Testing the Difference Between Two Means (e) decide whether to reject or fail to reject the null hypothesis, 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: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis states that there is no difference in the maximum aerobic speed (MAS) before and after training (H₀: μd = 0), while the alternative hypothesis states that the MAS after training is greater than before training (Hₐ: μd > 0).
2. Step 2: Calculate the differences (d) between the MAS before and after training for each athlete. For each pair of values, subtract the 'MAS before training' from the 'MAS after training'.
3. Step 3: Compute the mean difference (d̄) and the standard deviation of the differences (sₐ). Use the formulas: d̄ = (Σd) / n and sₐ = sqrt((Σ(d - d̄)²) / (n - 1)), where n is the number of paired observations.
4. Step 4: Calculate the test statistic (t) using the formula: t = d̄ / (sₐ / sqrt(n)). This formula accounts for the mean difference, the standard deviation of the differences, and the sample size.
5. Step 5: Compare the calculated t-value to the critical t-value from the t-distribution table at α = 0.10 with n-1 degrees of freedom. If the calculated t-value is greater than the critical t-value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

Key Concepts

Null Hypothesis

The null hypothesis is a statement that there is no effect or no difference, serving as a default position in hypothesis testing. In this context, it posits that sprint interval training does not significantly improve the maximum aerobic speed (MAS) of trained athletes. Researchers aim to gather evidence to either reject or fail to reject this hypothesis based on statistical analysis.

Dependent Samples

Dependent samples refer to pairs of observations that are related or matched in some way, such as measurements taken from the same subjects before and after an intervention. In this scenario, the MAS of athletes is measured before and after sprint interval training, making the samples dependent. This relationship is crucial for applying the appropriate statistical tests, such as the paired t-test.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. In this case, α is set at 0.10, meaning there is a 10% risk of concluding that a difference exists when there is none. This level influences the interpretation of the p-value obtained from the statistical test, guiding the decision on whether the evidence is strong enough to support the researcher's claim.