Question

A factory producing 50,000 pairs of shoes daily. From a sample of 500 pairs 2% were found to be substandard quality. Estimate the number of pairs that can be reasonably expected to be spoiled in the daily production and assign limits at 95% level of confidence.

Answer 1

Determination of Confidence Limit For Population Proportion

Because you want a 95% confidence interval, your z-value is 1.96.

From a sample of 500 pairs 2% were found to be substandard quality. So

$\hat{p} =2 \; \%=0.02$

The formula for a confidence interval for a population proportion is

$\hat{p}±z \times \sqrt{\frac{\hat{p}(1- \hat{p})}{n}}$

Find

$\hat{p}\frac{1 - \hat{p}}{n} = 0.02 \frac{1-0.02}{500} = 3.92 \times 10^{-5}$

Take the square root to get 0.00626

Multiply your answer by z.

This step gives you the margin of error.

$1.96 \times 0.00626 = 0.0122$

Confidence interval:

0.02±0.012 or (0.008, 0.032)

The number of pairs that can be reasonably expected to be spoiled in the daily production:

$50000 \times 0.02 ± 50000 \times 0.012$ or $(50000 \times 0.008, 50000 \times 0.032)$

1000 ± 600 or (400, 1600)