Eight SmartCars were randomly selected and the highway mileage of each was noted. The highway mileage per gallon for each car is shown below: $\begin{array} { l l l l l l l l l } 32 & 42 & 29 & 34 & 41 & 37 & 38 & 39 & 40 \end{array}$
It was desired to determine if the median miles per gallon of all SmartCars exceeded 35 miles per gallon. Find the p-value when conducting the Sign Test.
A) $\mathrm { p } = 0.145$
B) $p = 0.855$
C) $p = 0.965$
D) $0.035$

Question

Quizplus · Accepted Answer

The Sign Test is used to test the median of a distribution. In this case, we are testing if the median miles per gallon for SmartCars exceeds 35 miles per gallon. We count the number of observations above the median (35 mpg) and those below. Here, 5 observations are above 35 (37, 38, 39, 40, 41, 42) and 3 are below (29, 32, 34). The Sign Test looks at the smaller of these two counts (in this case, 3) and calculates the probability of observing this count or a more extreme count under the null hypothesis that the median is 35 mpg. Using the binomial distribution (with n=8 trials and a probability of success of 0.5 under the null hypothesis), the p-value is calculated. The correct p-value is 0.145, indicating the probability of observing 3 or fewer cars below 35 mpg (or equivalently 5 or more cars above 35 mpg) by chance if the true median is 35 mpg.

Eight SmartCars Were Randomly Selected and the Highway Mileage of Each