Preliminary data analyses indicate that use of a paired...

Preliminary data analyses indicate that use of a paired t-test is reasonable. Perform each hypothesis test by using either the critical-value approach or the P-value approach. Measuring Treadwear. R. Stichler ct al. compared two methods of measuring treadwear in their paper "Measurement of Treadwear of Commercial Tires" (Rubber Age, Vol. 73:2). Eleven tires were each measured for treadwear by two methods, one based on weight and the other on groove wear. The following are the data, in thousands of miles. At the $5 \%$ significance level, do the data provide sufficient evidence to conclude that, on average, the two measurement methods give different results?

Preliminary data analyses indicate that use of a paired t-test is reasonable. Perform each hypothesis test by using either the critical-value approach or the P-value approach.
Measuring Treadwear. R. Stichler ct al. compared two methods of measuring treadwear in their paper "Measurement of Treadwear of Commercial Tires" (Rubber Age, Vol. 73:2). Eleven tires were each measured for treadwear by two methods, one based on weight and the other on groove wear. The following are the data, in thousands of miles.
At the $5 \%$ significance level, do the data provide sufficient evidence to conclude that, on average, the two measurement methods give different results?

Key Concepts

P-value Approach

The P-value approach calculates the probability of obtaining a test statistic as extreme as the one observed, assuming that the null hypothesis is true. If this probability is less than or equal to the predetermined significance level, it provides evidence to reject the null hypothesis in favor of the alternative.

Critical-value Approach

The critical-value approach involves comparing the test statistic with a critical value determined from the sampling distribution of the test statistic under the null hypothesis. If the test statistic falls in the rejection region defined by the critical value, the null hypothesis is rejected.

Significance Level

The significance level (typically represented as ?, such as 0.05) is the threshold for deciding whether an observed effect is statistically significant. It represents the probability of making a Type I error, which occurs when a true null hypothesis is incorrectly rejected.

Paired t-test

The paired t-test is used when comparing two related samples, where each subject or entity is measured twice under different conditions. This test evaluates whether the mean difference between paired observations is statistically significant, accounting for within-subject variability.

Hypothesis Testing

Hypothesis testing is a formal process used to decide whether there is enough evidence in a sample to infer that a certain condition holds true for the entire population. It involves stating a null hypothesis (typically a statement of 'no effect' or 'no difference') and an alternative hypothesis, then using data to decide whether to reject the null hypothesis.

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