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

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.

Formulate the null and alternative hypotheses for testing whether a six-sided die is fair, given observed counts $9,11,3,15,8,9$ for faces 1–6.

At significance level $\alpha=0.05$, conclude the chi-squared test for fairness of the die based on the statistic $\chi^2\approx8.39$ 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.

At significance level $\alpha=0.01$, conclude the chi-squared test for fairness of the die based on the statistic $\chi^2\approx8.39$ with 5 degrees of freedom.

Calculate the expected frequency for face 3 by hand and verify this expectation using statistical software, given 55 rolls of a fair die.

Compute the chi-squared test statistic by hand for the observed counts $9,11,3,15,8,9$ under the hypothesis of a fair die.

Using statistical software, determine the p-value for the chi-squared statistic computed from the observed counts $9,11,3,15,8,9$ with 5 degrees of freedom.

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.

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

Determine the smallest significance level $\alpha$ at which the fair-die hypothesis would be rejected for a test statistic $\chi^2\approx8.39$ with 5 degrees of freedom.