Let's go through the hypothesis test step by step:
1. Define the hypotheses:
- Null hypothesis [tex]\(H_0\)[/tex]: [tex]\(p = 0.6\)[/tex], where [tex]\(p\)[/tex] is the true proportion of women who will see skin improvement.
- Alternative hypothesis [tex]\(H_a\)[/tex]: [tex]\(p > 0.6\)[/tex] (we are testing if the proportion of improvement is greater than 60%).
2. Given data:
- Sample size ([tex]\(n\)[/tex]): 42
- Number of successes ([tex]\(x\)[/tex]): 36
- Sample proportion ([tex]\(\hat{p}\)[/tex]): [tex]\(\hat{p} = \frac{x}{n} = \frac{36}{42} = 0.8571\)[/tex]
- Null hypothesis proportion ([tex]\(p_0\)[/tex]): 0.6
- Significance level ([tex]\(\alpha\)[/tex]): 0.05
3. Calculate the standard error:
- Standard error ([tex]\(SE\)[/tex]): [tex]\[ SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} \][/tex]
[tex]\[ SE = \sqrt{\frac{0.6 \times (1 - 0.6)}{42}} \approx 0.0756 \][/tex]
4. Calculate the z-test statistic:
- [tex]\(z\)[/tex]-statistic: [tex]\[ z = \frac{\hat{p} - p_0}{SE} \][/tex]
[tex]\[ z = \frac{0.8571 - 0.6}{0.0756} \approx 3.4017 \][/tex]
5. Determine the critical value for a one-tailed test at [tex]\(\alpha = 0.05\)[/tex]:
- The critical value ([tex]\(z^\)[/tex]) for a one-tailed test at [tex]\(\alpha = 0.05\)[/tex] is approximately 1.645.
6. Make the decision:
- If the [tex]\(z\)[/tex]-statistic is greater than the critical value [tex]\(z^\)[/tex], we reject the null hypothesis. In this case:
[tex]\[ z = 3.4017 > 1.645 \][/tex]
Because the z-test statistic (3.4017) is greater than the critical value (1.645), we reject the null hypothesis.
Conclusions:
(a) Test statistic: [tex]\( z = 3.4017 \)[/tex]
(b) Critical Value: [tex]\( z^* = 1.645 \)[/tex]
(c) The final conclusion is:
A. We can reject the null hypothesis that [tex]\( p = 0.6 \)[/tex] and accept that [tex]\( p > 0.6 \)[/tex]. That is, the cream can improve the skin of more than 60% of women over 50.
