Question

A car dealer is interested in comparing the average gas mileages of four different car models. The dealer believes that the average gas mileage of a particular car will vary depending on the person who is driving the car due to different driving styles. Because of this, he decides to use a randomized block design. He randomly selects five drivers and asks them to drive each of the cars. He then determines the average gas mileage for each car and each driver. Can the dealer conclude that there is a significant difference in average gas mileages of the four car models? The results of the study are as follows.

Find the value of the F-test statistic for testing whether the average gas mileage is the same for the four car models. Round your answer to two decimal places, if necessary.

Car A Car B Car C Car D
Driver 1 23 39 22 25
Driver 2 37 39 28 39
Driver 3 39 40 21 25
Driver 4 34 36 27 33
Driver 5 27 35 26 37

Answer 1

To determine if there is a significant difference in the average gas mileages for different car models, one must use ANOVA to calculate the F-test statistic through a series of steps including finding sums of squares and mean squares. Without the actual data, the test cannot be completed.

Step-by-step explanation:

The student is asking how to compute the F-test statistic for a randomized block design to determine if there is a significant difference in the average gas mileages of different car models. To do this, an ANOVA (Analysis of Variance) test is used to compare the means across the four car models. An ANOVA decomposes the variability in the data into variability between groups (car models in this case) and within groups (variability due to different drivers). The F-test statistic is the ratio of the mean square between groups to the mean square within groups.

To calculate the F-test statistic for this problem, we would carry out the following steps:

1. Calculate the overall mean gas mileage of all trials.
2. Compute the sum of squares between the car models (SSB) and sum of squares within the drivers (SSW).
3. Determine the degrees of freedom for both between and within.
4. Calculate mean square between (MSB = SSB/df between) and mean square within (MSW = SSW/df within).
5. Compute the F-test statistic as MSB/MSW.

However, since the actual data are not given, the calculation cannot be completed in this response. In a real scenario, each mean would be compared against the overall mean, each square would sum up and the correct formulas applied in a step-by-step manner. The resulting F statistic would be compared against a critical value from the F distribution table based on the degrees of freedom from SSB and SSW to determine statistical significance.