Problem

The students in a group of schools can choose a club to join. There are 4 clubs available: Music, Art, Sports and Computers. The director collected information about the number of students in each club, using a random sample of 88 students from across the schools. The results are given in Table 1 below.

	Music	Art	Sports	Computers
No. of students	14	28	27	19

The director uses a chi-squared test to determine whether or not the students are uniformly distributed across the 4 clubs.

(a) (i) Find the expected frequencies he should use.

Given that the test statistic he calculated was 6.09 (to 3 significant figures)

(ii) use a 5% level of significance to complete the test. You should state the degrees of freedom and the critical value used.

The director wishes to examine the situation in more detail and takes a second random sample of 88 students. The director assumes that within each school, students select their clubs independently. The students come from 3 schools and the distribution of the students from each school amongst the clubs is given in Table 2 below.

School	Music	Art	Sports	Computers
School A	3	10	9	8
School B	1	11	13	5
School C	11	6	7	4

The director wishes to test for an association between a student’s school and the club they choose.

(b) State hypotheses suitable for such a test.

(c) Calculate the expected frequency for School C and the Computers club.

The director calculates the test statistic to be 7.29 (to 3 significant figures) with 4 degrees of freedom.

(d) Explain clearly why his test has 4 degrees of freedom.

(e) Complete the test using a 5% level of significance and stating clearly your critical value.