample proportion:

Οƒp^=βˆ‘p^2Kβˆ’ΞΌp^2=120/935βˆ’(47)2=120315βˆ’1649=8147\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:

Οƒp^=8147\sigma_{\hat{p}}= \frac{8}{147}


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