S3 June 2017 Q4

EdexcelOld spec14 marksIncludes hypothesis testingChi-Squared Tests

4. A psychologist carries out a survey of the perceived body weight of 150 randomly chosen people. He asks them if they think they are underweight, about right or overweight. His results are summarised in the table below.

	Underweight	About right	Overweight
Male	20	22	30
Female	16	28	34

The psychologist calculates two of the expected frequencies, to 2 decimal places, for a test of independence between perceived body weight and gender. These results are shown in the table below.

	Underweight	About right	Overweight
Male	17.28
Female	18.72

(a) Complete the table of expected frequencies shown above. (2)

(b) Test, at the 10% level of significance, whether or not perceived body weight is independent of gender. State your hypotheses clearly. (7)

The psychologist now combines the male and female data to test whether or not body weight types are chosen equally.

(c) Find the smallest significance level, from the tables in the formula booklet, for which there is evidence of a preference. (5)

Mark scheme (a)

Scheme	Marks
\(\dfrac{72 \times 50}{150} = 24,\ \dfrac{78 \times 50}{150} = 26\)	M1
\(\dfrac{72 \times 64}{150} = 30.72,\ \dfrac{78 \times 64}{150} = 33.28\)	A1
	(2)

Notes

M1 For one correct \(\frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}\); can be implied by correct answers.

A1 24, 26, 30.72, 33.28 only.

Mark scheme (b)

Scheme

Marks

\(\mathrm{H}_0 :\) Perceived (body) weight is independent of gender (no association)
\(\mathrm{H}_1 :\) Perceived (body) weight is not independent of gender (association)

\(O\)	\(E\)	\(\dfrac{(O-E)^2}{E}\)	\(\dfrac{O^2}{E}\)
20	17.28	0.428148	23.14815
22	24	0.166667	20.16667
30	30.72	0.016875	29.29688
16	18.72	0.395214	13.67521
28	26	0.153846	30.15385
34	33.28	0.015577	34.73558
150	150	1.176327	151.1763

M1A1

\(\displaystyle\sum \frac{(O-E)^2}{E}\) or \(\displaystyle\sum \frac{O^2}{E} - 150 = 1.18\)

\(\nu = (3-1)(2-1) = 2,\ \chi^2_2(10\%) = 4.605\)

B1B1ft

(Accept \(\mathrm{H}_0\)) Perceived (body) weight is independent of gender (no association)

A1ft

(7)

Notes

B1 Both hypotheses required. Must mention “Perceived”, “weight” and “gender” at least once.
Use of “relationship” or “correlation” or “connection” or “link” award B0.

M1 for at least 2 correct terms (as in 3^rd or 4^th column) or correct expressions.
A1 for all correct. Accept 2sf accuracy. Allow truncation e.g. 1.17…

A1 Awrt 1.18-1.19

B1B1ft 2 can be implied by 4.605 seen

A1ft A correct comment in context - must mention “weight” and “gender”. Condone “relationship” or “connection” here but not “correlation”.
Follow through from their test stat and cv, but hypotheses must be correct.

Mark scheme (c)

Scheme

Marks

\(O\)	\(E\)	\(\dfrac{(O-E)^2}{E}\)	\(\dfrac{O^2}{E}\)
36	50	3.92	25.92
50	50	0	50
64	50	3.92	81.92
150	150	7.84	157.84

B1
M1A1

\(\displaystyle\sum \frac{(O-E)^2}{E}\) or \(\displaystyle\sum \frac{O^2}{E} - 150 = 7.84\)

\(\nu = 2,\ \chi^2_2(2.5\%) = 7.378\)

(5)

(14 marks)

Notes

B1 for \(E_i = 50\), could be implied.

M1 for combining values and for attempting \(\dfrac{(O-E)^2}{E}\) or \(\dfrac{O^2}{E}\) with at least 2 correct expressions or values.

A1 for all correct, can be implied by correct answer below.

A1 Awrt 7.84

A1 0.025 or 2.5%