Performing a Runs Test In Exercises 15 – 20, (e) interpret the decision in the context of the original claim. Use α = 0.05
Coin Toss A coach records the results of the coin toss at the beginning of each football game for a season. The results are shown, where H represents heads and T represents tails. The coach claimed the tosses were not random. Test the coach’s claim.
H T T T H T H H T T T T H T H H

Verified step by step guidance

1. Step 1: Understand the Runs Test. The Runs Test is a non-parametric test used to determine if a sequence of data points is random. A 'run' is a sequence of similar outcomes (e.g., consecutive H's or T's). The null hypothesis (H₀) assumes the sequence is random, while the alternative hypothesis (H₁) assumes the sequence is not random.
2. Step 2: Identify the sequence and count the number of runs. In the given sequence (H T T T H T H H T T T T H T H H), identify each run of consecutive H's or T's. For example, the first run is 'H', the second run is 'T T T', and so on. Count the total number of runs in the sequence.
3. Step 3: Calculate the expected number of runs under the null hypothesis. Use the formula for the expected number of runs (E[R]) in a sequence of two categories: $E[R]\; =\; \backslash frac\{2n\_1n\_2\}\{n\_1\; +\; n\_2\}\; +\; 1$, where $n\_1$ is the number of H's, $n\_2$ is the number of T's, and $n\_1\; +\; n\_2$ is the total number of observations.
4. Step 4: Calculate the standard deviation of the number of runs. Use the formula for the standard deviation (σ[R]): $\backslash sigma[R]\; =\; \backslash sqrt\{\backslash frac\{2n\_1n\_2(2n\_1n\_2\; -\; n\_1\; -\; n\_2)\}\{(n\_1\; +\; n\_2)^2(n\_1\; +\; n\_2\; -\; 1)\}\}$. This will help determine the z-score for the observed number of runs.
5. Step 5: Perform the hypothesis test. Compute the z-score using the formula $z\; =\; \backslash frac\{R\; -\; E[R]\}\{\backslash sigma[R]\}$, where R is the observed number of runs. Compare the z-score to the critical value for a two-tailed test at α = 0.05 (±1.96). If the z-score falls outside this range, reject the null hypothesis and conclude that the sequence is not 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 consecutive identical elements, to assess whether the observed pattern deviates from what would be expected in a random sequence. In this context, it helps evaluate the coach's claim about the randomness of coin toss results.

Hypothesis Testing

Hypothesis testing is a statistical method that uses sample data to evaluate a hypothesis about a population parameter. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using a significance level (α) to determine whether to reject H0. In this case, the null hypothesis would state that the coin tosses are random, while the alternative would support the coach's claim of non-randomness.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether the results of a statistical test are statistically significant. Commonly set at 0.05, it represents a 5% risk of concluding that a difference exists when there is none (Type I error). In this scenario, using α = 0.05 means that if the p-value from the Runs Test is less than 0.05, the coach's claim of non-randomness would be supported.