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Correlation and Regression Exam questions

Syllabus: Edexcel ALevel Questions: 10 Time: 1 hour 33 minutes Total Marks: 77

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Correlation and Regression Exam questionsQuestions: 10 Time: 1 hour 33 minutes Total Marks: 77

Q1 ID: 7109

Q2 ID: 5560

The blood pressures, p mmHg, and the ages, t years, of 7 hospital patients are shown in the table below.

Patient t p

A 45 101

B 70 129

C 53 122

D 31 84

E 55 180

F 31 76

G 60 136

45 8281 28 05454 p 3061 t = 3 t2 = 1 p = 8 p2 = 1 t = 4(a) Find S and S for these data. tt Spp tp

(b) Calculate the product moment correlation coefficient for these data. (c) Interpret the correlation coefficient.

The volume of a sample of gas is kept constant. The gas is heated and the pressure p, is measured at 12 different temperatures, t °C. The results are summarised below.

11 8149 13 7514 t 6850 p = 4 p2 = 3 t = 2 t2 = 2 p = 2(a) Find S and S for these data. pp pt

Given that S 3733 25, tt = 2(b) calculate the product moment correlation coefficient. (c) Give an interpretation of your answer to part (b).







Q3 ID: 6100

Q4 ID: 5543

Q5 ID: 4366

A city council attempted to reduce traffic congestion by introducing a congestion charge. The charge was set at £3 for the first year and then was increased by £2 each year. For each of the first eight years, the council recorded the average number of vehicles entering the city centre per day. The results are shown in the table.

Charge £x

Average number, y million 3 2 3

5 2 7

7 2 2

9 2 3

11 1 9

13 1 7

15 1 6

17 1 6

n 0 68 6 3 4 33 y 50 7 = 8 x = 8 x2 = 9 y = 1 y2 = 3 x = 1Calculate the product moment correlation coefficient for these data.

Crickets make a noise. The pitch, v kHz, of the noise made by a cricket was recorded at 18 different temperatures, t °C. These data are summarised below.

0923 24 2 3365 00 4 5 21 v 77 905 t2 = 1 v2 = 4 t = 4 v = 2 t = 6(a) Find S for these data. tt Svv Stv(b) Find the product moment correlation coefficient between t and v. (c) State, with a reason, which variable is the explanatory variable. (d) Give a reason to support fitting a regression model of the form v t to these data. = a + b(e) Find the value of a and the value of b. Give your answers to 3 significant figures. (f) Using this model, predict the pitch of the noise at 23 °C.

A sample of bivariate data was taken and the results were summarised as follows: n 0 78 1068 27 3903 y 6639 = 1 x = 7 x2 = 6 y = 7 y2 = 5 x = 5Show that the value of the the product moment correlation coefficient is 0 104, correct to 3 significant figures.






Q6 ID: 2477

Q7 ID: 3136

A scatter graph is drawn for a data set as shown above. (a) State the correlation coefficient. (b) The two data points shown are D (30 ) and D (60 ). Find the 1 8 2 5relationship between x and y. (c) Make a prediction for y when x 0 and comment. = 8

A second hand car dealer has 10 cars for sale. She decides to investigate the link between the age of the cars, x years, and the mileage, y thousand miles. The data collected from the cars are shown in the table below.

Age (years) x

Mileage (1000)y 5 5 22

2 5 26

4 5 22

2 42

5 5 49

3 25

2 42

5 46

3 45

6 5 41

You may assume that 9 5 60 80 25 y 429 x = 3 y = 3 x2 = 1 x = 1(a) Find S and S xx xy

(b) Find the equation of the regression line of y on x in the form y x. = a + bGive the values of a and b to 2 decimal places. (c) Give a practical interpretation of the slope b. (d) Using your answer to part (b), find the mileage predicted by the regression line for a 3 year old car.






