An insurance company is working with two auto repair shops to compare their prices. Officials randomly select 12 damaged cars, take each one to both shops, and request a repair cost estimate. They subtract the cost quoted by the first shop from that of the second shop for every car. The most appropriate pair of hypotheses for them to use would be: C) H0: μd = 0 and Ha: μd ≠ 0

The most appropriate pair of hypotheses for the insurance company to use in this situation is indeed option C) H0: μd = 0 and Ha: μd ≠ 0. This is because the hypothesis test aims to determine if there is a significant difference in the average cost estimates provided by the two auto repair shops.

Here's the reasoning in a step-by-step approach:

- The company is interested in comparing the two shops. Each car serves as its own control because the same car is being estimated for repair costs at both shops.

- The null hypothesis H0: μd = 0 implies that there is no difference in the average cost estimates between the two repair shops for the cars selected. In other words, it suggests that both shops, on average, quote the same price for the repair of the cars.

- The alternative hypothesis Ha: μd ≠ 0 states that there is a difference in the average estimates; this could mean that, on average, one shop is either more expensive or less expensive than the other, but it doesn't specify which direction the difference goes.

- The subscript 'd' corresponds to the differences in cost estimates for each car between the two shops. Therefore, μd represents the mean of these differences.

- Since every car is taken to both shops and the estimate from the first shop is subtracted from that of the second shop, the resulting values are differences for each of the 12 cars.

- The test will analyze these differences to determine if collectively they are significantly different from zero, which would imply a systematic difference in pricing between the two shops.

This paired sample t-test (also known as the dependent sample t-test) is an appropriate statistical method when comparing two related samples, such as the same group tested twice, or in this case, the same cars tested at two different shops.