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To test a drug intended to increase memory, 80 subjects were randomly given a pill that either contained the drug or was a placebo. After 10 minutes, the subjects were asked to look at a poster with pictures of 20 common objects for 5 minutes. After waiting 10 minutes, the subjects were asked to list as many of the objects as they could recall. For the 44 subjects who took the drug, [tex]$75\%$[/tex] were able to list at least half of the objects. For the 36 subjects who took the placebo, [tex]$50\%$[/tex] were able to list at least half of the objects.

Test the alternative hypothesis that the population proportion of subjects who took the drug and were able to list at least half of the objects is different from the population proportion of subjects who took the placebo and were able to do so. The critical values for the level of significance [tex]\alpha = 0.01[/tex] are [tex]z_{0.005} = -2.576[/tex] and [tex]z_{0.995} = 2.576[/tex]. The test statistic is [tex]z = 2.31[/tex].

Identify all of the appropriate conclusions to the hypothesis test.

Select all that apply:
A. Reject the null hypothesis.
B. Fail to reject the null hypothesis.
C. The conclusion of the hypothesis test is that there is sufficient evidence to support the claim that the population proportion of subjects who were able to list at least half of the objects is different for the groups.
D. The conclusion of the hypothesis test is that there is insufficient evidence to support the claim that the population proportion of subjects who were able to list at least half of the objects is different for the groups.



Answer :

GinnyAnswer
### Step-by-Step Solution:

1. Formulate the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population proportion of subjects who took the drug and were able to list at least half of the objects ([tex]\(p_1\)[/tex]) is equal to the population proportion of subjects who took the placebo and were able to list at least half of the objects ([tex]\(p_2\)[/tex]).
[tex]\[ H_0: p_1 = p_2 \][/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The population proportion of subjects who took the drug and were able to list at least half of the objects ([tex]\(p_1\)[/tex]) is different from the population proportion of subjects who took the placebo and were able to list at least half of the objects ([tex]\(p_2\)[/tex]).
[tex]\[ H_a: p_1 \neq p_2 \][/tex]

2. Determine the Significance Level ([tex]\(\alpha\)[/tex]):
- The significance level is given as [tex]\(\alpha = 0.01\)[/tex].

3. State the Critical Values:
- The critical values for [tex]\(\alpha = 0.01\)[/tex] are [tex]\(z_{0.005} = -2.576\)[/tex] and [tex]\(z_{0.995} = 2.576\)[/tex].

4. Calculate the Test Statistic:
- The test statistic is provided as [tex]\(z = 2.31\)[/tex].

5. Decision Rule:
- If the test statistic [tex]\(z\)[/tex] is less than [tex]\(z_{0.005} = -2.576\)[/tex] or greater than [tex]\(z_{0.995} = 2.576\)[/tex], reject the null hypothesis.
- Otherwise, fail to reject the null hypothesis.

6. Compare the Test Statistic to the Critical Values:
- The test statistic [tex]\(z = 2.31\)[/tex] is between the critical values [tex]\(-2.576\)[/tex] and [tex]\(2.576\)[/tex].

7. Conclusion:
- Since [tex]\(z = 2.31\)[/tex] does not exceed the critical values [tex]\(-2.576\)[/tex] and [tex]\(2.576\)[/tex], we fail to reject the null hypothesis.

8. Interpretation:
- The conclusion of the hypothesis test is that there is insufficient evidence to support the claim that the population proportion of subjects who were able to list at least half of the objects is different between the groups (those who took the drug and those who took the placebo).

### Appropriate Conclusions:
- Fail to reject the null hypothesis.
- The conclusion of the hypothesis test is that there is insufficient evidence to support the claim that the population proportion of subjects who were able to list at least half of the objects is different for the groups.

Thus, the appropriate conclusions to the hypothesis test are:
- Fail to reject the null hypothesis.
- The conclusion of the hypothesis test is that there is insufficient evidence to support the claim that the population proportion of subjects who were able to list at least half of the objects is different for the groups.

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