Do wet suits allow a swimmer to swim faster? Researchers...

Do wet suits allow a swimmer to swim faster? Researchers measured the speed (in meters per second) of swimmers both with and without a wetsuit. The results of the study are shown in the table. Conduct the appropriate test to determine whether the data suggest that wet suits allow a swimmer to swim faster. Be sure to check all model requirements. Use an $\alpha=0.05$ level of significance. If you reject the null hypothesis, estimate the average difference in speed of the swimmer with the wet suit with $95 \%$ confidence.

Key Concepts

Model Assumptions

Statistical tests like the paired t-test rely on certain assumptions to produce valid results. For the paired t-test, it is essential that the differences between paired observations are approximately normally distributed. Verifying these assumptions helps ensure that the conclusions drawn from the test are reliable.

Confidence Interval

A confidence interval is a range of values derived from sample statistics that is likely to contain the true population parameter. For example, a 95% confidence interval means that if the same population is sampled numerous times, approximately 95% of the calculated intervals would contain the true mean difference.

Hypothesis Testing

Hypothesis testing is a formal procedure used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves establishing a null hypothesis (typically representing no effect or no difference) and an alternative hypothesis, then using a test statistic to determine whether to reject the null hypothesis based on a predetermined significance level.

Paired t-test

The paired t-test is used to compare the means of two related groups, such as measurements taken on the same subjects under two different conditions. This test focuses on the differences within each pair of observations and assumes that these differences are normally distributed, providing a more efficient test when the measurements are dependent.

Significance Level (?)

The significance level, denoted by alpha (?), is the threshold used to determine whether the observed results are statistically significant. A common choice for alpha is 0.05, which indicates a 5% risk of concluding that there is an effect when there is none. It sets the criterion for rejecting the null hypothesis in hypothesis testing.

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