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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.

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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<0.4

To 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.

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