A car dealer recommends that the oil change in a car be performed every 3,000 mi

Question

A car dealer recommends that the oil change in a car be performed every 3,000 miles. To see whether the customers are adhering to this recommendation, the dealer selects a sample of 35 customers and finds that the average mileage of the automobiles serviced is 3,156. The standard deviation of the sample is 158.4 miles.
a. Determine whether the owners are having their transmissions serviced at 3,000 miles using traditional approach to hypothesis testing. Use a=0.10.
b. Solve the problem using p-value approach
c. Do you think that a=0.10 is an appropriate significance level?

Explanation / Answer

a)
SD of average = 158.4 / 35 = 26.71
t = (3156-3000)/26.71 = 5.841
Critical t at a=0.1 with 34 df: 1.307
5.841 is bigger than the critical t so reject the null hypothesis.
Conclusion: Owners are not having their transmissions serviced at 3000 miles.

b) Calculate t=5.841 as in part a)
p-value for t=5.841 with 34 df: 0.0000
0.0000 < 0.10 so reject the null hypothesis.

c)
If a=0.1 then there is a 1/10 chance that we'd think owners were not having their transmissions serviced at 3000 miles when in fact they were. Whether this is tolerable depends on what the dealer plans to do with this knowledge. If for instance the dealer is thinking about embarking on an expensive campaign to encourage owners to check their transmissions more often, then he would probably want more certainty before committing to this campaign. If there's some terrible consequences that would befall if people aren't checking their transmissions and it's not expensive to address this problem then we might want to have a generous a value of 0.1

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