Testing Laboratory Gloves The New York Times ran an article about a study in which Professor Denise Korniewicz and other Johns Hopkins researchers subjected laboratory gloves to stress. Among 240 vinyl gloves, 63% leaked viruses. Among 240 latex gloves, 7% leaked viruses. At the 0.005 significance level, test the claim that vinyl gloves have a larger virus leak rate than latex gloves.

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To test the claim that vinyl gloves have a larger virus leak rate than latex gloves, we can use a hypothesis test. Here, we want to compare the proportion of leaking viruses in vinyl gloves (p1) to the proportion of leaking viruses in latex gloves (p2).

Let's set up our null and alternative hypotheses:

Null Hypothesis (H0): The virus leak rate in vinyl gloves is equal to or smaller than the virus leak rate in latex gloves. p1 <= p2
Alternative Hypothesis (H1): The virus leak rate in vinyl gloves is larger than the virus leak rate in latex gloves. p1 > p2

Next, we need to calculate the test statistic and determine the critical value. For hypothesis testing involving proportions, we use the z-test statistic. The formula for the z-test statistic is:

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

where:
p1 = proportion of virus leak rate in vinyl gloves
p2 = proportion of virus leak rate in latex gloves
n1 = sample size of vinyl gloves
n2 = sample size of latex gloves

In this case, p1 = 0.63 (63%), p2 = 0.07 (7%), n1 = 240, and n2 = 240.

Using these values, we can calculate the z-test statistic:

z = (0.63 - 0.07) / sqrt((0.63 * (1-0.63) / 240) + (0.07 * (1-0.07) / 240))

Once we calculate the test statistic, we can determine the critical value at the 0.005 significance level. For a one-tailed test, the critical value would be the z-score that corresponds to a 0.005 (or 0.5%) tail probability. We can look this up in the standard normal distribution table or use a calculator to find the critical value.

Comparing the test statistic to the critical value, if the test statistic is greater than the critical value, we reject the null hypothesis. If the test statistic is smaller than the critical value, we fail to reject the null hypothesis.

I hope this explanation helps you understand how to test the claim comparing the virus leak rates in vinyl and latex gloves!