Question Type 5: Categorizing data to ensure a specific expected frequency Exercises

Compare the two $\chi^2$ (chi-square) tests in questions 3 and 4 in terms of sensitivity and category selection. Which grouping would you prefer and why?

Conduct a chi-square test of independence at the $5\%$ level of significance. State the null and alternative hypotheses, the test statistic, the degrees of freedom, and the conclusion of the test.

The critical value for this test is $3.84$.

In a goodness-of-fit test for a distribution with 4 categories where 2 parameters have been estimated from the data, what is the appropriate degrees of freedom?

Data on the number of defects per unit yields the following observed frequencies:

Assuming a Poisson distribution with the mean estimated from the data, perform a $\chi^2$ goodness-of-fit test at the 1% significance level. Calculate the expected frequencies, the $\chi^2$ test statistic, the degrees of freedom, and state your conclusion.

Determine the degrees of freedom for a $\chi^2$ test of independence between a categorical variable with $k$ levels and another with $m$ levels.

Outline the procedure to select appropriate age categories for a chi-square test of independence when given raw age data and a binary preference variable, ensuring that all expected frequencies exceed 5.

A $3 \times 2$ contingency table yields the observed frequencies below:

Carry out the chi-square test of independence at the $5\%$ significance level. Compute the $\chi^2$ statistic, the degrees of freedom, and state your conclusion.

A set of continuous data is grouped into 5 intervals with observed frequencies $(5, 12, 20, 8, 5)$. The sample mean and variance are estimated as $\bar{x} = 50$ and $s^2 = 100$. A chi-square goodness-of-fit test is to be conducted at a $5\%$ significance level to determine if the data follows a normal distribution.

Outline the steps required to compute the expected frequencies from the normal model, state the degrees of freedom for this test, and describe the calculation of the $\chi^2$ test statistic.

The following observed frequencies for a categorical variable with 4 equally likely outcomes were recorded: $O = (10, 15, 14, 11)$. Carry out a goodness-of-fit test at the $\alpha = 0.05$ significance level. Compute the test statistic, the number of degrees of freedom, and the conclusion of the test. [6 marks]

Given a detailed contingency table with age in single years and preference counts, describe how you would combine adjacent age classes to achieve the expected frequency condition $E_{ij} \ge 5$. What criteria guide your decisions?

When performing a goodness-of-fit test to a Poisson model, you estimated the mean from the data and used 6 observed categories ($0, 1, 2, 3, 4, \ge 5$). State the degrees of freedom for the test.

In a trial of a die rolled 120 times, the observed face counts for the six faces are $(18, 23, 19, 20, 21, 19)$. Test whether the die is fair at the 1% significance level.