Some people believe that different octane gasoline result in different miles per gallon in a vehicle. The following data is a sample of 11 people which were asked to drive their car only using 10 gallons of gas and record their mileage for each 87 Octane and 92 Octane. \begin{tabular}{|c|c|c|} \hline Person & \begin{tabular}{c} Miles with \\ 87 Octane \end{tabular} & \begin{tabular}{c} Miles with \\ 92 Octane \end{tabular} \\ \hline 1 & 234 & 237 \\ \hline 2 & 257 & 238 \\ \hline 3 & 243 & 229 \\ \hline 4 & 215 & 224 \\ \hline 5 & 114 & 119 \\ \hline 6 & 287 & 297 \\ \hline 7 & 315 & 351 \\ \hline 8 & 229 & 241 \\ \hline 9 & 192 & 186 \\ \hline 10 & 204 & 209 \\ \hline 11 & 547 & 562 \\ \hline \end{tabular} Do the data support that different octanes produce different miles per gallon at the $\alpha=0.02$ level of significance? Note: A normal probability plot of difference in car mileage between Octane 87 and Octane 92 indicates the population could be normal and a boxplot indicated no outliers. a. Express the null and alternative hypotheses in symbolic form for this claim. Assume $\mu_{d}=\mu_{1}-\mu_{2}$, where $\mu_{1}$ is the population mean mileage for Octane 87 and $\mu_{2}$ is the mean mileage for Octane 92 . \[ \begin{array}{l} H_{0}: \mu_{d} \text { Select an answer } \vee \\ H_{1}: \mu_{d} \text { Select an answer } v \end{array} \] b. What is the significance level? \[ \alpha= \] c. What is the test statistic? Round to 3 decimal places. \[ ? \vee= \] d. What is the $p$-value? Round to 5 decimal places. \[ p= \] e. Make a decision. Do not reject the null Reject the null f. What is the conclusion? There is not sufficient evidence to support the claim that different octanes produce different miles per gallon. There is sufficient evidence to support the claim that different octanes produce different miles per gallon. Submit Question
Solution
Step 1 ：Express the null and alternative hypotheses in symbolic form for this claim. Assume \(\mu_{d}=\mu_{1}-\mu_{2}\), where \(\mu_{1}\) is the population mean mileage for Octane 87 and \(\mu_{2}\) is the mean mileage for Octane 92. The null and alternative hypotheses in symbolic form for this claim are: \[ \begin{array}{l} H_{0}: \mu_{d} = 0 \ H_{1}: \mu_{d} \neq 0 \end{array} \]
Step 2 ：The significance level is given as 0.02, so \(\alpha=0.02\).
Step 3 ：Calculate the test statistic. The test statistic is approximately -1.135, so \(t = -1.135\).
Step 4 ：Calculate the p-value. The p-value is approximately 0.283, so \(p = 0.283\).
Step 5 ：Make a decision based on the p-value and the significance level. The p-value is greater than the significance level of 0.02, so we do not reject the null hypothesis.
Step 6 ：Draw a conclusion based on the decision. There is not sufficient evidence to support the claim that different octanes produce different miles per gallon.
