To approach this problem using hypothesis testing, we need to determine whether the observed results match the expected values based on a fair distribution of the dice rolls. Here’s a step-by-step breakdown for hypothesis testing:

Step 1: State the Hypotheses

• Null Hypothesis ($H_0$): The dice are fair, meaning the observed counts follow the expected distribution.
• Alternative Hypothesis ($H_a$): The dice are not fair, meaning the observed counts do not follow the expected distribution.

Step 2: Choose the Significance Level

Select a significance level ($\alpha$), typically 0.05.

Step 3: Determine the Test Statistic

This scenario requires a Chi-Square Goodness-of-Fit Test, where we calculate:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

Where:

• $O_i$ = Observed count
• $E_i$ = Expected count

From the table:

• $E_0 = 58, E_1 = 34.5, E_2 = 7, E_3 = 0.5$
• $O_0 = 47, O_1 = 35, O_2 = 15, O_3 = 3$

Step 4: Calculate the Test Statistic

$\chi^2 = \frac{(47 - 58)^2}{58} + \frac{(35 - 34.5)^2}{34.5} + \frac{(15 - 7)^2}{7} + \frac{(3 - 0.5)^2}{0.5}$

Breaking it down:

1. For 0 sixes: $\frac{(47 - 58)^2}{58} = \frac{(-11)^2}{58} = \frac{121}{58} \approx 2.086$

2. For 1 six: $\frac{(35 - 34.5)^2}{34.5} = \frac{(0.5)^2}{34.5} = \frac{0.25}{34.5} \approx 0.007$

3. For 2 sixes: $\frac{(15 - 7)^2}{7} = \frac{(8)^2}{7} = \frac{64}{7} \approx 9.143$

4. For 3 sixes: $\frac{(3 - 0.5)^2}{0.5} = \frac{(2.5)^2}{0.5} = \frac{6.25}{0.5} = 12.5$

Adding these: $\chi^2 = 2.086 + 0.007 + 9.143 + 12.5 = 23.736$

Step 5: Determine Degrees of Freedom

Degrees of freedom ($df$) is calculated as: $df = k - 1$ where $k$ is the number of categories. Here, $k = 4$, so: $df = 4 - 1 = 3$

Step 6: Find the Critical Value

Using a Chi-Square table and $\alpha = 0.05$ with $df = 3$, the critical value is: $\chi^2_{critical} = 7.815$

Step 7: Compare and Make a Decision

• If $\chi^2_{calculated} > \chi^2_{critical}$, reject $H_0$.
• Here, $23.736 > 7.815$.

Thus, we reject the null hypothesis. The observed data significantly differ from the expected distribution, suggesting the dice may not be fair.

Step 8: Conclusion

The results indicate that the dice are likely not fair, as there is a significant difference between observed and expected frequencies.

Do you have further questions or need clarification on any steps? Here are some related topics to explore:

1. How to interpret Chi-Square test results in practical scenarios?
2. What are the assumptions of the Chi-Square Goodness-of-Fit Test?
3. How does sample size affect the reliability of the test?
4. What other statistical tests can be used to assess fairness in dice?
5. How does altering the significance level ($\alpha$) change the outcome?

Tip: Always verify the expected values meet the Chi-Square test requirement ($E_i \geq 5$ for all categories).