Question

A newspaper published an article about a study in which researchers subjected laboratory gloves to stress. Among 254 vinyl​ gloves, 69​% leaked viruses. Among 254 latex​ gloves, 12​% leaked viruses. Using the accompanying display of the technology​ results, and using a 0.05 significance​ level, test the claim that vinyl gloves have a greater virus leak rate than latex gloves. Let vinyl gloves be population 1. What is the test statistic and p value?

Answer 1

1 denotes vinyl​gloves and 2 denotes latex​ gloves.

For sample 1, the sample proportion is p1 = 0.69, n1 = 254

For sample 2, the sample proportion is p2 = 0.12, n2 = 254

The value of the pooled proportion is computed as

Also, the given significance level is α = 0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p1​ = p2​

Ha: p1 ​> p2​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

The significance level is α = 0.05, and the critical value for a right-tailed test is zc ​= 1.64.

The rejection region for this right-tailed test is R = { z : z > 1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = 13.086 > zc ​= 1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 1.979575e-39 0, and since p = 0 < 0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is greater than p2​, at the 0.05 significance level.

Therefore, there is enough evidence to claim that the vinyl gloves have a greater virus leak rate than latex gloves at the 0.05 significance level.