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

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.

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

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

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

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

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

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. $\text{R}$ or Python) based on a uniform distribution model.

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

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.

Calculate the $\chi^2$ goodness of fit test statistic for these results, under the null hypothesis that the die is fair.

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.

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