Roulette Wheel A pit boss suspects that a roulette wheel is out of balance. A roulette wheel has 18 black slots, 18 red slots, and 2 green slots. The pit boss spins the wheel 500 times and records the following frequencies:

Is the wheel out of balance? Use the α = 0.05 level of significance.

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses. The null hypothesis (H0) assumes the roulette wheel is balanced, meaning the probabilities of landing on Black, Red, and Green are proportional to the number of slots: Black = 18/38, Red = 18/38, Green = 2/38. The alternative hypothesis (H1) is that the wheel is not balanced, so the observed frequencies differ significantly from the expected frequencies.
2. Step 2: Calculate the expected frequencies for each outcome based on the total number of spins (500) and the theoretical probabilities. Use the formula: \(E_i = n \times p_i\), where \(n = 500\) and \(p_i\) is the probability for each color (Black, Red, Green).
3. Step 3: Use the Chi-square goodness-of-fit test to compare observed and expected frequencies. The test statistic is calculated as: \(\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\), where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency for each category.
4. Step 4: Determine the degrees of freedom for the test. Since there are 3 categories, degrees of freedom = number of categories - 1 = 2.
5. Step 5: Find the critical value from the Chi-square distribution table at \(\alpha = 0.05\) significance level and 2 degrees of freedom. Compare the calculated \(\chi^2\) statistic to this critical value to decide whether to reject or fail to reject the null hypothesis.

Key Concepts

Chi-Square Goodness-of-Fit Test

This test determines if observed categorical data differ significantly from expected frequencies. It compares the observed counts to expected counts under a specific hypothesis, such as a fair roulette wheel, to assess if deviations are due to chance or indicate imbalance.

Expected Frequencies in Probability

Expected frequencies are calculated by multiplying the total number of trials by the theoretical probabilities of each outcome. For a balanced roulette wheel, these probabilities are based on the proportion of black, red, and green slots, guiding the expected counts for comparison.

Significance Level and Hypothesis Testing

The significance level (α = 0.05) sets the threshold for rejecting the null hypothesis, which assumes the wheel is balanced. If the test statistic exceeds the critical value at this level, it suggests the wheel is out of balance with 95% confidence.