Question

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

Answer 1

Answer:

$z=\frac{0.0943-0.00281}{\sqrt{0.0208(1-0.0208)(\frac{1}{700}+\frac{1}{2850})}}=15.197$  

$p_v =P(Z>15.197) \approx 0$  

If we compare the p value and using any significance level for example $\alpha=0.05$ always $p_v$ 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  

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

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

$n_{M}=700$ sample of male slected

$n_{W}=2850$ sample of female selected

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

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

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 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:$p_{M} \leq p_{W}$  

Alternative hypothesis:$p_{M} > p_{W}$  

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

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

Where $\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{66+8}{700+2850}=0.0208$

Calculate the statistic

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

$z=\frac{0.0943-0.00281}{\sqrt{0.0208(1-0.0208)(\frac{1}{700}+\frac{1}{2850})}}=15.197$  

Statistical decision

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

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

$p_v =P(Z>15.197) \approx 0$  

If we compare the p value and using any significance level for example $\alpha=0.05$ always $p_v$ 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.