**Chi-Square Distribution Analysis**Enter the critical value, along with the significance level and degrees of freedom $\chi^2_{(\alpha, df)}$ below the graph. (Graph is for illustration only. No need to shade.)**Graph Description:**The graph depicts a probability distribution labeled "Chi-Square Distribution" with probability on the y-axis and X on the x-axis. The curve starts low, peaks around a probability of 0.10, and then tapers off gradually towards zero.**Form Fields:**- $\chi^2$ ($\alpha$, df) = [Input Field] *(Round to three decimal places.)*- Test Statistic = [Input Field] *(Round to three decimal places.)*- p-value = [Input Field] *(Round to four decimal places.)***Decision:**- [Dropdown Menu: Select an answer]**Conclusion:**- [Dropdown Menu: Select an answer] the claim that all 6 categories are equally likely to be selected. **Chi-Square Test for Fairness of a Six-Sided Dice**Test the claim that a six-sided dice is fair at the 0.01 significance level. Complete the table. Round all answers to three decimal places.| Number rolled on dice | Observed Frequency | Expected Frequency | $\frac{{(O - E)^2}}{E}$ ||-----------------------|---------------------|--------------------|---------------------------|| 1 | 15 | | || 2 | 11 | | || 3 | 5 | | || 4 | 12 | | || 5 | 15 | | || 6 | 16 | | |**Hypotheses**- $H_0: p_1 = p_2 = p_3 = p_4 = p_5 = p_6$- $H_1:$ at least one is different**Original claim =** [Select an answer]**Instructions:** Fill in the expected frequencies and calculate the chi-square statistic using the formula $\frac{{(O - E)^2}}{E}$ for each outcome. Determine if the dice is fair based on the calculated statistic and the significance level.

Answered: Test the claim that a six-sided dice is…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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**Chi-Square Distribution Analysis**

Enter the critical value, along with the significance level and degrees of freedom $\chi^2_{(\alpha, df)}$ below the graph. (Graph is for illustration only. No need to shade.)

**Graph Description:**
The graph depicts a probability distribution labeled "Chi-Square Distribution" with probability on the y-axis and X on the x-axis. The curve starts low, peaks around a probability of 0.10, and then tapers off gradually towards zero.

**Form Fields:**

- $\chi^2$ ($\alpha$, df) = [Input Field]

*(Round to three decimal places.)*

- Test Statistic = [Input Field]

*(Round to three decimal places.)*

- p-value = [Input Field]

*(Round to four decimal places.)*

**Decision:**
- [Dropdown Menu: Select an answer]

**Conclusion:**
- [Dropdown Menu: Select an answer] the claim that all 6 categories are equally likely to be selected.

**Chi-Square Test for Fairness of a Six-Sided Dice**

Test the claim that a six-sided dice is fair at the 0.01 significance level. Complete the table. Round all answers to three decimal places.

| Number rolled on dice | Observed Frequency | Expected Frequency | $\frac{{(O - E)^2}}{E}$ |
|-----------------------|---------------------|--------------------|---------------------------|
| 1 | 15 | | |
| 2 | 11 | | |
| 3 | 5 | | |
| 4 | 12 | | |
| 5 | 15 | | |
| 6 | 16 | | |

**Hypotheses**

- $H_0: p_1 = p_2 = p_3 = p_4 = p_5 = p_6$
- $H_1:$ at least one is different

**Original claim =** [Select an answer]

**Instructions:** Fill in the expected frequencies and calculate the chi-square statistic using the formula $\frac{{(O - E)^2}}{E}$ for each outcome. Determine if the dice is fair based on the calculated statistic and the significance level.

Expert Solution

Step 1

Here, the claim of the test is that six-sided dice is fair. Hence, the original claim is that six-sided dice is fair. That is, the claim is that all 6 categories are equally likely to be selected.

Here, n is 74 and p=1/6=0.167.

The table is,

Number rolled on dice	Observed Frequency	Expected frequency (E=n*p)	(O-E)^2	(O-E)^2/E
1	15	12.358	6.980	0.565
2	11	12.358	1.844	0.149
3	5	12.358	54.140	4.381
4	12	12.358	0.128	0.010
5	15	12.358	6.980	0.565
6	16	12.358	13.264	1.073
Total	74			6.744

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