Step 1 :First, we need to calculate the differences between the heights of the presidents and their main opponents. The differences are: 7, -6, 11, 8, 14, -11 cm.
Step 2 :Next, we calculate the mean of these differences. The sum of the differences is 23, and there are 6 pairs of data, so the mean is \(\frac{23}{6} \approx 3.83 \) cm.
Step 3 :The null hypothesis for the hypothesis test is that the mean difference \(\mu_{d}\) is equal to 0 cm, or \(H_{0}: \mu_{d} = 0\).
Step 4 :The alternative hypothesis is that the mean difference \(\mu_{d}\) is greater than 0 cm, or \(H_{1}: \mu_{d} > 0\).
Step 5 :Since we are testing the claim that the mean difference is greater than 0 cm, we are conducting a one-tailed test. The significance level is 0.01.
Step 6 :We would then proceed to calculate the test statistic and the p-value, and compare the p-value with the significance level to decide whether to reject the null hypothesis. However, since the problem only asks for the null and alternative hypotheses, we stop here.
Step 7 :The null and alternative hypotheses for the hypothesis test are \(H_{0}: \mu_{d} = 0\) and \(H_{1}: \mu_{d} > 0\), respectively. \(\boxed{H_{0}: \mu_{d} = 0, H_{1}: \mu_{d} > 0}\)