Testing the Difference Between Two Means (b) find the critical value(s) and identify the rejection region(s), 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 mean passing play percentages for home and away games. The alternative hypothesis (H₁) states that there is a difference in the mean passing play percentages for home and away games.
2. Step 2: Calculate the differences between the paired observations (home passing play percentage minus away passing play percentage) for each college team. This will create a new dataset of differences.
3. Step 3: Compute the mean and standard deviation of the differences. Use the formulas for the sample mean and sample standard deviation: $\mu =\frac{\sum d}{n}$ and $s=\sqrt{\frac{\sum (d-\mu {)}^2}{n-1}}$, where d represents the differences.
4. Step 4: Determine the test statistic using the formula for a paired t-test: $t=\frac{\mu -0}{\frac{s}{\sqrt{n}}}$, where μ is the mean of the differences, s is the standard deviation of the differences, and n is the number of pairs.
5. Step 5: Find the critical t-value(s) for α = 0.20 and degrees of freedom (df = n - 1). Use a t-distribution table or statistical software to identify the rejection region(s). Compare the calculated t-statistic to the critical t-value(s) to determine whether to reject or fail to reject the null hypothesis.

Key Concepts

Dependent Samples

Dependent samples refer to pairs of observations that are related or matched in some way. In this context, the passing play percentages for home and away games are dependent because they come from the same teams. This relationship allows for the use of paired statistical tests, which account for the correlation between the two sets of data.

Critical Value

A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α) and the distribution of the test statistic. In this case, with α = 0.20, the critical value will help identify the rejection region for the test, indicating where the observed difference in means would be considered statistically significant.

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. It is determined by the critical value and represents the outcomes that are unlikely to occur if the null hypothesis is true. In this scenario, identifying the rejection region is crucial for assessing whether the difference in passing play percentages between home and away games is statistically significant.