Answer:
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
Step-by-step explanation:
From the question we are told that
The sample size of vinyl gloves [tex]n_1 = 216[/tex]
The sample size of latex gloves is [tex]n_2 = 216[/tex]
The sample proportion of vinyl gloves that leaked virus is [tex]\^ p_1 = 0.68[/tex]
The sample proportion of latex gloves that leaked virus is [tex]\^ p_2 = 0.06[/tex]
The significance level is [tex]\alpha =0.05[/tex]
The null hypothesis is [tex]H_o : p_1 = p_2[/tex]
The alternative hypothesis is [tex]H_a: p_1 > p_2[/tex]
Generally the number of vinyl gloves that leaked virus is
[tex]x_1 = \^ p_1 * n_1[/tex]
=> [tex]x_1 = 0.68 * 216[/tex]
=> [tex]x_1 = 146.88[/tex]
Generally the number of latex gloves that leaked virus is
[tex]x_2 = \^ p_2 * n_2[/tex]
=> [tex]x_2 = 0.06 * 216[/tex]
=> [tex]x_2 = 12.96[/tex]
Generally the pooled population proportion is mathematically represented as
[tex]\^ p = \frac{x_1 + x_2}{ n_1 + n_2}[/tex]
=> [tex]\^ p = \frac{146.88 +12.96 }{ 216 + 216}[/tex]
=> [tex]\^ p = 0.37[/tex]
Generally the test statistics is mathematically represented as
[tex]z = \frac{ \^ p_1 - \^ p_2}{ \sqrt{\^ p (1- \^ p ) [\frac{1}{n_1} + \frac{1}{n_2} ]} }[/tex]
=> [tex]z = \frac{ 0.68 - 0.06}{ \sqrt{0.37 (1- 0.37 ) [\frac{1}{216} + \frac{1}{216} ]} }[/tex]
=> [tex]z =13.35[/tex]
From the z table the area under the normal curve to the right corresponding to 13.35 is
[tex]P( Z > 13.35) = 0.00[/tex]
Generally the p-value is mathematically represented as
[tex]p-value = 2 * P(Z > 13.35)[/tex]
=> [tex]p-value = 2 * 0.00[/tex]
=> [tex]p-value = 0.00[/tex]
From the value obtained we see that [tex]p-value < \alpha[/tex] 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
