Chi-squared test for independence Problem set A

A survey asks 237 students how they usually travel to school and which year group they are in. 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 sport chosen by 154 pupils. If the sport chosen is independent of gender, find the expected frequency of girls who chose tennis.

Find the degrees of freedom for a chi-squared test for independence on a contingency table with: (a) 5 rows and 3 columns; (b) 2 rows and 6 columns; (c) 4 rows and 4 columns.

A teacher records how often 175 students hand in homework on time, by gender. 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 5 degrees of freedom. The shaded right tail has area 0.05.

1. (a) Find the critical value.
2. (b) A test gives \(\chi^2=12.4\). Find the p-value.

A tour company asks 287 customers which type of holiday they chose and which age group they are in. At the 5% significance level, test whether holiday type is independent of age group.

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 shop records 250 sales of two kinds of phone case, by the colour chosen. Test, at the 5% significance level, whether the case chosen is independent of the colour.

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

A gym asks 328 members how many days a week they visit and which membership they hold. 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 doctor records whether 90 patients took a vitamin and whether they caught a cold in winter.

1. (a) Find the expected frequency for patients who took the vitamin and caught a cold 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 favourite music is independent of age group gives a p-value of 0.032.

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