Question

A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of 67 women over the age of 50 used the new cream for 6 months. Of those 67 women, 40 of them reported skin improvement(as judged by a dermatologist). Is this evidence that the cream will improve the skin of more than 60% of women over the age of 50? Test using ?=0.01

(a) Test statistic: ?=

(b) Critical Value: ?∗=

(c) The final conclusion is

A. We can reject the null hypothesis that ?=0.6p=0.6 and accept that ?>0.6. That is, the cream can improve the skin of more than 60% of women over 50.
B. There is not sufficient evidence to reject the null hypothesis that ?=0.6. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than 60% of women over 50.

Answer 1

a. Here claim is that p>0.60

So hypothesis is vs

So test statistics is

b. The z-critical value for a right-tailed test, for a significance level of α=0.01 is

zc​=2.33

Graphically

c. As test statistics do not fall in the rejection region, we fail to reject the null hypothesis

B. There is not sufficient evidence to reject the null hypothesis that ?=0.6. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than 60% of women over 50.