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

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.

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 $E_{ij}\ge5$. What criteria guide your decisions?

Determine the degrees of freedom for a chi-square test of independence between a categorical variable with $k$ levels and another with $m$ levels when no parameters are estimated.

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?

When performing a goodness-of-fit test to a Poisson model, you estimated the mean from the data and used 6 observed categories ($0,1,2,3,4,\ge5$). State the degrees of freedom for the test.

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

In the table below, conduct the chi-square test of independence using age categories $A_1$ and $A_2$. Compute the test statistic, degrees of freedom and conclude at $\alpha=0.05$:

A $3\times2$ contingency table yields the observed frequencies below:

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

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 $\alpha=0.05$. Compute the test statistic, degrees of freedom and conclusion.

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

Data on the number of defects per unit yields counts: $O=(50,30,15,4,1)$ for defect counts $x=(0,1,2,3,4)$. Assuming a Poisson distribution with the mean estimated from the data, perform a goodness-of-fit test at the 1% level. Compute expected frequencies, $\chi^2$, degrees of freedom and conclusion.

Given continuous data grouped into 5 intervals with observed frequencies $(5,12,20,8,5)$ and a sample mean and variance estimated as $\bar x=50$, $s^2=100$, conduct a chi-square goodness-of-fit test against a normal distribution at $\alpha=0.05$. Outline the steps required to compute expected frequencies from the normal model, state the degrees of freedom and describe how to compute $\chi^2$.