Suppose you are trying to infer about what proportion of the population will vote for the a candidate in the next election. You sample 300 people and found that 75 are willing to vote for the candidate

Find a 95% confidence interval for the actual support for the candidate.

Test whether the level of support is 1/3 or not



### 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?

---

**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.

Learn how to calculate a 95% confidence interval for a proportion of support based on a sample size of 300 people with 75 supporters. Also, test whether the level of support differs significantly from 1/3 using hypothesis testing. Get step-by-step explanations and formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation