S3 June 2008 Q2

EdexcelOld spec11 marksIncludes hypothesis testingChi-Squared Tests

2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results

		Course
		Arts	Science	Humanities
Gender	Boy	30	50	35
Gender	Girl	40	20	42

Showing your working clearly, test, at the 1% level of significance, whether or not there is an association between gender and the type of course taken. State your hypotheses clearly. (11)

Mark scheme

Scheme

Marks

\(\dfrac{115 \times 70}{217} = 37.0967\ldots\) or \(\dfrac{1150}{31}\) (etc) \(\dfrac{1265}{31}, \dfrac{1020}{31}, \dfrac{1122}{31}\)

Expected (Obs)	A	S	H
Boy	37.1 (30)	37.1 (50)	40.8 (35)
Girl	32.9 (40)	32.9 (20)	36.2 (42)

A1A1

\(\mathrm{H}_0\) : There is no association between course and gender
\(\mathrm{H}_1\) : There is some association between course and gender (both)

\(\displaystyle\sum \frac{(O - E)^2}{E} = \frac{(37.1 - 30)^2}{37.1} + \frac{(32.9 - 40)^2}{32.9} + \ldots + \frac{(36.2 - 42)^2}{36.2}\)

M1A1ft

\(= 1.358 + 4.485 + 0.824 + 1.532 + 5.058 + 0.929 = 14.189\ldots\) awrt 14.2

\(\nu = (3 - 1)(2 - 1) = 2\), \(\chi^2_2(1\%)\) critical value is 9.210 (condone 9.21)

B1, B1ft

Significant result or reject null hypothesis

There is evidence of an association between course taken and gender
[Correct answers only score full marks]

A1ft

(11)

(11 marks)

Alternative

Scheme	Marks
\(\displaystyle\sum \frac{O^2}{E} - N = \frac{30^2}{37.1} + \frac{40^2}{32.9} + \ldots + \frac{42^2}{36.2} - 217\)	M1A1ft

Notes

1^st M1 for some use of the \(\dfrac{\text{row total} \times \text{col total}}{\text{grand total}}\) formula

1^st A1 for one correct row or one correct column of expected frequencies to nearest integer
2^nd A1 for all expected frequencies correct to awrt 1 dp (Allow exact fractions)

1^st B1 for hypotheses. Independence is OK. Must mention courses and gender at least once.
Use of \(\rho\) or “correlation” is B0 but allow ISW.

2^nd M1 for an attempt to calculate test statistic. At least one correct expression, ft expected freq.
3^rd A1 follow through expected frequencies for at least 3 expressions

3^rd M1 for a correct statement relating their test statistic and their cv (may be implied by comment)
5^th A1 for a contextualised comment relating their test statistic and their cv. Ignore their \(\mathrm{H}_0\) or \(\mathrm{H}_1\) or assume that they were correct. Must mention courses and gender