Testing the Difference Between Two Means (b) find the critical value(s) and identify the rejection region(s), Assume the samples are random and dependent, and the populations are normally distributed.
Interval Training
A researcher claims that sprint interval training improves running performance in trained athletes. The table shows the maximum aerobic speed (MAS), in kilometers per hour, of trained athletes before and after six sessions of sprint interval training. At , α=0.10 is there enough evidence to support the researcher’s claim? (Adapted from National Strength and Conditioning Association)

Verified step by step guidance

1. Step 1: Identify the type of test to be performed. Since the samples are dependent (paired data), use a paired t-test to compare the means of the two related groups (MAS before and after training).
2. Step 2: Calculate the differences between the paired observations (MAS before training - MAS after training) for each athlete. This will give you a new dataset of differences.
3. Step 3: Compute the mean and standard deviation of the differences. Use the formulas for mean and standard deviation: Mean = (Σd)/n and Standard Deviation = sqrt(Σ(d - mean)^2 / (n-1)), where d represents the differences and n is the number of pairs.
4. Step 4: Determine the test statistic using the formula for the paired t-test: t = (mean difference) / (standard deviation of differences / sqrt(n)).
5. Step 5: Find the critical t-value for α=0.10 and degrees of freedom (df = n-1). Compare the calculated t-value to the critical t-value to identify the rejection region. If the calculated t-value falls in the rejection region, reject the null hypothesis; otherwise, fail to reject it.

Key Concepts

Dependent Samples

Dependent samples, also known as paired samples, occur when the same subjects are measured under different conditions. In this case, the maximum aerobic speed (MAS) of athletes is measured before and after sprint interval training. This design allows for a direct comparison of the effects of the training on the same individuals, controlling for individual variability.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to support a specific claim about a population parameter. In this scenario, the null hypothesis (H0) would state that there is no difference in MAS before and after training, while the alternative hypothesis (H1) posits that there is a significant improvement. The significance level (α) of 0.10 indicates the threshold for rejecting the null hypothesis.

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level and the distribution of the test statistic. The rejection region is the range of values for the test statistic that would lead to rejecting the null hypothesis. For a paired t-test, this involves calculating the t-statistic from the differences in MAS and comparing it to the critical t-value based on the degrees of freedom.