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.


b. Use the binomial probability distribution to determine the P-value for the hypothesis test from part (a).

Verified step by step guidance

1. Identify the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \). Typically, \( H_0 \) assumes the probability of the stock market increasing after a Super Bowl win is \( p = 0.5 \) (no effect), and \( H_a \) assumes \( p > 0.5 \) (market increases more often than not).
2. Define the binomial random variable \( X \) as the number of years the stock market increased after a Super Bowl win by an NFL team. Here, \( X = 28 \) successes out of \( n = 31 \) trials.
3. Calculate the P-value, which is the probability of observing \( X \) or more successes under the null hypothesis. This is given by the sum of binomial probabilities from \( X = 28 \) to \( X = 31 \): \[ P\text{-value} = P(X \geq 28) = \sum_{k=28}^{31} \binom{31}{k} (0.5)^k (0.5)^{31-k} \]
4. Use the binomial probability formula for each term in the sum: \[ P(X = k) = \binom{31}{k} (0.5)^k (0.5)^{31-k} = \binom{31}{k} (0.5)^{31} \]
5. Add the probabilities for \( k = 28, 29, 30, 31 \) to find the total P-value. This P-value tells you how likely it is to observe such an extreme or more extreme result if the null hypothesis were true.

Key Concepts

Binomial Probability Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is used when outcomes are binary (e.g., win or lose). The probability of observing a specific number of successes is calculated using combinations and the success probability raised to appropriate powers.

Hypothesis Testing and P-value

Hypothesis testing evaluates a claim about a population parameter using sample data. The P-value measures the probability of obtaining results at least as extreme as the observed, assuming the null hypothesis is true. A small P-value suggests evidence against the null hypothesis, guiding decision-making.

Small Sample Considerations

When sample sizes are small, normal approximations may be inaccurate, so exact distributions like the binomial are preferred. Small samples require careful interpretation of results, as variability is higher and assumptions must be checked to ensure valid conclusions.