After 2 months of the previous polling, you sample 2100 people and found that 630 of them say they support the Candidate

Find a 95% confidence interval for the increase in support in two months.

Test whether the null hypothesis of the support between the two time positions is the same vs. the alternative hypothesis of the support changing in two months

### 1. Confidence Interval for the Increase in Support

To calculate the 95% confidence interval for the increase in support, we'll first determine the confidence intervals for the proportions at each time point, and then compute the interval for the difference between these proportions.

#### **First Polling Data:**
- Sample size \(n_1 = 300\)
- Supporters \(X_1 = 75\)
- Proportion \( \hat{p}_1 = \frac{75}{300} = 0.25 \)

#### **Second Polling Data:**
- Sample size \(n_2 = 2100\)
- Supporters \(X_2 = 630\)
- Proportion \( \hat{p}_2 = \frac{630}{2100} = 0.30 \)

#### **Difference in Proportions:**
- Difference in proportions \( \hat{p}_2 - \hat{p}_1 = 0.30 - 0.25 = 0.05 \)

#### **Standard Error of the Difference:**
The standard error for the difference between two independent proportions is:

\[
\text{SE}_{\text{diff}} = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}
\]

Substituting the values:

\[
\text{SE}_{\text{diff}} = \sqrt{\frac{0.25 \times 0.75}{300} + \frac{0.30 \times 0.70}{2100}} \approx \sqrt{0.000625 + 0.0001} \approx \sqrt{0.000725} \approx 0.0269
\]

#### **Margin of Error for the Difference:**
\[
\text{ME}_{\text{diff}} = 1.96 \times 0.0269 \approx 0.053
\]

#### **Confidence Interval for the Difference:**
\[
\text{CI}_{\text{diff}} = 0.05 \pm 0.053
\]
So the 95% confidence interval for the increase in support is:
\[
[-0.003, 0.103]
\]

### 2. Hypothesis Test for Change in Support

#### **Hypotheses:**
- Null Hypothesis (\(H_0\)): The support has not changed, i.e., \( p_1 = p_2 \).
- Alternative Hypothesis (\(H_a\)): The support has changed, i.e., \( p_1 \neq p_2 \) (Two-tailed test).

#### **Test Statistic:**
The test statistic for comparing two proportions is given by:

\[
Z = \frac{\hat{p}_2 - \hat{p}_1}{\text{SE}_{\text{diff}}}
\]

Substituting the values:

\[
Z = \frac{0.30 - 0.25}{0.0269} \approx \frac{0.05}{0.0269} \approx 1.86
\]

#### **Decision Rule:**
- For a 95% confidence level, the critical value is \( Z_{\text{critical}} = \pm 1.96 \).
- Since \( Z = 1.86 \) is less than 1.96, we do not reject the null hypothesis at the 95% confidence level.

### Conclusion:
- **Confidence Interval**: The 95% confidence interval for the increase in support is approximately \([-0.003, 0.103]\), indicating that the increase in support may not be statistically significant.
- **Hypothesis Test**: The test does not provide sufficient evidence to reject the null hypothesis that the level of support remained the same over the two months.

Would you like more details on any step or have any questions?

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**Related Questions:**
1. How would the hypothesis test result change if we used a 90% confidence level?
2. What is the interpretation of a confidence interval that includes zero in this context?
3. How might the results differ if the sample size for the second poll was smaller?
4. Can we make any assumptions about the population based on these results?
5. What would be the implications if the second poll showed a higher support increase?

**Tip:** When comparing two proportions, the confidence interval for the difference is crucial for understanding whether a change has occurred. A confidence interval that includes zero suggests that the observed difference may not be statistically significant.

Learn how to calculate a 95% confidence interval for the increase in support and perform a hypothesis test to determine if the support has changed over two months. This problem involves statistical concepts such as confidence intervals, hypothesis testing, and proportions. Suitable for advanced undergraduate students and professionals.

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