S4 June 2014 Q7

EdexcelOld spec14 marksIncludes hypothesis testingt-Tests & Further Tests

7. Two groups of students take the same examination.

A random sample of students is taken from each of the groups.

The marks of the 9 students from Group 1 are as follows

30 29 35 27 23 33 33 35 28

The marks, \(x\), of the 7 students from Group 2 gave the following statistics

\[\bar{x} = 31.29 \qquad s^2 = 12.9\]

A test is to be carried out to see whether or not there is a difference between the mean marks of the two groups of students.

You may assume that the samples are taken from normally distributed populations and that they are independent.

(a) State one other assumption that must be made in order to apply this test and show that this assumption is reasonable by testing it at a 10% level of significance. State your hypotheses clearly. (7)

(b) Stating your hypotheses clearly, test, using a significance level of 5%, whether or not there is a difference between the mean marks of the two groups of students. (7)

Mark scheme (a)

Scheme	Marks
The variance of the two group’s marks must be the same.	B1
\(\mathrm{H}_0 : {\sigma_1}^2 = {\sigma_2}^2 \quad \mathrm{H}_1 : {\sigma_1}^2 \ne {\sigma_2}^2\)	B1
\({s_1}^2 = 16.25\)	B1
\(\left(F_{8,6} =\right) \dfrac{16.25}{12.9} = (1.2597\ldots)\) \(\left(\dfrac{1}{\mathrm{F}_{8,6}} = \dfrac{12.9}{16.25} = 0.7938\ldots\right)\)	M1A1
\(\mathrm{F}_{8,6}\) 5% (two-tail) cv \(= \mathbf{4.15}\) (0.241) (or prob. = awrt 0.39)	B1
Not significant so can accept the assumption that variances are equal.	A1
	(7)

Notes

2nd B1 allow \(\sigma\) or \(\sigma^2\)

3rd B1 allow awrt 16.3 or \(s_1 = \text{awrt } 4.03\)

M1 for use of the correct test statistic

5th B1 allow “assumption is correct”

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \mu_1 = \mu_2 \quad \mathrm{H}_1 : \mu_1 \ne \mu_2\)	B1
\({s_p}^2 = \dfrac{8 \times 16.25 + 6 \times 12.9}{14},= 14.814\ldots\) or \(s_p = 3.8489\ldots\)	M1
\(\left(t_{14} =\right)(\pm)\dfrac{30.33 - 31.29}{s_p\sqrt{\frac{1}{9} + \frac{1}{7}}} = (\pm)0.494927\ldots = \text{awrt } \underline{\mathbf{0.49}}\)	B1 M1A1
\(t_{14}(0.025)\) two-tail cv \(= \mathbf{2.145}\)	B1
There is insufficient evidence to reject \(\mathrm{H}_0\). There is no evidence of a significant difference between the mean marks of the two groups	A1
	(7)
	(14 marks)

Notes

1st M1 for attempting \(s_p\) or \({s_p}^2\)

1st B1 for 30.33

2nd M1 for use of a correct test statistic

2nd A1 for awrt 0.49 (accept \(\pm\)) or 0.495

2nd B1 for 2.145 (allow \(\pm\) 1.761 for one-tailed \(\mathrm{H}_1\))