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.
[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: Identify the claim and hypotheses. The claim is that the passing play percentage is different for home and away games. The null hypothesis (H₀) states that there is no difference in passing play percentages between home and away games (μ₁ = μ₂). The alternative hypothesis (Hₐ) states that there is a difference in passing play percentages (μ₁ ≠ μ₂).
2. Step 2: Verify assumptions. Ensure that the samples are random and dependent (paired data), and confirm that the populations are normally distributed. The table shows paired data for 10 colleges, which satisfies the dependency requirement.
3. Step 3: Calculate the differences between paired observations. For each college, subtract the away passing play percentage from the home passing play percentage to obtain the differences (d). For example, for College 1: d = 54.3 - 52.9 = 1.4.
4. Step 4: Compute the mean and standard deviation of the differences (d̄ and 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.
5. Step 5: Perform a t-test for paired samples. Calculate the test statistic using the formula: t = d̄ / (s_d / sqrt(n)). Compare the calculated t-value to the critical t-value at α = 0.20 and degrees of freedom (df = n - 1). If |t| > critical t-value, reject H₀; otherwise, fail to reject H₀.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters 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 hypotheses will assess whether the passing play percentages for home and away games differ significantly.

Dependent Samples

Dependent samples refer to pairs of observations that are related or matched in some way, often arising from the same subjects measured under different conditions. In this case, the passing play percentages for home and away games are dependent because they come from the same teams. This relationship affects the choice of statistical tests, as it requires methods that account for the paired nature of the data.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining 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 scenario, α is set at 0.20, indicating a willingness to accept a 20% chance of concluding that there is a difference in passing play percentages when there is none.