Question

In a study of red/green color blindness, 600 men and 2000 women are randomly selected and tested. Among the men, 52 have red/green color blindness. Among the women, 5 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_m not=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:

(c) The test statistic is

(d) Is there sufficient evidence to support the claim that men have a higher rate of red/green color blindness than women? Use a 1 % significance level.

A. Yes
B. No

(e) Construct the 99% confidence interval for the difference between the color blindness rates of men and women.

_______ <(pm−pw)<_______

Answer 1

Part a)

To Test :-

H0 :- Pm = Pw

Part b)
H1 :- Pm > Pw

Part c)

p̂1 = 52 / 600 = 0.0867
p̂2 = 5 / 2000 = 0.0025

Test Statistic :-
Z = ( p̂1 - p̂2 ) / √( p̂ * q̂ * (1/n1 + 1/n2) ))
p̂ is the pooled estimate of the proportion P
p̂ = ( x1 + x2) / ( n1 + n2)
p̂ = ( 52 + 5 ) / ( 600 + 2000 )
p̂ = 0.0219
q̂ = 1 - p̂ = 0.9781
Z = ( 0.0867 - 0.0025) / √( 0.0219 * 0.9781 * (1/600 + 1/2000) )
Z = 12.3483

Part d)

Test Criteria :-
Reject null hypothesis if Z > Z(α)
Z(α) = Z(0.01) = 2.326
Z > Z(α) = 12.3483 > 2.326, hence we reject the null hypothesis
Conclusion :- We Reject H0

There is sufficient evidence to support the claim that men have a higher rate of red/green color blindness than women.