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Classic Car Repairs

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Classic car show enthusiasts often ask what type of maintenance is required to keep the vehicles in display condition. In a random sample of owners of cars at a classic car show, each owner was asked what kind of car they owned and to identify the type of repair that occurred most frequently. Below are data for three different car models and five different types of repairs:

Car Model Repair Type
Brakes Transmission Electrical Engine mechanics Exterior body work TOTAL
Pontiac GTO 6 11 12 8 25 62
Chevy Corvair 9 19 8 5 13 54
Ford Falcon 31 10 6 26 11 84
TOTAL 46 40 26 39 49 200

A chi-squared test of independence is performed on this data to discover if evidence suggests that some types of repairs are significantly more likely to be performed on one car model than another. In order to perform this test, expected values are calculated for each cell in the table above.

Here is an example of how to calculate an expected value: 54 out of 200 owners in the sample owned a Chevy Corvair. There were a total of 40 owners that mentioned transmission repairs occurring most frequently. 54 out of 200 is 27%, so it would be expected that 27% of the 40 transmission repairs, or 10.8 repairs, would have occurred to Corvair owners, under the assumption that each car model behaves identically.

Since 19 repairs were ​observed in this sample, there was a large difference between the observed and expected counts. The chi-squared statistic is calculated by taking the difference between the observed and expected counts, squaring the difference, and then dividing by the expected count. Expected counts are not rounded to the nearest whole number during this process.

The table below displays the chi-squared statistics calculated for all but one of the corresponding cells in the table above:

Car Model Repair Type
Brakes Transmission Electrical Engine mechanics Exterior body work
Pontiac GTO 4.78 0.16 1.38 6.33
Chevy Corvair 0.94 6.23 0.14 2.90 0.01
Ford Falcon 7.06 2.75 2.22 5.65 4.46

Decide whether the following statements are true or false about this scenario.

True

False

True

False

The expected number of Ford Falcon repairs that are from exterior body work is greater than 15.

True

False

The missing chi-squared value is greater than 2.

True

False

The Chevy Corvair typically sees the greatest variation between the number of repairs expected and the number of repairs observed in this sample.

True

False

As the total of the chi-squared statistics for a particular car model increasesin comparison to the chi-squared statistics total for another car model, it becomes more reasonable to conclude that certain repairs tend to happen toone of the two car models more frequently than to the other.