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Sagot :
### 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.
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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