Problem

A scientist is investigating whether the ability to remember depends on age. A random sample of 150 students in different age groups is chosen. Each student is shown a set of 20 objects for thirty seconds and then asked to list as many as they can remember. The students are graded A or B according to how many objects they remembered correctly: grade A for 16 or more correct and grade B for fewer than 16 correct. The results are shown in the table.

	Age of students
	11-12 years	13-14 years	15-16 years
Grade A	25	16	19
Grade B	28	45	17

(a) Carry out a $\chi^2$-test at the 2.5% significance level to test whether grade is independent of age of student.

The scientist decides instead to use three grades: grade A for 16 or more correct, grade B for 10 to 15 correct and grade C for fewer than 10 correct. The results are shown in the following table.

	Age of students
	11-12 years	13-14 years	15-16 years
Grade A	25	16	19
Grade B	12	27	11
Grade C	16	18	6

With this second set of data, the test statistic is calculated as 10.91.

(b) Complete the $\chi^2$-test at the 2.5% significance level for this second set of data.

(c) State, with a reason, whether you would prefer to use the result from part (a) or part (b) to investigate whether the ability to remember depends on age.