Combining categories in a chi-squared test Problem set A

A theatre records the seat bought by 119 customers and the customer's age group. The seat type and the age group are assumed to be independent. The observed counts are shown.

1. (a) Find the expected frequency for Stalls and Under 16.
2. (b) Find all the expected frequencies that are 5 or less.

182 runners in a marathon are grouped by age and by finishing time. A chi-squared test for independence is planned. The expected frequencies must be greater than 5 for a chi-squared test.

1. (a) Use the calculator to find the expected frequencies. Which are 5 or less?
2. (b) Combine neighbouring columns so that every expected frequency is greater than 5. Give a reason for your choice and the new table.
3. (c) State the degrees of freedom for the new table.

129 students are asked how often they cycle to school. Test, at the 5% significance level, whether this is independent of year group. Combine categories if you need to. Give the hypotheses, the degrees of freedom, the p-value and your conclusion.

A cafe chain collects 186 customer ratings from three branches. Test, at the 5% significance level, whether the rating is independent of the branch. Combine categories if you need to. Give the hypotheses, the degrees of freedom, the p-value and your conclusion.

160 pupils are grouped by the club they joined and their age. Test, at the 5% significance level, whether club and age are independent. Combine categories if you need to. Give the hypotheses, the degrees of freedom, the p-value and your conclusion.

A school team played 162 matches over three terms. Test, at the 5% significance level, whether the result is independent of the term. Combine categories if you need to. Give the hypotheses, the degrees of freedom, the p-value and your conclusion.

The number of days 152 students were absent last term is recorded by year group. A student wants to test, at the 5% significance level, whether days absent is independent of year group.

1. (a) The student combines 5 or more days = 10 or more days + 5 to 9 days. Show that the condition for the test is still not met.
2. (b) Combine more categories until the condition is met. State the groups you choose and carry out the test.

A market stall records how 111 customers paid and the customer's age group. Combine categories first. Test whether payment method is independent of age group, at the 5% level and then at the 1% level.

1. (a) Carry out the test at the 5% significance level.
2. (b) State the conclusion at the 1% significance level, and explain why the two conclusions differ.