A pit boss is concerned that a pair of dice being used in a craps game is not fair. The distribution of the expected sum of two fair dice is as follows:

The pit boss rolls the dice 400 times and records the sum of the dice. The table shows the results. Do you think the dice are fair? Use the α = 0.01 level of significance.

Verified step by step guidance

1. Step 1: Identify the hypotheses for the chi-square goodness-of-fit test. The null hypothesis (H0) is that the dice are fair, meaning the observed frequencies follow the expected probabilities. The alternative hypothesis (H1) is that the dice are not fair.
2. Step 2: Calculate the expected frequencies for each sum of two dice by multiplying the total number of rolls (400) by the corresponding probability for each sum. For example, for sum 2, the expected frequency is \$400 \times \frac{1}{36}$.
3. Step 3: Use the chi-square test statistic formula: \[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\] where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency for each sum.
4. Step 4: Determine the degrees of freedom for the test, which is the number of categories minus 1. Since there are 11 possible sums (2 through 12), the degrees of freedom is \$11 - 1 = 10$.
5. Step 5: Compare the calculated chi-square statistic to the critical value from the chi-square distribution table at \(\alpha = 0.01\) significance level and 10 degrees of freedom. If the test statistic exceeds the critical value, reject the null hypothesis; otherwise, do not reject it.

Key Concepts

Probability Distribution of Two Dice Sums

The probability distribution shows the likelihood of each possible sum when rolling two fair dice. Each sum from 2 to 12 has a specific probability based on the number of combinations that produce that sum, with 7 being the most likely. Understanding this distribution is essential to compare observed results against expected probabilities.

Chi-Square Goodness-of-Fit Test

This statistical test compares observed frequencies with expected frequencies to determine if there is a significant difference. It helps assess whether the dice are fair by testing if the observed sums follow the expected probability distribution at a given significance level (α = 0.01).

Significance Level and Hypothesis Testing

The significance level (α) defines the threshold for rejecting the null hypothesis, which in this case states that the dice are fair. A 0.01 level means there is a 1% risk of incorrectly rejecting fairness. Understanding this helps interpret the test results and make decisions about the fairness of the dice.