Question

A newspaper published an article about a study in which researchers subjected laboratory gloves to stress. Among 215 vinyl​ gloves, 70​% leaked viruses. Among 215 latex​ gloves, 7​% leaked viruses.
1. Using a 0.01 significance​ level, test the claim that vinyl gloves have a greater virus leak rate than latex gloves.

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

Answer:

$z=\frac{0.7-0.07}{\sqrt{0.385(1-0.385)(\frac{1}{215}+\frac{1}{215})}}=13.42$  

$p_v =P(Z>13.42)\approx 0$  

So the p value is a very low value and using any significance level given $\alpha=0.01$ we se that $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that vinyl gloves have a greater virus leak rate than latex gloves at 1% of significance.

Step-by-step explanation:

Data given and notation  

$n_{V}=215$ sample of vinyl gloves selected

$n_{L}=215$ sample of latex gloves selected

$p_{V}=0.7$ represent the proportion of vynil gloves with leaked viruses

$p_{L}=0.07$ represent the proportion of latex gloves with leaked viruses

z would represent the statistic (variable of interest)  

$p_v$ represent the value for the test (variable of interest)

Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the vinyl gloves have a greater virus leak rate than latex gloves , the system of hypothesis would be:  

Null hypothesis:$p_{V} \leq p_{L}$  

Alternative hypothesis:$p_{V} > \mu_{L}$  

We need to apply a z test to compare proportions, and the statistic is given by:  

$z=\frac{p_{V}-p_{L}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{V}}+\frac{1}{n_{L}})}}$   (1)

Where $\hat p=\frac{X_{V}+X_{L}}{n_{V}+n_{L}}=\frac{0.7+0.07}{2}=0.385$

Calculate the statistic

Replacing in formula (1) the values obtained we got this:  

$z=\frac{0.7-0.07}{\sqrt{0.385(1-0.385)(\frac{1}{215}+\frac{1}{215})}}=13.42$  

Statistical decision

Since is a one right tailed test the p value would be:  

$p_v =P(Z>13.42)\approx 0$  

So the p value is a very low value and using any significance level given $\alpha=0.01$ we se that $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that vinyl gloves have a greater virus leak rate than latex gloves at 1% of significance.