1. Confidence Interval Calculation

To calculate the 95% confidence interval for the proportion of the population that will vote for the candidate, we use the following formula:

$\text{Confidence Interval} = \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Where:

• $\hat{p}$ is the sample proportion.
• $Z$ is the Z-score corresponding to the desired confidence level (for 95%, $Z = 1.96$).
• $n$ is the sample size.

Step-by-Step Calculation

1. Sample Proportion ($\hat{p}$): $\hat{p} = \frac{75}{300} = 0.25$

2. Standard Error (SE): $\text{SE} = \sqrt{\frac{0.25 \times (1 - 0.25)}{300}} = \sqrt{\frac{0.25 \times 0.75}{300}} \approx 0.025$

3. Margin of Error (ME): $\text{ME} = 1.96 \times 0.025 \approx 0.049$

4. Confidence Interval: $\text{CI} = 0.25 \pm 0.049$ So the 95% confidence interval is: $[0.201, 0.299]$

2. Hypothesis Test

Next, we perform a hypothesis test to determine whether the level of support is equal to $\frac{1}{3}$ (which is approximately 0.3333).

Hypotheses:

• Null Hypothesis ($H_0$): $p = \frac{1}{3}$
• Alternative Hypothesis ($H_a$): $p \neq \frac{1}{3}$ (Two-tailed test)

Test Statistic:

The test statistic for the proportion is given by:

$Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$

Where:

• $p_0 = \frac{1}{3} \approx 0.3333$
• $\hat{p} = 0.25$

1. Standard Error under $H_0$: $\text{SE}_{H_0} = \sqrt{\frac{0.3333 \times (1 - 0.3333)}{300}} \approx 0.0272$

2. Z-Score Calculation: $Z = \frac{0.25 - 0.3333}{0.0272} \approx -3.06$

Decision Rule:

• Compare the Z-score with the critical value for a 95% confidence level in a two-tailed test ($Z_{\text{critical}} = \pm 1.96$).
• Since $-3.06$ is less than $-1.96$, we reject the null hypothesis.

Conclusion:

• Confidence Interval: The 95% confidence interval for the actual support for the candidate is approximately $[0.201, 0.299]$.
• Hypothesis Test: The test suggests that the level of support is significantly different from $\frac{1}{3}$.

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

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Related Questions:

1. How would the confidence interval change with a larger sample size?
2. What is the interpretation of the confidence interval in the context of this problem?
3. How would the test change if the confidence level were 90% instead of 95%?
4. What would the outcome be if 150 out of 300 people supported the candidate?
5. How does the choice of significance level affect the hypothesis test?

Tip: Remember that the confidence interval provides a range of plausible values for the population parameter, not a definitive value.