Question

A newspaper published an article about a study in which researchers subjected laboratory gloves to stress. Among 225 vinyl? gloves, 60?% leaked viruses. Among 225 latex? gloves, 9?% 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.
Technology Results:::

Pooled? proportion: 0.35

Test? statistic, z: 11.2921

Critical? z: 1.6449

?P-value: 0.0000

90?% Confidence? interval:

0.4441890 < p1? p2 <0.5691444

A) What are the null and alternative hypotheses?

B) Identify the test statistic

C) Identify the P-value

D) What is the conclusion for the test?

Answer 1

Answer:

a)

H₀: ρ₁ ≤ ρ₂

H₁: ρ₁ > ρ₂

b)

Z= $\frac{('p_{1} - 'p_{2}) - (p_{1} - p_{2})}{\sqrt{'p(1-'p)[\frac{1}{n_1} + \frac{1}{n_2} ]} }$ ≈ N(0;1)

c)

p-value < .00001

d)

Reject the null hypothesis.

The population proportion vinyl gloves that leak viruses is greater than the population proportion of latex gloves that leak viruses.

Step-by-step explanation:

Hello!

The objective is to test if the population proportion of vinyl gloves that leak viruses (population 1) is greater then the population proportion of latex gloves that leak viruses (population 2).

Sample 1 (Vinyl)

n₁= 225

Sample proportion 'ρ₁= 0.60

Sample 2 (Latex)

n₂= 225

Sample proportion 'ρ₂= 0.09

Pooled proportion: 'ρ= (x₁+x₂)/(n₁+n₂)= ( 'ρ₁ + 'ρ₂)/2= (0.60 + 0.09)/2= 0.345 ≅0.35

The hypothesis is:

H₀: ρ₁ ≤ ρ₂

H₁: ρ₁ > ρ₂

α: 0.05

The statistic to use is a pooled Z

Z= $\frac{('p_{1} - 'p_{2}) - (p_{1} - p_{2})}{\sqrt{'p(1-'p)[\frac{1}{n_1} + \frac{1}{n_2} ]} }$ ≈ N(0;1)

The critical region is one-tailed to the right (positive), the critical value is:

$Z_{1-\alpha } = Z_{0.95} = 1.64$

If Z ≥ 1.64, you will reject the null hypothesis.

If Z < 1.64, you will not reject the null hypothesis.

Z= $\frac{(0.6 - 0.09) - (0}{\sqrt{0.35(1-0.35)[\frac{1}{225} + \frac{1}{225} ]} }$ = 11.34

Since the calculated Z- value is greater than the critical value, the decision is to reject the null hypothesis. The population proportion vinyl gloves that leak viruses is greater than the population proportion of latex gloves that leak viruses.

P-value approach:

p-value < .00001

If you compare with the level of signification, you reach the same decision.

I hope it helps!