Testing the Difference Between Two Means (a) identify the claim and state Ho and Ha ,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 claim and hypotheses. The claim is that sprint interval training improves running performance in trained athletes. The null hypothesis (H₀) states that there is no difference in maximum aerobic speed (MAS) before and after training, i.e., μ₁ = μ₂. The alternative hypothesis (Hₐ) states that sprint interval training improves MAS, i.e., μ₁ < μ₂.
2. Step 2: Calculate the differences between the paired data points (MAS before training - MAS after training) for each athlete. This will give you a set of differences (d).
3. Step 3: Compute the mean of the differences (d̄) and the standard deviation of the differences (s_d). Use the formulas: d̄ = (Σd) / n and s_d = sqrt((Σ(d - d̄)²) / (n - 1)), where n is the number of paired observations.
4. Step 4: Perform a paired t-test. Calculate the test statistic using the formula: t = d̄ / (s_d / sqrt(n)). This test statistic will help determine whether the observed difference is statistically significant.
5. Step 5: Compare the calculated t-value to the critical t-value at α = 0.10 and degrees of freedom (df = n - 1). If the calculated t-value is less than the critical t-value, reject the null hypothesis (H₀) and conclude that there is enough evidence to support the claim that sprint interval training improves MAS.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which states there is no effect or difference, and the alternative hypothesis (Ha), which suggests there is an effect or difference. In this context, the null hypothesis would assert that sprint interval training does not improve running performance, while the alternative would claim that it does.

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 maximum aerobic speed (MAS) of the same athletes is measured before and after sprint interval training, making the samples dependent. This relationship is crucial for selecting the appropriate statistical test, such as the paired t-test, to analyze the data.

Significance Level (α)

The significance level, denoted as α, is the threshold used to determine whether to reject the null hypothesis in hypothesis testing. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this case, α is set at 0.10, meaning there is a 10% risk of concluding that sprint interval training improves performance when it does not. This level influences the interpretation of the test results and the strength of evidence required to support the researcher's claim.