To address this problem, let’s set up the null and alternative hypotheses based on the claim provided:

• The claim states that the mean difference ($\mu_d$) in heights between presidents and their main opponents is greater than 0 cm.

Hypotheses

1. Null Hypothesis ($H_0$): $\mu_d \leq 0 \, \text{cm}$ (There is no significant difference, or presidents are not taller than their opponents.)
2. Alternative Hypothesis ($H_1$): $\mu_d > 0 \, \text{cm}$ (Presidents are taller than their opponents.)

The hypotheses can be written as:

$H_0: \mu_d = 0 \, \text{cm}$ $H_1: \mu_d > 0 \, \text{cm}$

Let me know if you’d like to proceed with analyzing the data (e.g., calculate the mean, standard deviation, test statistic, and p-value) or if you have specific questions.

Here are some follow-up questions:

1. Would you like me to calculate the mean and standard deviation of the differences for the paired data?
2. Do you need an explanation on how to perform a hypothesis test for this situation?
3. Should we calculate the test statistic (e.g., $t$-statistic) using the provided data?
4. Are you interested in checking if the results meet the 0.05 significance level?
5. Do you want a step-by-step walkthrough of the hypothesis testing process?

Tip: When performing paired data hypothesis testing, always ensure to check the assumptions, such as normality of the differences or use a non-parametric test if necessary.