Question

A bag contains chips of which 27.5 percent are blue. A random sample of 5 chips will be selected one at a time and with replacement. What are the mean and standard deviation of the sampling distribution of the sample proportion of blue chips for samples of size 5 ?

Answer 1

Answer:

$\mu _{\hat{p}}= 0.275\\\\ \sigma_{\hat{p}}=0.1997$

Step-by-step explanation:

We know that the mean and standard deviation of the sampling distribution of the sample proportion($\hat{p}$) is given by :-

$\mu _{\hat{p}}=p\\\\ \sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{n}}$

, where p= Population proportion and n = sample size.

Let p be the proportion of blue chips.

As per given , we have

p= 0.275

n= 5

Now , the mean and standard deviation of the sampling distribution of the sample proportion of blue chips for samples of size 5 will be :

$\mu _{\hat{p}}= 0.275\\\\ \sigma_{\hat{p}}=\sqrt{\dfrac{ 0.275(1- 0.275)}{5}}\\\\=0.19968725547\approx0.1997$

Hence, the mean and standard deviation of the sampling distribution of the sample proportion of blue chips for samples of size 5 are :

$\mu _{\hat{p}}= 0.275\\\\ \sigma_{\hat{p}}=0.1997$