Testing the Difference Between Two Means (c) calculate d̄ and Sd, 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: Calculate the differences (d) between the home and away passing play percentages for each college. For each pair of values in the table, subtract the away percentage from the home percentage.
2. Step 2: Compute the mean of the differences (d̄). Add all the differences calculated in Step 1 and divide by the total number of colleges (n = 10). Use the formula: $d¯=\frac{\sum d}{n}$
3. Step 3: Calculate the standard deviation of the differences (Sd). Use the formula: $S$_d=∑(d-d¯)2n-1
4. Step 4: Perform a t-test for dependent samples to test the claim that the passing play percentage is different for home and away games. Use the formula for the t-statistic: $t=\frac{d¯}{\frac{S}{}}$_dn
5. Step 5: Compare the calculated t-statistic to the critical t-value at α = 0.20 and degrees of freedom (df = n - 1 = 9). If the absolute value of the t-statistic exceeds the critical t-value, reject the null hypothesis and conclude that there is enough evidence to support the claim that passing play percentage is different for home and away games.

Key Concepts

Dependent Samples

Dependent samples, also known as paired samples, occur when the samples are related or matched in some way. In this context, the passing play percentages for home and away games are collected from the same teams, making them dependent. This relationship is crucial for applying statistical tests that compare means, as it accounts for the inherent correlation between the paired observations.

Difference of Means

The difference of means refers to the comparison of the average values of two related groups. In this case, it involves calculating the mean passing play percentage for home games and the mean for away games, then determining the difference. This concept is essential for hypothesis testing, as it helps assess whether the observed differences are statistically significant or due to random variation.

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 scenario, the null hypothesis would state that there is no difference in passing play percentages between home and away games, while the alternative hypothesis suggests a difference exists. The significance level (α = 0.20) indicates the threshold for rejecting the null hypothesis, guiding the decision-making process.