https://file.aiquickdraw.com/mathbot/file/1727583491345cuifz12j.jpg

construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

To construct the confidence interval for the hypothesis test described in part (a), we need to follow these steps:

### 1. Mean Difference Calculation
First, calculate the mean of the differences \( d \), where \( d = \text{President's height} - \text{Opponent's height} \) for each pair. From the given dataTo construct the confidence interval for the hypothesis test described in part (a), we need to follow these steps:

### 1. Mean Difference Calculation
First, calculate the mean of the differences \( d \), where \( d = \text{President's height} - \text{Opponent's height} \) for each pair. From the given 

| Height (cm) of President | 183 | 179 | 168 | 189 | 189 | 170 |
|--------------------------|-----|-----|-----|-----|-----|-----|
| Height (cm) of Opponent   | 177 | 171 | 173 | 174 | 195 | 183 |
| Difference \( d \)        | 6   | 8   | -5  | 15  | -6  | -13 |

Now, we compute the mean of these differences:
\[
\bar{d} = \frac{6 + 8 - 5 + 15 - 6 - 13}{6}
\]

### 2. Standard Deviation of Differences
Calculate the standard deviation of these differences \( s_d \).

### 3. Confidence Interval Formula
The formula for the confidence interval for paired differences is:
\[
\bar{d} \pm t_{\alpha/2} \left(\frac{s_d}{\sqrt{n}}\right)
\]
where:
- \( \bar{d} \) is the mean of the differences.
- \( t_{\alpha/2} \) is the critical value from the t-distribution for the chosen confidence level (typically 95%).
- \( s_d \) is the standard deviation of the differences.
- \( n \) is the sample size (in this case, 6 pairs).

### 4. Conclusion Based on Confidence Interval
If the confidence interval contains 0, it indicates that there is no statistically significant difference between the heights of the presidents and their opponents. This would align with the conclusion from part (a), where the null hypothesis \( H_0: \mu_d = 0 \) was not rejected, and there was insufficient evidence to support the claim that presidents are taller than their opponents.

Let me calculate the values for you.The calculations for the confidence interval are as follows:

- The mean difference \( \bar{d} \) is approximately 0.83 cm.
- The standard deviation of the differences \( s_d \) is approximately 10.50 cm.
- The 95% confidence interval for the mean difference is approximately \( (-10.18 \, \text{cm}, 11.85 \, \text{cm}) \).

### Conclusion:
Since the confidence interval contains 0, this suggests there is no statistically significant difference between the heights of the presidents and their opponents. This aligns with the conclusion from part (a), where the null hypothesis \( H_0: \mu_d = 0 \) was not rejected, indicating insufficient evidence to support the claim that presidents tend to be taller than their opponents.

Do you have any further questions or need more details?

---

Here are five related questions:
1. How is the t-critical value determined for a given confidence level?
2. What would change in the confidence interval if we increased the sample size?
3. How does the standard deviation affect the width of the confidence interval?
4. Why is it important that the confidence interval contains 0 in hypothesis testing?
5. What happens if we use a different confidence level, such as 90%?

**Tip**: The margin of error decreases as the sample size increases, leading to a narrower confidence interval.

Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

This problem demonstrates how to construct a confidence interval for paired sample data and use it in hypothesis testing. Using data from president and opponent heights, this solution covers key statistical concepts such as the t-distribution, mean difference, and standard deviation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation