Testing the Difference Between Two Means (c) calculate d̄ and Sd, 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: Calculate the differences (d) between the MAS before training and MAS after training for each athlete. Use the formula d = MAS_before - MAS_after for each pair of values in the table.
2. Step 2: Compute the mean of the differences (d̄). This is done by summing all the differences calculated in Step 1 and dividing by the total number of athletes (n = 8). The formula is d̄ = (Σd) / n.
3. Step 3: Calculate the standard deviation of the differences (Sd). Use the formula Sd = sqrt((Σ(d - d̄)^2) / (n - 1)), where d̄ is the mean difference calculated in Step 2, and n is the number of athletes.
4. Step 4: Perform a paired t-test to determine if the mean difference is significantly different from 0. The test statistic is t = d̄ / (Sd / sqrt(n)), where d̄ is the mean difference, Sd is the standard deviation of the differences, and n is 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 degrees of freedom (df = n - 1). If the calculated t-value exceeds the critical t-value, reject the null hypothesis and conclude that there is enough evidence to support the researcher's claim.

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 the same athletes is measured before and after sprint interval training. This design helps control for individual variability, allowing for a more accurate assessment of the training's effect.

Difference of Means

The difference of means refers to the calculation of the average difference between paired observations in dependent samples. In this scenario, it involves subtracting the MAS before training from the MAS after training for each athlete. This value is crucial for determining whether the training had a statistically significant effect on performance.

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 case, the null hypothesis would state that there is no difference in MAS before and after training, while the alternative hypothesis posits that there is a significant improvement. The significance level (α=0.10) indicates the threshold for rejecting the null hypothesis based on the calculated p-value.