A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights ( in centimeters ) of randomly selected presidents along with the heights of their main opponents. Complete parts (a) and (b) with the data here. The height (cm) of president is : 183, 179, 168, 189, 189, 170. the height ( cm) of main opponent is : 177,171,173,174,195,183. Part A asks : use the sample data with a 0.05 significance level to test the claim that for the population of heights for presidents and their main opponents, the differences have a mean greater than 0 cm. In this example, ųd is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the president’s height minus their main opponent’s height. what are the null and alternative hypotheses for the hypothesis test?

To solve this hypothesis test problem, we'll first identify the null and alternative hypotheses based on the claim provided. The claim is that the differences in heights between presidents and their main opponents have a mean greater than 0 cm.

Let \( \mu_d \) represent the population mean of the differences in heights (president’s height minus opponent’s height).

### Step 1: Define the Hypotheses
- **Null Hypothesis \( H_0 \)**: The null hypothesis will reflect the assumption that there is no advantage in terms of height difference. In this case, the mean difference \( \mu_d \) is 0.
  \[
  H_0: \mu_d = 0
  \]

- **Alternative Hypothesis \( H_1 \)**: The alternative hypothesis will reflect the claim that the mean difference in heights is greater than 0.
  \[
  H_1: \mu_d > 0
  \]

This is a **right-tailed** test, since the claim is that the mean difference is greater than zero.

### Summary of the hypotheses:
- **Null Hypothesis \( H_0 \):** The mean difference in heights is 0 (i.e., \( \mu_d = 0 \)).
- **Alternative Hypothesis \( H_1 \):** The mean difference in heights is greater than 0 (i.e., \( \mu_d > 0 \)).

Would you like to proceed with calculating the test statistic and p-value for the next part? Let me know if you need further clarification! 

Here are five related questions:
1. How do you calculate the test statistic in a hypothesis test involving paired differences?
2. What is the critical value for a one-tailed test at a 0.05 significance level?
3. How can you interpret the p-value in hypothesis testing?
4. How is the standard deviation of differences calculated for paired data?
5. How do we check if the normality assumption holds for small sample sizes in paired t-tests?

**Tip:** In a paired t-test, it's important to keep track of the differences between paired data points, as it simplifies the analysis.

A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (in centimeters) of randomly selected presidents along with the heights of their main opponents. The height (cm) of president is: 183, 179, 168, 189, 189, 170. The height (cm) of main opponent is: 177, 171, 173, 174, 195, 183. Part A asks: use the sample data with a 0.05 significance level to test the claim that for the population of heights for presidents and their main opponents, the differences have a mean greater than 0 cm. In this example, ųd is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the president’s height minus their main opponent’s height. What are the null and alternative hypotheses for the hypothesis test?

Explore a hypothesis test for the mean difference in heights between U.S. presidential candidates and their main opponents using paired t-test methodology. This problem covers significance levels, paired data analysis, and t-distribution. Data includes heights of six presidents and their opponents, with a significance level of 0.05.

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