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Question

A researcher wants to know if the octane level of gasoline affects the gas mileage of a car. She randomly selects 10 cars and puts 5 gallons of 87-octane gasoline in the tank. On a closed track, each car is driven at 50 miles per hour until it runs out of gas. The experiment is repeated, with each car getting 5 gallons of 92-octane gasoline. The miles per gallon for each car are then computed. The results are shown. Would you recommend purchasing 92 octane? Why or why not?$\begin{matrix} \text{87 Octane} & \text{92 Octane} & \text{87 Octane} & \text{92 Octane}\\ \text{18.0} & \text{18.5} & \text{23.4} & \text{22.8}\\ \text{23.2} & \text{23.1} & \text{23.1} & \text{23.5}\\ \text{31.5} & \text{31.9} & \text{19.0} & \text{19.5}\\ \text{24.9} & \text{26.7} & \text{26.8} & \text{26.2}\\ \text{24.1} & \text{25.1} & \text{31.8} & \text{30.7}\\ \end{matrix}$

Solution

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Let's redo the given table, but with computed differences in the matched-pairs observations and the ranks (from smallest to largest) of the absolute value of the differences.

$\begin{bmatrix}\text{Car} & \text{87 Octane, }X & \text{92 Octane, Y} & D=Y-X & \text{Signed Ranks}\\ \hline \\ 1 & 18.0 & 18.5 & \phantom{-}0.5 & \phantom{-}4.5\\ 2 & 23.2 & 23.1 & -0.1 & -1\\ 3 & 31.5 & 31.9 & \phantom{-}0.4 & \phantom{-}2.5\\ 4 & 24.9 & 26.7 & \phantom{-}1.8 & \phantom{-}10\\ 5 & 24.1 & 25.1 & \phantom{-}1.0 & \phantom{-}8\\ 6 & 23.4 & 22.8 & -0.6 & -6.5\\ 7 & 23.1 & 23.5 & \phantom{-}0.4 & \phantom{-}2.5\\ 8 & 19.0 & 19.5 & \phantom{-}0.5 & \phantom{-}4.5\\ 9 & 26.8 & 26.2 & -0.6 & -6.5\\ 10 & 31.8 & 30.7 & -1.1 & -9 \end{bmatrix}$

Since we expect the 92 Octane to be better than 87 Octane, most of the differences $D$ should be positive. This results in small value of $T_{-}$. Now, let's compute the value of $T_{-}$.

\begin{aligned} T_{-}&=-1+(-6.5)+(-6.5)+(-9)\\\\ &=-\pmb{23} \end{aligned}

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