Question

a newspaper published an article about a study in which researchers subjected laboratory gloves to stress. among 254 vinyl gloves, 62% leaked viruses. among 254 latex gloves, 6% 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 are the null and alternative hypotheses? a. $h_0: p_1 = p_2$, $h_1: p_1 \
eq p_2$ b. $h_0: p_1 > p_2$, $h_1: p_1 = p_2$ c. $h_0: p_1 = p_2$, $h_1: p_1 > p_2$ d. $h_0: p_1 \
eq p_2$, $h_1: p_1 = p_2$ e. $h_0: p_1 < p_2$, $h_1: p_1 = p_2$ f. $h_0: p_1 = p_2$, $h_1: p_1 < p_2$ identify the test statistic. (round to two decimal places as needed) technology results pooled proportion: 0.34 test statistic z: 13.3133 critical z: 1.6449 p - value: 0.0000 90% confidence interval: 0.5033215 < $p_1 - p_2$ < 0.6147887

Explanation:

First Sub - Question (Null and Alternative Hypotheses)

The claim is that vinyl gloves (population 1) have a greater virus leak rate than latex gloves (population 2). The null hypothesis \(H_0\) for a two - proportion test of this type is that the two proportions are equal (\(p_1 = p_2\)). The alternative hypothesis \(H_1\) reflects the claim, so \(p_1>p_2\). Looking at the options, option C (\(H_0:p_1 = p_2\), \(H_1:p_1>p_2\)) matches this.

Step 1: Locate the test statistic

From the "Technology Results" box, the test statistic (z - statistic) is given as \(z = 13.3133\).

Step 2: Round to two decimal places

Rounding \(13.3133\) to two decimal places, we look at the third decimal place which is \(3\). Since \(3<5\), we round down. So \(13.3133\approx13.31\).

Answer:

C. \(H_0:p_1 = p_2\), \(H_1:p_1>p_2\)

Second Sub - Question (Test Statistic)