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.
[APPLET] Passing Play Percentages The passing play percentages of 10 randomly selected NCAA Division 1A college football teams for home and away games in the 2020–2021 season are shown in the table. At , α=0.20 is there enough evidence to support the claim that passing play percentage is different for home and away games? (Source: TeamRankings)

Verified step by step guidance

1. Step 1: Formulate the null and alternative hypotheses. The null hypothesis (H₀) states that there is no difference in the passing play percentages for home and away games (mean difference = 0). The alternative hypothesis (H₁) states that there is a difference in the passing play percentages for home and away games (mean difference ≠ 0).
2. Step 2: Calculate the differences between the home and away passing play percentages for each college team. For example, for College 1, the difference is 54.3 - 52.9 = 1.4. Repeat this for all 10 colleges.
3. Step 3: Compute the mean and standard deviation of the differences. Use the formulas for the sample mean and sample standard deviation: Mean = (Σd) / n, where d represents the differences, and Standard Deviation = sqrt(Σ(d - Mean)² / (n - 1)).
4. Step 4: Perform a paired t-test. Calculate the test statistic using the formula: t = (Mean difference) / (Standard deviation of differences / sqrt(n)), where n is the number of paired observations. The degrees of freedom for the test are n - 1.
5. Step 5: Compare the calculated t-value to the critical t-value at α = 0.20 and degrees of freedom = n - 1. If the absolute value of the calculated t-value exceeds 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 between groups in a statistical test. In this context, it posits that there is no difference in passing play percentages between home and away games for the selected college football teams. Testing this hypothesis involves comparing sample data to determine if observed differences are statistically significant.

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 under different conditions. In this scenario, the passing play percentages for home and away games are dependent because they come from the same teams, allowing for a paired t-test to assess differences effectively.

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 0.20, meaning there is a 20% risk of concluding that a difference exists when there is none. This relatively high significance level indicates a willingness to accept more false positives in the analysis of the passing play percentages.