In Exercises 9–12, use the runs test with a significance level of . (All data are listed in order by row.)




Baseball World Series Victories Test the claim that the sequence of World Series wins by American League and National League teams is random. Given are recent results, with A = American League and N = National League.

Verified step by step guidance

1. Step 1: Understand the runs test. The runs test is a non-parametric statistical test used to determine whether a sequence of data points is random. A 'run' is defined as a sequence of similar elements (e.g., consecutive A's or N's). The test evaluates the number of runs in the sequence compared to what would be expected under randomness.
2. Step 2: Count the total number of A's (American League wins) and N's (National League wins) in the sequence. This will help determine the expected number of runs under randomness.
3. Step 3: Identify and count the number of runs in the sequence. A run is formed whenever the sequence switches from A to N or N to A. For example, the sequence 'A N N A' has three runs: 'A', 'N N', and 'A'.
4. Step 4: Use the runs test formula to calculate the expected number of runs and the standard deviation of runs under randomness. The formulas are: \( E(R) = \frac{2n_1n_2}{n_1 + n_2} + 1 \) and \( \sigma_R = \sqrt{\frac{2n_1n_2(2n_1n_2 - n_1 - n_2)}{(n_1 + n_2)^2(n_1 + n_2 - 1)}} \), where \( n_1 \) is the number of A's, \( n_2 \) is the number of N's, and \( R \) is the number of runs.
5. Step 5: Compare the observed number of runs to the expected number of runs using a z-test. The test statistic is \( z = \frac{R - E(R)}{\sigma_R} \). Determine whether the z-value falls within the critical region for the given significance level (e.g., 0.05). If it does, reject the null hypothesis that the sequence is random; otherwise, fail to reject the null hypothesis.

Key Concepts

Runs Test

The Runs Test is a non-parametric statistical test used to determine the randomness of a sequence of data points. It analyzes the occurrence of runs, which are sequences of similar items, to assess whether the observed pattern deviates from what would be expected under a random distribution. This test is particularly useful for binary data, such as the sequence of wins by American League (A) and National League (N) teams.

Significance Level

The significance level, often denoted as alpha (α), is the threshold used to determine whether to reject the null hypothesis in hypothesis testing. Commonly set at 0.05, it represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In the context of the Runs Test, the significance level helps to assess whether the observed runs are statistically significant or could have occurred by chance.

Null Hypothesis

The null hypothesis (H0) is a statement that assumes no effect or no difference in a statistical test. In the context of the Runs Test for World Series victories, the null hypothesis would state that the sequence of wins by American League and National League teams is random. The goal of the test is to determine if there is enough evidence to reject this hypothesis in favor of an alternative hypothesis, which suggests a non-random pattern.