Chi-squared goodness of fit Problem set B

A radio station says that its listeners prefer pop, rock and dance music in the proportions \(\frac12\), \(\frac13\), \(\frac16\). You test this with a goodness of fit test. Write \(H_0\) and \(H_1\).

State the degrees of freedom of a goodness of fit test for each table.

1. (a) The twelve months of the year.
2. (b) Data grouped into 6 classes.
3. (c) A table with 8 groups.

A packet of 250 flower seeds is said to give white, pink, yellow and red flowers in the proportions 0.36, 0.28, 0.20 and 0.16. Find the expected frequency of each colour.

A teacher records 96 late arrivals across the four lessons of a morning. Assuming that a late arrival is equally likely in each lesson, find the expected frequency for each lesson.

A goodness of fit test is used to check that a school's five lunchtime clubs are equally popular. For 150 pupils the observed numbers are 27, 36, 33, 22, 32.

1. (a) State the degrees of freedom and the expected frequency of each club.
2. (b) Use the calculator to find \(\chi^2\) and the p-value.

A bakery says that 45% of its sales are bread, 25% cakes, 20% pastries and 10% other items. In a sample of 300 sales, 120, 85, 60 and 35 are bread, cakes, pastries and other items. Test the bakery's claim at the 5% significance level.

1. (a) Write \(H_0\) and \(H_1\).
2. (b) State the degrees of freedom and find the expected frequencies.
3. (c) Use the calculator to find \(\chi^2\) and the p-value.
4. (d) Write the conclusion in context.

A hotel has four types of room that the manager thinks are equally popular. Of 240 bookings, 45, 75, 68 and 52 are for the four room types. Test, at the 5% level, whether the four are equally popular. The critical value of \(\chi^2\) with df = 3 at the 5% level is 7.815.

1. (a) Write \(H_0\) and \(H_1\), and state the expected frequency.
2. (b) Find \(\chi^2\) on the calculator.
3. (c) Compare with the critical value and write the conclusion in context.

A game uses a wheel with six equal sectors, numbered 1 to 6. It is spun 150 times. The bar chart shows the number of times each number came up. The dashed line shows the expected frequency for a fair wheel.

1. (a) Find the expected frequency of each number.
2. (b) Before any test, do the results look consistent with a fair wheel? Give a reason.

A school records 200 student absences from Monday to Friday in one week. If absences were equally likely on each day the expected frequency would be 40 for each day. The table shows the observed frequencies.

1. (a) Find \(\chi^2\) and the p-value from the calculator.
2. (b) Write the conclusion at the 5% level.
3. (c) Suggest a reason, from the context, for the result.

A student spins a four-colour spinner 200 times for a project. The table shows the observed frequencies. The \(\chi^2\) test for equal probabilities gives a p-value very close to 1.

1. (a) Find \(\chi^2\) and the p-value from the calculator.
2. (b) What does such a large p-value say about the observed frequencies compared with the expected frequencies?
3. (c) Suggest a reason for it.