Question

Q1: A new cream that advertises that it can reduce wrinkles and improve skin was subject to a recent study. A sample of 56 women over the age of 50 used the new cream for 6 months. Of those 56 women, 30 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.

(a) Test statistic: z=

(b) Critical Value: z∗=

(c) The final conclusion is

A. We can reject the null hypothesis that p=0.4 and accept that p>0.4. That is, the cream can improve the skin of more than 40% of women over 50.
B. There is not sufficient evidence to reject the null hypothesis that p=0.4. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than 40% of women over 50.

Q2: A newspaper conducted a statewide survey concerning the 1998 race for state senator. The newspaper took a SRS of n=1300 registered voters and found that 670 would vote for the Republican candidate. Let pp represent the proportion of registered voters in the state who would vote for the Republican candidate.
We test

H0:p=.50
Ha:p>.50

(a) What is the z-statistic for this test?  

(b) What is the P-value of the test?  

Answer 1

(Q1)

(a)

H0: Null Hypothesis: p 0.40 ( The cream will not improve the skin of more than 40% of women over the age of 50)

HA: Alternative Hypothesis: p > 0.40 ( The cream will improve the skin of more than 40% of women over the age of 50) (Claim)

n = Sample Size = 56

p = Population Proportion = 0.40

q = 1 - p = 0.60

SE =

= Sample Proportion = 30/56 = 0.5357

Test Statistic is given by:
Z = (0.5357 - 0.40)/0.0655

= 2.0718

(b)

= 0.01

One Side - Right Side Test

From Table, critical value of Z = 2.33

(c)

Since calculated value of Z = 2.0718 is less than critical value of Z = 2.33, the difference is not significant. Fail to reject null hypothesis.

Conclusion:
The data do not support the claim that the cream will improve the skin of more than 40% of women over the age of 50.

So,

Correct option:

B. There is not sufficient evidence to reject the null hypothesis that p=0.4. That is, there is not sufficient evidence to reject that the cream can improve the skin of more than 40% of women over 50.

(Q2)

(a)

n = Sample Size = 1300

p = Population Proportion = 0.50

q = 1 - p = 0.50

SE =

= Sample Proportion = 670/1300 = 0.5154

Test Statistic is given by:
Z = (0.5154 - 0.5)/0.0139

= 1.1068

(b)

By Technology, the Cumulative Area Under the Standard Normal Curve = 0.8658

So,

P - Value = 1 - 0.8658 = 0.1342

So,

P - value = 0.1342