"Small Sample Hypothesis Test: Super Bowl InvestingFrom Super Bowl I (1967) through Super Bowl XXXI (1997), the stock market increased if an NFL team won the Super Bowl and decreased if an AFL team won. This condition held 28 out of 31 years.


a. Suppose the likelihood of predicting the direction of the stock market (increasing or decreasing) in any given year is 0.50. Decide on the appropriate null and alternative hypotheses to test whether the outcome of the Super Bowl can be used to predict the direction of the stock market.
"

Verified step by step guidance

1. Step 1: Identify the parameter of interest. Here, we are interested in the probability \( p \) that the stock market direction can be correctly predicted based on the Super Bowl outcome in any given year.
2. Step 2: Formulate the null hypothesis \( H_0 \). Since the likelihood of correctly predicting the market direction by chance is 0.50, the null hypothesis assumes that the Super Bowl outcome does not help predict the market direction better than random guessing. Mathematically, \( H_0: p = 0.50 \).
3. Step 3: Formulate the alternative hypothesis \( H_a \). Because the problem suggests that the market increased 28 out of 31 times when an NFL team won (which is more than expected by chance), the alternative hypothesis should reflect that the probability of correct prediction is greater than 0.50. Thus, \( H_a: p > 0.50 \).
4. Step 4: Clarify the type of test. Since the alternative hypothesis is one-sided (greater than), this will be a right-tailed test.
5. Step 5: Summarize the hypotheses clearly: \[ H_0: p = 0.50 \quad \text{vs.} \quad H_a: p > 0.50 \]

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the default assumption, usually stating no effect or no difference. The alternative hypothesis (H1) reflects the claim we want to test, suggesting an effect or difference exists. Formulating these correctly is essential to guide the statistical test and interpret results.

Binomial Probability Model

The binomial model describes the number of successes in a fixed number of independent trials, each with the same probability of success. Here, predicting the stock market direction correctly or incorrectly can be modeled as a binomial process with probability 0.5 under the null hypothesis.

Small Sample Hypothesis Testing

When sample sizes are small, exact or specialized methods like the binomial test are used instead of normal approximations. This ensures accurate p-values and conclusions, especially when testing if observed success rates (e.g., 28 out of 31) significantly differ from chance.