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In a survey of 100 randomly selected people in City A, 73 support increased government spending on education. In a survey of 100 randomly selected people in City B, 81 support such spending. Test the alternative hypothesis that the population proportion of people in City A that support such spending is different from the population proportion of people in City B. Use the level of significance [tex]\alpha=0.05[/tex]. The test statistic is [tex]z \approx -1.34[/tex], and the [tex]p[/tex]-value is approximately 0.180. Identify all of the appropriate conclusions to the hypothesis test below.

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 people in City A that support such spending is different from the population proportion of people in City B.
D. The conclusion of the hypothesis test is that there is insufficient evidence to support the claim that the population proportion of people in City A that support such spending is different from the population proportion of people in City B.



Answer :

GinnyAnswer
To test the hypothesis that the population proportion of people in City A who support increased government spending on education is different from the population proportion in City B, follow these steps:

### Step-by-Step Solution:

1. State the Null and Alternative Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population proportion of people in City A that support increased government spending on education is equal to the population proportion in City B.
[tex]\[ H_0: p_A = p_B \][/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The population proportion of people in City A that support increased government spending is different from the population proportion in City B.
[tex]\[ H_1: p_A \neq p_B \][/tex]

2. Given Values:
- Significance Level ([tex]\(\alpha\)[/tex]): 0.05
- Test Statistic (z): [tex]\(-1.34\)[/tex]
- p-value: 0.180

3. Decision Rule:
- To determine whether to reject or fail to reject the null hypothesis, compare the p-value to the significance level ([tex]\(\alpha = 0.05\)[/tex]).
- If the p-value is less than [tex]\(\alpha\)[/tex], reject the null hypothesis.
- If the p-value is greater than or equal to [tex]\(\alpha\)[/tex], fail to reject the null hypothesis.

4. Comparison with [tex]\(\alpha\)[/tex]:
- p-value (0.180) is greater than [tex]\(\alpha (0.05)\)[/tex].

5. Conclusion Based on p-value:
- Since the p-value is greater than the significance level ([tex]\(\alpha = 0.05\)[/tex]), we fail to reject the null hypothesis.

6. Interpret the Result:
- Failing to reject the null hypothesis means there is not enough evidence to support the claim that the population proportion of people in City A who support increased government spending on education is different from the population proportion in City B.

### 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 people in City A that support such spending is different from the population proportion of people in City B.

### Final Answer:
- 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 people in City A that support such spending is different from the population proportion of people in City B.

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