Exam-style Questions: Bivariate Data

1. Sketch scatter diagrams to show possible forms (or describe in words) for the following values of the product moment correlation coefficient, r.

a) r = 0.78

b) r = -1

c) r = -0.25

(3 marks)

d) calculate the regression line x on y for the following summarised data

n = 12 Σx = 1800 Σx2 = 336296 Σy = 36.0 Σy2 = 126.34 Σxy = 6348.6

(5 marks)

(Marks available: 8)

Answer

Answer outline and marking scheme for question: 1

a) a good positive correlation

b) a perfect negative correlation - all points lie on line

c) a poor negative correlation - a fair degree of scatter

(3 marks)

d) need Sxy = 6348.6 - 1800 x 36 = 948.6

12

Syy = 126.34 - 362 = 18.34

12

so b' = 948.6 = 51.7

18.34

a' = 1800 - 51.7 x 36 = -5.17

12 12

so regression line x on y is x = 51.7y - 5.17

(5 marks)

(Marks available: 8)

2. A wine expert grades 10 bottles of wine on a scale from 0 to 50. He records the results next to their ages

Wine A B C D E F G H I J
Age 15 21 24 28 30 34 36 40 42 44
Score 18 11 14 25 27 17 33 22 37 41

a) calculate the product moment correlation coefficient for the age of the wine against the grade given

b) calculate Spearman's rank correlation coefficient for the same data and comment on the results

(Marks available: 9)

Answer

Answer outline and marking scheme for question: 2

a) data obtained from calculator

n = 10 Σx = 314 Σx2 = 10678 Σy = 245 Σy2 = 6907 Σxy = 8347 and from calculator r = 0.760 (3sf)

if you prefer to use the formulae then you should obtain the following

Sxy = 654 Sxx = 818.4 Syy = 904.5 and r = 654 = 0.760

Copyright S-cool

b) ranking

Wine A B C D E F G H I J
Age x 15 21 24 28 30 34 36 40 42 44
Score y 18 11 14 25 27 17 33 22 37 41
Rank x 10 9 8 7 6 5 4 3 2 1
Rank y 7 10 9 4 5 8 3 6 2 1
d 3 1 1 3 1 3 1 3 0 0
d2 9 1 1 9 1 9 1 9 0 0

n = 10 and Σd2 = 40 so rs = 1 - 6 x 40 = 0.758 (3sf)
10(102-1)

commenting; approximation rs of r, is extremely close. Both show a good positive correlation meaning the older the wine the better quality it is according to this particular expert.

(Marks available: 9)

3. A physicist wants to find out what happens to a length of metal rod (cm) when put under various temperatures (°C).
She carries out an experiment and her data is as follows

Temp t 105 110 115 120 125 130
Length x 200.1 201.8 201.8 202 204.3 205.1

The physicist would like to calculate a line of regression for this data.

a) advise the physicist on which line to use 'x on t' or 't on x'

b) calculate this line of regression and use it to estimate the length of a bar at 145°C

(Marks available: 9)

Answer

Answer outline and marking scheme for question: 3

a) as the temperature appears to be controlled and independent we should calculate the line 'x on t'

b) summarised data from calc

n = 6 Σ t = 705 Σt2 = 83275 Σx = 1214.1 Σx2 = 245692.39 Σtx = 142746

Stx = 89.25 Stt = 437.5 hence b = 89.25 = 0.204
437.5

so a = 1214.1 - 0.204 x 705 = 178.38
6 6

equation of 'x on t' is x = 178.38 + 0.204t

at 1450C x = 178.38 + 0.204 x -145 = 207.96cm

(Marks available: 9)