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}{lllllll} \hline Height $(\mathrm{cm})$ of President & 181 & 178 & 176 & 189 & 202 & 182 \\ \hline Height $(\mathrm{cm})$ of Main Opponent & 168 & 187 & 179 & 183 & 187 & 171 \\ \hline \end{tabular} \[ \begin{array}{l} \mathrm{H}_{0}: \mu_{\mathrm{d}}=0 \mathrm{~cm} \\ \mathrm{H}_{1}: \mu_{\mathrm{d}}>0 \mathrm{~cm} \end{array} \] (Type integers or decimals. Do not round.) Identify the test statistic. $\mathrm{t}=1.41$ (Round to two decimal places as needed.) Identify the P-value. P-value $=\square$ (Round to three decimal places as needed $)$

Step 1 ：The question is asking for the P-value given a t-statistic of 1.41. The P-value is the probability that, given the null hypothesis is true, we would observe a statistic as extreme as the one calculated from our sample data. In this case, the null hypothesis is that the mean difference in height between presidents and their main opponents is 0 cm. The alternative hypothesis is that the mean difference is greater than 0 cm. This is a one-tailed t-test.

Step 2 ：To find the P-value, we need to use the cumulative distribution function (CDF) of the t-distribution. However, since we are dealing with a one-tailed test, we need to find the complement of the CDF at the given t-statistic. This can be done using the Survival Function (SF), which is 1 - CDF.

Step 3 ：Given the t-statistic of 1.41, the P-value is calculated to be approximately 0.10880301204041529.

Step 4 ：Rounding to three decimal places, the final P-value is \(\boxed{0.109}\).