A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of 70 women over the age of 50 used the new cream for 6 months. Of those 70 women, 35 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=0.01.

Answer:Yes, evidence shows that the cream will improve the skin of more than 40% of women over the age of 50 in 0.01 significance level. Step-by-step explanation:We need to make a hypothesis test.Let p be the proportion of women who used cream report skin improvement.[tex]H_{0}[/tex]: p=0.4[tex]H_{0}[/tex]: p<0.4To test the hypothesis, we need to calculate z-score of the sample mean and compare its probability with the significance level. z=[tex]\frac{p(s)-p}{\sqrt{\frac{p*(1-p)}{N} } }[/tex] where p(s) is the sample proportion of women reported improvement (0.5)p is the proportion assumed under null hypothesis. (0.4)N is the sample size (70)Putting the numbers:z=[tex]\frac{0.5-0.4}{\sqrt{\frac{0.4*0.6}{70} } }[/tex] ≈ 1.71 And P(z<1.71) ≈ 0.955. Since 0.955>0.01 we fail to reject the null hypothesis that the cream will improve the skin of more than 40% of women.