Chi-squared test for independence Problem set B

A school canteen records the main lunch chosen by 272 pupils from five houses. The results are shown. A chi-squared test for independence is carried out.

1. (a) State the null and alternative hypotheses.
2. (b) Find the number of degrees of freedom.

The table shows the exhibition chosen by 155 visitors to a museum. If the exhibition chosen is independent of ticket type, find the expected frequency of adult visitors who chose Space.

Find the degrees of freedom for a chi-squared test for independence on a contingency table with: (a) 3 rows and 7 columns; (b) 6 rows and 2 columns; (c) 5 rows and 5 columns.

A shop company asks 143 customers how satisfied they were, by branch. Use your calculator to find the expected frequencies, \(\chi^2\), the p-value and the degrees of freedom for a test of independence.

The curve shows the \(\chi^2\) distribution with 3 degrees of freedom. The shaded right tail has area 0.10.

1. (a) Find the critical value.
2. (b) A test gives \(\chi^2=9.1\). Find the p-value.

A library asks 306 members which type of book they borrowed and which age band they are in. At the 5% significance level, test whether book type is independent of age band.

1. (a) State the hypotheses.
2. (b) Write down the number of degrees of freedom.
3. (c) Use your calculator to find \(\chi^2\) and the p-value.
4. (d) State the conclusion in context.

A farmer records 236 apples from two orchards by size class. Test, at the 5% significance level, whether the size of an apple is independent of the orchard.

1. (a) Find the value of \(\chi^2\).
2. (b) The critical value is 7.81. State the conclusion in context.

A café asks 323 customers which drink they usually order and how often they visit. The results are shown.

1. (a) Find the expected frequencies.
2. (b) Find the cell that adds most to \(\chi^2\), and give its value of \((O-E)^2/E\).
3. (c) Compare the observed and expected frequencies in that cell. Describe what this shows in context.

A school records whether 109 students ate breakfast and whether they arrived late.

1. (a) Find the expected frequency for students who ate breakfast and arrived late if the variables are independent.
2. (b) State the degrees of freedom. Use your calculator to find \(\chi^2\), then use the critical value 3.84 to state the conclusion in context, at the 5% significance level.

A test of whether the choice of pet is independent of the type of home gives a p-value of 0.018.

1. (a) State the conclusion at the 5% significance level, in context.
2. (b) State the conclusion at the 1% significance level, in context.