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, 36 of them reported skin improvement(as judged by a dermatologist). Is this evidence that the cream will improve the skin of more than 50% 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=0.5. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than 50% of women over 50. B. We can reject the null hypothesis that p=0.5 and accept that p>0.5. That is, the cream can improve the skin of more than 50% of women over 50.
n = 42, x = 36
p̄ = x/n = 0.8571
α = 0.05
Null and Alternative hypothesis:
Ho : p = 0.5
H1 : p > 0.5
a) Test statistic:
z = (p̄ -p)/√(p*(1-p)/n) = (0.8571 - 0.5)/√(0.5 * 0.5/42) = 4.6285
b) Critical value :
Right tailed critical value, z crit = ABS(NORM.S.INV(0.05)) = 1.645
c) Decision:
z = 4.6285 > 1.645, Reject the null hypothesis
Conclusion:
B. We can reject the null hypothesis that p=0.5 and accept that p>0.5. That is, the cream can improve the skin of more than 50% of women over 50.
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