You are given a six-faced die and asked to determine with 95% confidence if it is “fair.”

Answer: the null H0: the die is fair, p1 = p2 = … = p6 = 1/6; the alternative Ha: the die is not fair, at least one pi ≠ 1/6.

Answer: degrees of freedom df = 6 – 1 = 5. From a table or calculator, χ0.052 =11.071.

Common student mistake #1:

using an incorrect value to enter table or calculator. Some tables give the area under the curve to the right of the critical value; others to the left. The same is true for calculators. You must check the table or calculator description and/or graph of the Chi-square distribution associated with it. Hint: remember that we want the area under the curve to the right of the critical value to be equal to alpha = 0.05 which is 1 – the 95% confidence level.

You toss the die 120 times and record the numbers of 1’s, 2’s, …, and 6’s (shown in the table in the column labeled “Observed”). Determine the chi-square statistic.


Chi-square statistic
Chi-square statistic


Empty Table





Common student mistake#2: Incorrect “Expected values.” Hint: remember the expected frequency of each die face assuming the die is fair is 1/6 times the total number of tries.

The completed table:

Completed Table





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