A consumer products testing group is evaluating two competing brands of tires, Brand 1 and Brand 2. Though the two brands have been comparable in the past, some technological advances were recently made in the Brand 1 manufacturing process, and the consumer group is testing to see if Brand 1 will outperform Brand 2. Tread wear can vary considerably depending on the type of car, and the group is trying to eliminate this effect by installing the two brands on the same random sample of 10 cars. In particular, each car has one tire of each brand on its front wheels, with half of the cars chosen at random to have Brand 1 on the left front wheel, and the rest to have Brand 2 there. After all of the cars are driven over the standard test course for 20,000 miles, the amount of tread wear (in inches) is recorded, as shown in the table below. Car 1 2 3 4 5 6 7 8 9
A consumer products testing group is evaluating two competing brands of tires, Brand 1 and Brand 2. Though the two brands have been comparable in the past, some technological advances were recently made in the Brand 1 manufacturing process, and the consumer group is testing to see if Brand 1 will outperform Brand 2. Tread wear can vary considerably depending on the type of car, and the group is trying to eliminate this effect by installing the two brands on the same random sample of 10 cars. In particular, each car has one tire of each brand on its front wheels, with half of the cars chosen at random to have Brand 1 on the left front wheel, and the rest to have Brand 2 there. After all of the cars are driven over the standard test course for 20,000 miles, the amount of tread wear (in inches) is recorded, as shown in the table below.
| Car |
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
|---|---|---|---|---|---|---|---|---|---|---|
| Brand 1 |
0.18
|
0.22
|
0.21
|
0.17
|
0.24
|
0.30
|
0.31
|
0.19
|
0.18
|
0.31
|
| Brand 2 |
0.16
|
0.18
|
0.26
|
0.31
|
0.32
|
0.35
|
0.39
|
0.28
|
0.23
|
0.27
|
| Difference (Brand 1 - Brand 2) |
0.02
|
0.04
|
−0.05
|
−0.14
|
−0.08
|
−0.05
|
−0.08
|
−0.09
|
−0.05
|
0.04
|
Based on these data, can the consumer group conclude, at the 0.01level of significance, that the mean tread wear of Brand 2 exceeds that of Brand 1? Answer this question by performing a hypothesis test regarding μd (which is μ with a letter "d" subscript), the population mean difference in tread wear for the two brands of tires. Assume that this population of differences (Brand 1 minus Brand 2) is
a. State the null hypothesis H0 and the alternative hypothesis H1.
b. Find the value of the test statistic.
c. Find the critical value at the 0.01level of significance.
d. At the 0.01level, can the consumer group conclude that mean tread wear of Brand 2 exceeds that of Brand 1?
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