Q8 ID: 4372

Q9 ID: 5932

It is thought that the pH value of sand may affect the extent to which a particular species of plant will grow in that sand. A botanist wished to determine whether there was any correlation between the ph value of the sand on certain sand dunes and the amount of each of two species of plant growing there. She chose random sections of equal area on each of 8 sand dunes and measured the pH values. She then measures the area within each section that was covered by each of the two species. The results were as follows:

Dune pH value x

area species P area species Q

A 8 5 145 148

B 8 5 145 22

C 9 5 575 55

D 8 5 335 207

E 6 5 54 89

F 7 5 16 17

G 8 5 358 0

H 9 336 5

The results for Species P can be summarised by n 6 5 58 75 964 29132 y 7313 = 8 x = 6 x2 = 5 y = 1 y2 = 7 x = 1(a) Calculate the equation of the regression line for Species P, in the form y x where a and b are given to three significant figures. = a + b(b) Estimate the value for Species P on sand where the pH value is 7.0. (c) State with a reason whether the regression line of y on x for Species P will provide a reliable estimate of the y value when the pH value is (i) 8, (ii) 4.

Roseen is a selfemployed decorator who wishes to estimate the times that it will take her to decorate bedrooms based upon their floor areas. She records the floor area, x , and the m2

decorating time, y hours, for each of 10 bedrooms she has recently decorated.

x y

9 5 15 5

20 35 5

9 14 5

20 22

11 5 24 5

16 5 16 5

12 5 15 5

15 5 25 5

20 22

22 34

(a) Plot a scatter diagram of these data. (b) Calculate the equation of the least squares regression line of y on x. (c) Draw your regression line on the diagram. (d) Use your regression equation estimate the time that Roseen will take to decorate a bedroom with a floor area of 17 3 . m2

(e) Making reference to the diagram, comment on the likely reliability of your estimate in part (d)





Q10 ID: 6099

A city council attempted to reduce traffic congestion by introducing a congestion charge. The charge was set at £4 for the first year and then was increased by £3 each year. For each of the first eight years, the council recorded the average number of vehicles entering the city centre per day. The results are shown in the table.

Charge £x

Average number, y million 4 2 3

7 2 7

10 2

13 2 4

16 2 1

19 1 7

22 1 7

25 1 7

n 16 060 6 6 5 42 y 25 1 = 8 x = 1 x2 = 2 y = 1 y2 = 3 x = 2(a) Explain why x is the independent variable. (b) Calculate the equation of the regression line y on x. (c) Use your equation to estimate the average number of vehicles which will enter the city centre per day when the congestion charge is raised to £28 (d) Comment on the reliability of your estimate.






Correlation and Regression Exam questions Mark Scheme

A1 ID: 7109

A2 ID: 5560

(a)

(b)

(c)

Stt

S pp

S tp

r

interpretation

=

=

=

=

=

8281 277 4286 t2 −n

( )t 2= 1 −

7(345)2

= 1

05454 513 4286 p2 −n

( )p 2

= 1 −7

(828)2= 7

p 3061 252 4286 t −n

( )( )t p= 4 −

7345 288

= 2

727 Stp

S Stt pp

=2252 4286

1277 4286 513 42867= 0

moderately strong higher blood pressure associated with higher age

M1A1

A1

A1

M2A1

B1

(a)

(b)

(c)

Spp

S pt

r

interpretation

=

=

=

=

=

:

8149 p2 −n

( )p 2

= 3 −124112

24072 25

t 6850 p −n

( )( )p t= 2 −

12411 132

19554 75

818 Spt

S Stt pp

=19554 75

23733 25 4072 252= 0

strong positive correlation higher temperature is associated with higher pressure

M1

A1

A1

M1A1

B1







A3 ID: 6100

A4 ID: 5543

A5 ID: 4366

S xx

S yy

S xy

r

=

=

=

=

68 68 x2 −n

( )x 2

= 9 −8

(80)2= 1

4 33 1187 y2 −n

( )y 2

= 3 −8

(16 3)2= 1

y 50 7 2 3 x −n

( )( )x y= 1 −

880 6 31

= − 1

8972 Sxy

S Sxx yy

=−12 3

168 11871= − 0 M2A1

(a)

(b)

(c) (d)

(e)

(f)

Stt

S vv

S tv

r

explanatory variable reason

b

a

t 3 = 2

=

=

=

=

= :

