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A newspaper published an article about a study in which researchers subjected laboratory gloves to stress. Among 22 vinyl gloves, [tex]$60\%$[/tex] leaked viruses. Among 222 latex gloves, [tex]$5\%$[/tex] leaked viruses. Using the accompanying display of the technology results, and using a 0.10 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. [tex]$H_0: P_1 = P_2$[/tex]
[tex]$H_1: P_1 \ \textless \ P_2$[/tex]

B. [tex]$H_0: P_1 \neq P_2$[/tex]
[tex]$H_1: P_1 = P_2$[/tex]

C. [tex]$H_0: P_1 \ \textless \ P_2$[/tex]
[tex]$H_1: P_1 = P_2$[/tex]

D. [tex]$H_0: P_1 = P_2$[/tex]
[tex]$H_1: P_1 \ \textgreater \ P_2$[/tex]

E. [tex]$H_0: P_1 = P_2$[/tex]
[tex]$H_1: P_1 \neq P_2$[/tex]

F. [tex]$H_0: P_1 \ \textgreater \ P_2$[/tex]
[tex]$H_1: P_1 = P_2$[/tex]

Identify the test statistic:
13.43
(Round to two decimal places as needed)

Identify the P-value:
[tex]$\square$[/tex]
(Round to three decimal places as needed)


Sagot :

To test the claim that vinyl gloves have a higher virus leak rate than latex gloves at a [tex]\( \alpha = 0.10 \)[/tex] significance level, we will set up the hypotheses and conduct a hypothesis test for the difference in proportions.

### Step-by-Step Solution:

1. Formulate the Hypotheses:

Null Hypothesis [tex]\( H_0 \)[/tex]: The proportion of virus leaks in vinyl gloves ([tex]\( P_1 \)[/tex]) is equal to the proportion of virus leaks in latex gloves ([tex]\( P_2 \)[/tex]).
[tex]\[ H_0: P_1 = P_2 \][/tex]

Alternative Hypothesis [tex]\( H_1 \)[/tex]: The proportion of virus leaks in vinyl gloves ([tex]\( P_1 \)[/tex]) is greater than the proportion of virus leaks in latex gloves ([tex]\( P_2 \)[/tex]).
[tex]\[ H_1: P_1 > P_2 \][/tex]

Based on the problem, the correct hypotheses are:
[tex]\[ \text{D:} \quad H_0: P_1 = P_2 \quad \text{and} \quad H_1: P_1 > P_2 \][/tex]

2. Identify the Test Statistic:

The test statistic for comparing two proportions is calculated using the z-score formula for proportions. The provided test statistic (z-score) is:
[tex]\[ z = 8.2173 \quad \text{(rounded to four decimal places)} \][/tex]

When rounded to two decimal places:
[tex]\[ z \approx 8.22 \][/tex]

3. Identify the P-value:

The p-value is determined based on the calculated z-score. For a one-tailed test to the right (since [tex]\( H_1: P_1 > P_2 \)[/tex]):
[tex]\[ p\text{-value} = P(Z > 8.2173) = 0.0 \][/tex]

When rounded to three decimal places:
[tex]\[ p\text{-value} = 0.000 \][/tex]

### Summary:

- Null and Alternative Hypotheses:
[tex]\[ \text{D:} \quad H_0: P_1 = P_2 \quad \text{and} \quad H_1: P_1 > P_2 \][/tex]

- Test Statistic:
[tex]\[ z \approx 8.22 \quad \text{(rounded to two decimal places)} \][/tex]

- P-value:
[tex]\[ p\text{-value} = 0.000 \quad \text{(rounded to three decimal places)} \][/tex]

Given the extremely low p-value (0.0), which is less than any reasonable significance level, including [tex]\( \alpha = 0.10 \)[/tex], we reject the null hypothesis. This strongly indicates that vinyl gloves have a significantly higher virus leak rate compared to latex gloves.