Question

A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of 54 women over the age of 50 used the new cream for 6 months. Of those 54 women, 42 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. 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.

B. 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.

Answer 1

n = 54, x = 42

p̄ = x/n = 0.7778

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) = (0.7778 - 0.4)/√(0.4 * 0.6/54) = 5.6670

b) Critical value :

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

c) Decision:

z = 5.667 >1.645, Reject the null hypothesis

Conclusion:

B. 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.