=

=

0923 24 016 564 t2 −n

( )t 2= 1 −

18400 42 = 2

2 3365 028 v2 −n

( )v 2

= 4 −18

25 212 = 7

v 77 905 17 123 t −n

( )( )t v= 6 −

18400 4 5 212

= 1

984 Stv

S Stt vv

=117 123

2016 564 0287= 0

t because temp can be controlled high correlation

0581 Stt

Stv =117 1232016 564

= 0

108 18

25 21− b

18400 4

= 0

v 108 0581 3 44 = 0 + 0 2 = 1

M1A1

A1

A1

M1A2

M1A1 A1

M1A1

M1A1

A1

Sxx

S yy

S xy

r

=

=

=

=

1068 39 6 x2 −n

( )x 2

= 6 −107782

= 5

3903 050 1 y2 −n

( )y 2

= 5 −107272

= 1

y 6639 8 4 x −n

( )( )x y= 5 −

10778 277

= 7

104 Sxy

S Sxx yy

=78 4

539 6 050 11= 0

B1

A1


http://www.mathsnetalevel.com/examprint.php?q=4366,5543,6100,5560,7109,2477,3136,4372,5932,6099&t=Correlation%20and%20Regression%20Exa… 10/12





A6 ID: 2477

A7 ID: 3136

A8 ID: 4372

(a)

(b)

(c)

correlation coefficient

gradient

equation

x 0 = 8

=

=

:

:

−1

x2 − x1

y2 − y1 =30−3

y (x 0) − 8 =30−3

− 3

y x 1 = −110

+ 1

y = 3unreliable because outside range

B1

M1A1

A1 A1

(a)

(b)

(c) (d)

Sxx

S xy

b

a

Comment x = 3

=

=

=

=

:

80 25 4 225 x2 −n

( )x 2

= 1 −10

(39 5)2= 2

y 429 x −n

( )( )x y= 1 −

1039 5 603

= 7

289 Sxx

Sxy= 7

24 225= 0

y x 289 4 858 − b =10360

− 01039 5

= 3

y 4 86 29x = 3 + 0A typical car will travel 290 miles every year y 4 86 29(3) 5 73 = 3 + 0 = 3mileage 5730 = 3

M1A1

A1

M1A1

M1A1 B1

M1A1

(a)

(b) (c)

Sxx

S xy

b

a

y x = 7

pH = 8pH = 4

=

=

=

=

=

58 75 9688 x2 −n

( )x 2

= 5 −8

(66 5)2= 5

y 7313 87 25 x −n

( )( )x y= 1 −

866 5 9641

= 9

65 Sxx

Sxy=5 9688987 25 = 1

y x 65 403 130 − b =8

1964− 1

866 5

= − 1

130 65x − 1 + 1y 1130 65(7) 8 4 = − + 1 = 2yes; r is high no; extrapolation

M1A1

M1A1 M1A1 B1 B1





A9 ID: 5932

(a)

(b)

(c) (d) (e)

scatter graph

S xx

S xy

b

a

regression line x 7 3 = 1estimate

:

=

=

=

=

:

=

see below

2656 25 07 025 −10

(156 5)2= 2

3758 75 29 675 −10

156 5 25 52= 2

109 Sxx

Sxy=207 025229 675

= 1

y x 109 194 − b =10

225 5− 1

10156 5

= 5

y 194 109x = 5 + 1see below y 194 109(17 3) 4 38 = 5 + 1 = 2unreliable because points widely scattered

B2

B2

B2

B2 B2 B2





A10 ID: 6099

(a)

(b)

(c) (d)

Reason

S xx

S xy

b

a

y x 8 = 2

comment

:

=

=

=

=

=

:

x is controlled

060 78 x2 −n

( )x 2

= 2 −8

(116)2= 3

y 25 1 5 6 x −n

( )( )x y= 2 −

8116 6 61

= − 1

0413 Sxx

Sxy=

378−15 6 = − 0

y x 0 0413 7 − b =8

16 6− −

8116

= 2

7 0413x 2 − 0y 7 0413(28) 1 5million = 2 − 0 = £only just outside given data so reliable

B1

M1A1

M1A1 M1A1 B1

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Whitby Community College - WordPress.comYour account expires on: 8 Nov, 2015 Correlation and Regression Exam questions Syllabus: Edexcel ALevel Questions: 10 Time: 1 hour 33 minutes