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 2 manufacturing process, and the consumer group is testing to see if Brand 2 will outperform Brand 1. 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 10 cars, chosen at random. 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 Brand 1 Brand 2 Difference (Brand 1 - Brand 2) Send data to calculator 1 2 4 5 7 89 10 0.64 0.34 0.64 0.32 0.59 0.54 0.32 0.53 0.42 0.53 0.50 0.40 0.33 0.18 0.57 0.49 0.33 0.51 0.41 0.40 0.14 0.06 0.31 0.14 0.02 0.05 -0.01 0.02 0.01 0.13 Based on these data, can the consumer group conclude, at the 0.10 level of significance, that the mean tread wear of Brand 1 exceeds that of Brand 2? Answer this question by performing a hypothesis test regarding μ (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 normally distributed.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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Question

A. Find the value of the test statistic. (Round to three or more decimal places.)

B. Find the critical value at the 0.10 level of significance. (Round to three or more decimal places.)

C. At the 0.10 level, can the consumer group conclude that the mean tread wear of Brand 1 exceeds that of Brand 2?

Transcribed Image Text: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 2 manufacturing process, and the consumer group is testing to see if Brand 2 will outperform Brand 1. 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 10 cars, chosen at random. 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 Brand 1 Brand 2 Difference (Brand 1 - Brand 2) Send data to calculator 1 2 0.50 3 4 5 6 0.64 0.34 0.64 0.32 0.59 0.54 0.32 0.40 0.33 0.18 0.57 7 0.49 0.33 8 9 10 0.53 0.42 0.53 0.51 0.41 0.40 0.14 0.06 0.31 0.14 0.02 0.05 -0.01 0.02 0.01 0.13 Based on these data, can the consumer group conclude, at the 0.10 level of significance, that the mean tread wear of Brand 1 exceeds that of Brand 2? Answer this question by performing a hypothesis test regarding μ (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 normally distributed.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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