Question

In a study of red/green color blindness, 950 men and 2800 women are randomly selected and tested. Among the men, 86 have red/green color blindness. Among the women, 6 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<p_w , for the proportion of men is smaller. )

(a) State the null hypothesis:

(b) State the alternative hypothesis:

Note that computation of the test statistic will produce z = 15.22

(c) Is there sufficient evidence to support the claim that men have a higher rate of red/green color blindness than women? Use a 10 % significance level.

A. Yes
B. No

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P_Men = P_Women
Alternative hypothesis: P_Men > P_Women

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.10. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)

p = 0.0245

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.0058045
z = (p₁ - p₂) / SE

z = 15.22

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

Since we have a one-tailed test, the P-value is the probability that the z-score is greater than 15.22

Thus, the P-value = less than 0.0001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.10), we have to reject the null hypothesis.

(c) Yes, there is sufficient evidence to support the claim that men have a higher rate of red/green color blindness than women? Use a 10 % significance level.