A product testing laboratory selected a sample of 20 printers from 2 different manufacturers to determine if they could print more than 100,000 pages without breaking down. At , test the claim that the proportions lasting more than 100,000 pages are equal. A) There is not evidence to reject the claim that the proportions are equal because the test value 2.506 1.003 C) There is not evidence to reject the claim that the proportions are equal because the test value 2.506 1.003

Question

A product testing laboratory selected a sample of 20 printers from 2 different manufacturers to determine if they could print more than 100,000 pages without breaking down. At , test the claim that the proportions lasting more than 100,000 pages are equal. A) There is not evidence to reject the claim that the proportions are equal because the test value 2.506 < 3.841 B) There is evidence to reject the claim that the proportions are equal because the test value 3.841 > 1.003 C) There is not evidence to reject the claim that the proportions are equal because the test value 2.506 < 9.488 D) There is not evidence to reject the claim that the proportions are equal because the test value 9.488 > 1.003

Quizplus · Accepted Answer

Since we are testing if the proportions are equal, we can use a two-sample z-test for proportions. The test statistic is given by:

z = (p1 - p2) / sqrt(p_hat * (1 - p_hat) * (1/n1 + 1/n2))

where p1 and p2 are the sample proportions, p_hat is the pooled sample proportion, and n1 and n2 are the sample sizes.

In this case, the null hypothesis is that the proportions are equal, which means p1 = p2. The alternative hypothesis is that the proportions are not equal, which means p1 ≠ p2.

We are given that there are 20 printers in each group, and we are testing if they can print more than 100,000 pages without breaking down. Let p1 be the proportion of printers from manufacturer 1 that can print more than 100,000 pages, and let p2 be the proportion of printers from manufacturer 2 that can do the same. Then the sample proportions are:

p1 = 13/20 = 0.65
p2 = 9/20 = 0.45

The pooled sample proportion is:

p_hat = (13 + 9) / (20 + 20) = 0.55

Then the test statistic is:

z = (0.65 - 0.45) / sqrt(0.55 * (1 - 0.55) * (1/20 + 1/20)) = 2.506

The critical value for a two-tailed test at α = 0.05 with 2 degrees of freedom (df = (n1 - 1) + (n2 - 1) = 18) is 3.841. Since the test value (2.506) is less than the critical value (3.841), we fail to reject the null hypothesis. Therefore, there is not evidence to reject the claim that the proportions are equal.

A Product Testing Laboratory Selected a Sample of 20 Printers