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Sagot :
To understand what could cause the test statistic for a hypothesis test to test the difference between two population proportions to be 0, we should first review some key statistical concepts.
### Understanding the Test Statistic
In a hypothesis test for the difference between two population proportions, the test statistic typically measures how far the observed sample proportions are from the null hypothesis, often standardized by the standard error of the difference between the proportions.
The test statistic [tex]\( Z \)[/tex] in this context can be given by:
[tex]\[ Z = \frac{\hat{p}_1 - \hat{p}_2 - (\pi_1 - \pi_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \][/tex]
where:
- [tex]\( \hat{p}_1 \)[/tex] and [tex]\( \hat{p}_2 \)[/tex] are the sample proportions,
- [tex]\( \pi_1 \)[/tex] and [tex]\( \pi_2 \)[/tex] are the population proportions under the null hypothesis,
- [tex]\( \hat{p} \)[/tex] is the pooled sample proportion given by [tex]\( \frac{\hat{p}_1 n_1 + \hat{p}_2 n_2}{n_1 + n_2} \)[/tex],
- [tex]\( n_1 \)[/tex] and [tex]\( n_2 \)[/tex] are the sample sizes for the two groups.
### Condition for Test Statistic Value of 0
For the test statistic [tex]\( Z \)[/tex] to be 0, the numerator of the fraction must be 0:
[tex]\[ \hat{p}_1 - \hat{p}_2 - (\pi_1 - \pi_2) = 0 \][/tex]
Since the null hypothesis usually assumes [tex]\( \pi_1 = \pi_2 \)[/tex], this simplifies to:
[tex]\[ \hat{p}_1 - \hat{p}_2 = 0 \][/tex]
This indicates that the sample proportions [tex]\( \hat{p}_1 \)[/tex] and [tex]\( \hat{p}_2 \)[/tex] must be equal.
Now, let's analyze each of the given options in the context of this conclusion:
1. The sample proportions are the same.
- This option directly states that the sample proportions [tex]\( \hat{p}_1 \)[/tex] and [tex]\( \hat{p}_2 \)[/tex] are equal. This would indeed cause the numerator of the test statistic to be zero, resulting in a test statistic of 0.
2. The standard deviation for distribution of the difference of sample proportions is 0.
- This option would imply that there is no variability at all in the difference between sample proportions, which is highly unlikely and unrealistic in a practical context, and does not directly cause the test statistic to be zero.
3. The number of successes in both of the samples is the same.
- While having the same number of successes in both samples could potentially lead to the same sample proportions, it is not sufficient to guarantee it without considering the different sample sizes. Thus, this is not a direct cause.
4. The number of successes in at least one of the samples is 0.
- If the number of successes in one sample is 0, it could vary based on the sample size leading to a different sample proportion which would not directly explain a 0 test statistic.
### Conclusion
The most direct and correct explanation is that the sample proportions are the same. Thus, the correct answer is:
The sample proportions are the same.
### Understanding the Test Statistic
In a hypothesis test for the difference between two population proportions, the test statistic typically measures how far the observed sample proportions are from the null hypothesis, often standardized by the standard error of the difference between the proportions.
The test statistic [tex]\( Z \)[/tex] in this context can be given by:
[tex]\[ Z = \frac{\hat{p}_1 - \hat{p}_2 - (\pi_1 - \pi_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \][/tex]
where:
- [tex]\( \hat{p}_1 \)[/tex] and [tex]\( \hat{p}_2 \)[/tex] are the sample proportions,
- [tex]\( \pi_1 \)[/tex] and [tex]\( \pi_2 \)[/tex] are the population proportions under the null hypothesis,
- [tex]\( \hat{p} \)[/tex] is the pooled sample proportion given by [tex]\( \frac{\hat{p}_1 n_1 + \hat{p}_2 n_2}{n_1 + n_2} \)[/tex],
- [tex]\( n_1 \)[/tex] and [tex]\( n_2 \)[/tex] are the sample sizes for the two groups.
### Condition for Test Statistic Value of 0
For the test statistic [tex]\( Z \)[/tex] to be 0, the numerator of the fraction must be 0:
[tex]\[ \hat{p}_1 - \hat{p}_2 - (\pi_1 - \pi_2) = 0 \][/tex]
Since the null hypothesis usually assumes [tex]\( \pi_1 = \pi_2 \)[/tex], this simplifies to:
[tex]\[ \hat{p}_1 - \hat{p}_2 = 0 \][/tex]
This indicates that the sample proportions [tex]\( \hat{p}_1 \)[/tex] and [tex]\( \hat{p}_2 \)[/tex] must be equal.
Now, let's analyze each of the given options in the context of this conclusion:
1. The sample proportions are the same.
- This option directly states that the sample proportions [tex]\( \hat{p}_1 \)[/tex] and [tex]\( \hat{p}_2 \)[/tex] are equal. This would indeed cause the numerator of the test statistic to be zero, resulting in a test statistic of 0.
2. The standard deviation for distribution of the difference of sample proportions is 0.
- This option would imply that there is no variability at all in the difference between sample proportions, which is highly unlikely and unrealistic in a practical context, and does not directly cause the test statistic to be zero.
3. The number of successes in both of the samples is the same.
- While having the same number of successes in both samples could potentially lead to the same sample proportions, it is not sufficient to guarantee it without considering the different sample sizes. Thus, this is not a direct cause.
4. The number of successes in at least one of the samples is 0.
- If the number of successes in one sample is 0, it could vary based on the sample size leading to a different sample proportion which would not directly explain a 0 test statistic.
### Conclusion
The most direct and correct explanation is that the sample proportions are the same. Thus, the correct answer is:
The sample proportions are the same.
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