Question

A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers. How large a sample do the buyers need in order to be 95% confident that the margin of error of their estimate is about 5%?

Answer 1

Answer: 385

Step-by-step explanation:

Formula to find the sample size is given by :-

$n=p(1-p)(\dfrac{z_c}{E})^2$

, where p = prior estimate of population proportion.

E= Margin of error.

$z_c$ = z-value for confidence interval of c.

When prior estimate of population proportion is not available , we take p= 0.5.

Then the above formula becomes ,

$n=(0.5)(1-0.5)(\dfrac{z_c}{E})^2$

$n=0.25(\dfrac{z_c}{E})^2$

Given : Confidence interval : 95%

From the z-value table , the z-value for 95% confidence interval = $z_c=1.96$

The current owners have never determined the percentage of outdated merchandise and cannot help the buyers.

i.e. prior estimate of population proportion is not available.

Margin of error : E= 5%=0.05

Now, the required minimum sample size would be :-

$n=0.25(\dfrac{1.96}{0.05})^2$

Simplify ,

$n=0.25\times1536.64=384.16\approx385$

Thus , the minimum sample size needed by the buyers = 385