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) below.
\begin{tabular}{|l|l|l|l|l|l|l}
\hline Height $(\mathbf{c m})$ of President & 181 & 174 & 177 & 187 & 189 & 164 \\
\hline Height $(\mathbf{c m})$ of Main Opponent 174 & 180 & 164 & 166 & 179 & 175 \\
\hline
\end{tabular}
a. Use the sample data with a 0.01 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 $0 \mathrm{~cm}$.
In this example, $\mu_{d}$ 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?
(Type integers or decimals. Do not round)
Answer
The null and alternative hypotheses for the hypothesis test are \(H_{0}: \mu_{d} = 0\) and \(H_{1}: \mu_{d} > 0\), respectively. \(\boxed{H_{0}: \mu_{d} = 0, H_{1}: \mu_{d} > 0}\)
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. \(\boxed{H_{0}: \mu_{d} = 0, H_{1}: \mu_{d} > 0}\)
