Combining categories in a chi-squared test Problem set B

A garage records the fuel used by 129 cars and the age of each car. The fuel type and the age of the car are assumed to be independent. The observed counts are shown.

1. (a) Find the expected frequency for Petrol and Under 3 years.
2. (b) Find all the expected frequencies that are 5 or less.

179 students record their daily screen time. The results are grouped by year group. 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.

154 pupils in three school clubs record how long they practise each week. Test, at the 5% significance level, whether practice time is independent of the club. Combine categories if you need to. Give the hypotheses, the degrees of freedom, the p-value and your conclusion.

177 students record how long they read each day. Test, at the 5% significance level, whether reading time 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 bookshop records the type and the format of 179 books sold. Test, at the 5% significance level, whether type of book and format are independent. Combine categories if you need to. Give the hypotheses, the degrees of freedom, the p-value and your conclusion.

A rail company checks 225 journeys on three routes. Test, at the 5% significance level, whether the outcome of a journey is independent of the route. Combine categories if you need to. Give the hypotheses, the degrees of freedom, the p-value and your conclusion.

The distance from home to school of 164 students is recorded by year group. A student wants to test, at the 5% significance level, whether distance to school is independent of year group.

1. (a) The student combines 1 to 5 km = 1 to 3 km + 3 to 5 km. 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 survey asks 172 commuters which transport they use. The commuters live in three towns. Combine categories first. Test whether transport used is independent of town, 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.