Question

A random sample of size n = 90 is obtained from a population whose size is

N = 10,000 and whose population proportion with a specified characteristic is P =0.8 Describe the sampling distribution of

Answer 1

Solution :

Given : n = 90 , N = 10000 , P = 0.8

Here,

n / N = 90 / 10000 = 0.009

Which is less than 0.05 so, finite population correction factor not necessary.

Let $url=https://latex.codecogs.com/gif.latex?%5Chat%20p&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=rPhekId3_kRHW_GnIPHqIA--~D$ be the sample proportion.

$url=https://latex.codecogs.com/gif.latex?%5Ctherefore&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=oMF_pyz5pCkYN9ix1tPExw--~D$ The sampling distribution of $url=https://latex.codecogs.com/gif.latex?%5Chat%20p&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=rPhekId3_kRHW_GnIPHqIA--~D$ is approximately normal with

mean = $\mu_{\hat p}$ =  p = 0.8

SD = $\sigma_{\hat p}$   =   $url=https://latex.codecogs.com/gif.latex?%5Csqrt%7Bp%281-p%29/n%7D&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=lEcnYtt_M7xILs1446V0Kw--~D$  

=

=  0.04216370213

SD = $\sigma_{\hat p}$ ​​​​​​​ =  0.04216370213