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


Testing for Randomness of Super Bowl Victories Listed below are the conference designations of teams that won the Super Bowl, where N denotes a team from the NFC and A denotes a team from the AFC. Do the results suggest that either conference is superior?

Verified step by step guidance

1. Step 1: Understand the runs test for randomness. The runs test is a non-parametric statistical test used to determine whether a sequence of data points is random. A 'run' is a sequence of similar elements (e.g., consecutive 'N's or 'A's). The test evaluates the number of runs in the sequence compared to what would be expected in a random sequence.
2. Step 2: Count the number of runs in the sequence. A run is defined as a consecutive sequence of the same symbol (either 'N' or 'A'). For example, if the sequence starts with 'NNN', that counts as one run. When the sequence switches to 'A', a new run begins. Continue counting runs throughout the sequence.
3. Step 3: Calculate the expected number of runs and the variance of runs for a random sequence. Use the formulas: \( E(R) = \frac{2n_1n_2}{n_1 + n_2} + 1 \) and \( \text{Var}(R) = \frac{2n_1n_2(2n_1n_2 - n_1 - n_2)}{(n_1 + n_2)^2(n_1 + n_2 - 1)} \), where \( n_1 \) is the number of 'N's, \( n_2 \) is the number of 'A's, and \( R \) is the number of runs.
4. Step 4: Compute the test statistic \( Z \) using the formula \( Z = \frac{R - E(R)}{\sqrt{\text{Var}(R)}} \). This standardizes the number of runs to determine how far it deviates from the expected value under the null hypothesis of randomness.
5. Step 5: Compare the test statistic \( Z \) to the critical value from the standard normal distribution at the given significance level (e.g., \( \alpha = 0.05 \)). If \( |Z| \) exceeds the critical value, reject the null hypothesis and conclude that the sequence is not random. Otherwise, fail to reject the null hypothesis and conclude that the sequence is random.

Key Concepts

Runs Test

The runs test is a non-parametric statistical test used to determine the randomness of a sequence of data. 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. In this context, it helps evaluate if the Super Bowl victories are randomly distributed between the NFC and AFC conferences.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether a statistical result is significant. Commonly set at 0.05, it represents the probability of rejecting the null hypothesis when it is actually true. In the context of the runs test, a significance level will help decide if the observed runs in the Super Bowl victories indicate a non-random pattern between the two conferences.

Null Hypothesis

The null hypothesis is a statement that assumes no effect or no difference exists in a given situation. In this case, the null hypothesis would state that there is no significant difference in the number of Super Bowl victories between the NFC and AFC conferences, implying that the victories are randomly distributed. Testing this hypothesis allows researchers to draw conclusions about the potential superiority of one conference over the other.