Question

A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of 42 women over the age of 50 used the new cream for 6 months. Of those 42 women, 30 of them reported skin improvement(as judged by a dermatologist). Is this evidence that the cream will improve the skin of more than 40% of women over the age of 50? Test using α=0.05.

(a) Test statistic: z=

(b) Critical Value: z∗=

(c) The final conclusion is

A. We can reject the null hypothesis that p=0.4 and accept that p>0.4. That is, the cream can improve the skin of more than 40% of women over 50.

B. There is not sufficient evidence to reject the null hypothesis that p=0.4. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than 40% of women over 50.

Answer 1

n = 42, x = 30

p̄ = x/n = 0.7143

α = 0.05

Null and Alternative hypothesis:

Ho : p ≤ 0.4 ; H1 : p > 0.4

a) Test statistic:

z = (p̄ -p)/√(p*(1-p)/n) = (0.7143 - 0.4)/√(0.4 * 0.6/42) = 4.1576

b) Critical value :

Right tailed critical value, z crit = ABS(NORM.S.INV(0.05)) = 1.645

Reject Ho if z > 1.645

c) A. We can reject the null hypothesis that p = 0.4 and accept that p > 0.4. That is, the cream can improve the skin of more than 40% of women over 50.