A new cream that advertises that it can reduce wrinkles and improve skin was sub

Question

A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of 58 women over the age of 50 used the new cream for 6 months. Of those 58 women, 40 of them reported skin improvements 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 alpha = 0.05. test statistics z = rejection region z > The final conclusion is A. 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. B. 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.

Explanation / Answer

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   <=   0.5
Ha:   p   >   0.5
As we see, the hypothesized po =   0.5      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.689655172      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.065653216      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    2.888741523   [ANSWER, TEST STATISTIC]

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As this is a    1   tailed test, then, at 0.05 level, the critical value is
          
zcrit = 1.645

Hence, reject Ho when z > 1.645. [ANSWER]

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As z = 2.8887 > 1.645, the final conclusion is OPTION A. [ANSWER, A]

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