In a study of red/green color blindness, 700 men and 2850 women are randomly selected and tested. Among the men, 66 have red/green color blindness. Among the women, 8 have red/green color blindness. Test the claim that men have a higher rate of red/green color blindness. (Note: Type ‘‘p_m″ for the symbol pm , for example, p_mnot=p_w for the proportions are not equal, p_m>p_w for the proportion of men with color blindness is larger, p_m

If we compare the p value and using any significance level for example [tex]\alpha=0.05[/tex] always [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion of men with red/green color blindness.is significanlty higher than the proportion of women with red/green color blindness.

Step-by-step explanation:

Data given and notation

[tex]X_{M}=66[/tex] represent the number of men with red/green color blindness.

[tex]X_{W}=8[/tex] represent the number of women with red/green color blindness.

[tex]n_{M}=700[/tex] sample of male slected

[tex]n_{W}=2850[/tex] sample of female selected

[tex]p_{M}=\frac{66}{700}=0.0943[/tex] represent the proportion of male with red/green color blindness.

[tex]p_{W}=\frac{8}{2850}=0.00281[/tex] represent the proportion of female with red/green color blindness.

z would represent the statistic (variable of interest)

[tex]p_v[/tex] 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 proportion of men with red/green color blindness. is higher than the proportionof women with red/green color blindness. , the system of hypothesis would be:

Null hypothesis:[tex]p_{M} \leq p_{W}[/tex]

Alternative hypothesis:[tex]p_{M} > p_{W}[/tex]

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

[tex]z=\frac{p_{M}-p_{W}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}[/tex] (1)

Where [tex]\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{66+8}{700+2850}=0.0208[/tex]

Calculate the statistic

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

[tex]z=\frac{0.0943-0.00281}{\sqrt{0.0208(1-0.0208)(\frac{1}{700}+\frac{1}{2850})}}=15.197[/tex]

Statistical decision

For this case we don't have a significance level provided [tex]\alpha[/tex], but we can calculate the p value for this test.

Since is a one side test the p value would be:

[tex]p_v =P(Z>15.197) \approx 0[/tex]

If we compare the p value and using any significance level for example [tex]\alpha=0.05[/tex] always [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion of men with red/green color blindness.is significanlty higher than the proportion of women with red/green color blindness.