Question

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 1

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.

Explanation:

We need to make a hypothesis test.

Let p be the proportion of women who used cream report skin improvement.

`H_(0)`: p=0.4

`H_(0)`: p<0.4

To test the hypothesis, we need to calculate z-score of the sample mean and compare its probability with the significance level.

z=
`\frac{p(s)-p}{\sqrt{(p*(1-p))/(N) } }` 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=
`\frac{0.5-0.4}{\sqrt{(0.4*0.6)/(70) } }` ≈ 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.