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

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

Find

\(\hat{p}\frac{1-\hat{p}}{n}=0.02\,\frac{1-0.02}{500}=3.92\times10^ {-5}\)

Take the square root to get 0.00626

Multiply your answer by z.

This step gives you the margin of error.

1.96 × 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 × 0.02 ± 50000 × 0.012 or

(50000 x 0.008,50000 x 0.032)

1000 ± 600 or (400, 1600)