Question: In a sample 80 workers from a factory in city A, it was found that 5% were unable to read, while in a sample of 50 workers in city B, 8% were unable to read. Can it be concluded that there is a difference in the proportions of nonreaders in the two cities? Use a = 0.10. Find the 90% confidence interval for the difference of the two proportions.

Solution: (a) We want to test the following null and alternative hypotheses:

We need to use a z-test for proportions. The level of significance is set at. From the information provided, we have that

and also, the pooled proportion is

With this information, the z-statistics is computed as

The critical value for for this test is. The rejection region is given by

Since, then we fail to reject the null hypothesis H₀.

Hence, we don't have enough evidence to support the claim that there is a difference in the proportions of nonreaders in the two cities.

(b) The 90% confidence interval for is computed as

Based on the information provided, we get that

The confidence interval is given by:

This means we are 90% confident that the interval contains the actual value of.