Question

Published an article about a study in which researchers subjected laboratory gloves to stress. Among 216 vinyl​ gloves, 68​% leaked viruses. Among 216 latex​ gloves, ​6% leaked viruses. Using the accompanying display of the technology​ results, and using a .05 significance​ level, test the claim that vinyl gloves have a greater virus leak rate than latex gloves. Let vinyl gloves be population 1.

Answer 1

The decision rule is

Reject the null hypothesis

The conclusion is

There is no sufficient evidence to show that the claim that the vinyl gloves have a greater virus leak rate than latex gloves

Explanation:

From the question we are told that

The sample size of vinyl gloves
`n_1 = 216`

The sample size of latex gloves is
`n_2 = 216`

The sample proportion of vinyl gloves that leaked virus is
`\^ p_1 = 0.68`

The sample proportion of latex gloves that leaked virus is
`\^ p_2 = 0.06`

The significance level is
`\alpha =0.05`

The null hypothesis is
`H_o : p_1 = p_2`

The alternative hypothesis is
`H_a: p_1 > p_2`

Generally the number of vinyl gloves that leaked virus is

`x_1 = \^ p_1 * n_1`

=>
`x_1 = 0.68 * 216`

=>
`x_1 = 146.88`

Generally the number of latex gloves that leaked virus is

`x_2 = \^ p_2 * n_2`

=>
`x_2 = 0.06 * 216`

=>
`x_2 = 12.96`

Generally the pooled population proportion is mathematically represented as

`\^ p = (x_1 + x_2)/( n_1 + n_2)`

=>
`\^ p = (146.88 +12.96 )/( 216 + 216)`

=>
`\^ p = 0.37`

Generally the test statistics is mathematically represented as

`z = \frac{ \^ p_1 - \^ p_2}{ \sqrt{\^ p (1- \^ p ) [(1)/(n_1) + (1)/(n_2) ]} }`

=>
`z = \frac{ 0.68 - 0.06}{ \sqrt{0.37 (1- 0.37 ) [(1)/(216) + (1)/(216) ]} }`

=>
`z =13.35`

From the z table the area under the normal curve to the right corresponding to 13.35 is

`P( Z > 13.35) = 0.00`

Generally the p-value is mathematically represented as

`p-value = 2 * P(Z > 13.35)`

=>
`p-value = 2 * 0.00`

=>
`p-value = 0.00`

From the value obtained we see that
`p-value < \alpha` hence we have that

The decision rule is

Reject the null hypothesis

The conclusion is

There is no sufficient evidence to show that the claim that the vinyl gloves have a greater virus leak rate than latex gloves