Testing the Difference Between Two Means (f) interpret the decision in the context of the original claim. 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 use. Since the samples are dependent (paired data), we will use a paired t-test to compare the means of the 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 us the differences for the paired data.
3. Step 3: Compute the mean and standard deviation of the differences. The mean difference will help us determine the average change in MAS, and the standard deviation will measure the variability of these differences.
4. Step 4: Formulate the null and alternative hypotheses. The null hypothesis (H₀) is that there is no difference in MAS before and after training (mean difference = 0). The alternative hypothesis (H₁) is that sprint interval training improves MAS (mean difference > 0).
5. Step 5: Calculate the t-statistic using the formula: t = (mean difference) / (standard deviation of differences / √n), where n is the number of paired observations. Compare the calculated t-statistic to the critical t-value at α = 0.10 to determine whether to reject the null hypothesis.

Key Concepts

Dependent Samples

Dependent samples, also known as paired samples, occur when the same subjects are measured under different conditions or at different times. In this context, 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.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether 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 an improvement. The significance level (α) of 0.10 indicates the threshold for rejecting the null hypothesis.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. In this analysis, the assumption of normality for the populations allows the use of parametric tests, which are more powerful than non-parametric tests when the data meet this assumption, facilitating valid conclusions about the training's effectiveness.