Exercises for Question Type 6: Initiating the chi-squared goodness of fit test with the hypothesis and critical region

Question 1

A chi-squared goodness-of-fit test is performed on a set of six categories with the following observed counts: 9, 11, 3, 15, 8, 9 . The null hypothesis is that the categories are equally likely. Using statistical software, determine the p-value for the chi-squared statistic, given that there are 5 degrees of freedom.

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Question 2

Determine the smallest significance level α at which the fair-die hypothesis would be rejected for a test statistic χ^{2} \approx 8.39 with 5 degrees of freedom.

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Question 3

The question requires the calculation of a critical value for the chi-squared distribution using the degrees of freedom and significance level.

Calculate the critical value χ_{0.05, 5}^{2} for a chi-squared distribution with 5 degrees of freedom at significance level α = 0.05 .

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Question 4

Calculate the expected frequency for each face when rolling a fair six-sided die a total of 55 times.

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Question 5

A loaded die is rolled 60 times with observed frequencies 5, 10, 15, 10, 10, 10 for faces 1-6. Test at α = 0.05 whether the die is fair by computing the chi-squared statistic and stating your conclusion.

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Question 6

At significance level α = 0.05, conclude the chi-squared test for fairness of the die based on the statistic χ^{2} \approx 8.39 with 5 degrees of freedom.

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Question 7

A fair six-sided die is rolled 55 times. Calculate the expected frequency for face 3. Show your calculation by hand and state the result as it would be verified using statistical software (e.g. R or Python) based on a uniform distribution model.

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Question 8

A six-sided die is rolled several times. The observed counts for the faces 1 to 6 are 9, 11, 3, 15, 8, and 9 respectively.

Formulate the null and alternative hypotheses for testing whether the die is fair.

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Question 9

Chi-squared goodness-of-fit test

Determine the degrees of freedom for the chi-squared goodness-of-fit test applied to a six-sided die with no parameters estimated from the data.

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Question 10

The following table shows the observed frequencies of each face when a six-sided die is rolled 55 times. Face 123456 Frequency 91131589

Face	1	2	3	4	5	6
Frequency	9	11	3	15	8	9

Calculate the χ^{2} goodness of fit test statistic for these results, under the null hypothesis that the die is fair.

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Question 11

For the observed counts 9, 11, 3, 15, 8, 9, calculate and interpret the Pearson residual for face 3 under the assumption of a fair die.

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Question 12

At significance level α = 0.01, conclude the chi-squared test for fairness of the die based on the statistic χ^{2} \approx 8.39 with 5 degrees of freedom.

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IB Mathematics Applications & Interpretation Question Type 6: Initiating the Chi-squared Goodness of Fit Test with the Hypothesis and Critical Region Exercises