# 1. Type Fast

```1.
The Type Fast secretarial training agency has a new computer software spreadsheet package. The agency
investigates the number of hours it takes people of varying ages to reach a level of proficiency using this
package. Fifteen individuals are tested and the results are summarised in the table below.
Age
(x)
32
40
21
45
24
19
17
21
27
54
33
37
23
45
18
Time
(in hours)
(y)
10
12
8
15
7
8
6
9
11
16
t
13
9
17
5
(a)
(i)
Given that Sy = 3.5 and Sxy = 36.7, calculate the product-moment correlation coefficient r for
this data.
(4)
(ii)
What does the value of the correlation coefficient suggest about the relationship between the
two variables?
(1)
(b)
Given that the mean time taken was 10.6 hours, write the equation of the regression line for y on x
in the form y = ax + b.
(3)
(c)
Use your equation for the regression line to predict
(i)
the time that it would take a 33 year old person to reach proficiency, giving your answer
correct to the nearest hour;
(2)
(ii)
the age of a person who would take 8 hours to reach proficiency, giving your answer correct
to the nearest year.
(2)
(Total 12 marks)
IB Questionbank Mathematical Studies 3rd edition
1
2.
Ten students were given two tests, one on Mathematics and one on English.
The table shows the results of the tests for each of the ten students.
Student
A
Mathematics (x)
8.6
English (y)
33
(a)
B
C
13.4 12.8
51
30
D
E
F
G
H
I
J
9.3
1.3
9.4
13.1
4.9
13.5
9.6
48
12
23
46
18
36
50
Given sxy (the covariance) is 35.85, calculate, correct to two decimal places, the product moment
correlation coefficient (r).
(6)
(b)
Use your result from part (a) to comment on the statement:
‘Those who do well in Mathematics also do well in English.’
(2)
(Total 8 marks)
IB Questionbank Mathematical Studies 3rd edition
2
3.
Ten students were asked for their average grade at the end of their last year of high school and their average grade at the end of
their last year at university. The results were put into a table as follows:
(a)
Studen
t
High School
y
1
2
3
4
5
6
7
8
9
10
90
75
80
70
95
85
90
70
95
85
3.2
2.6
3.0
1.6
3.8
3.1
3.8
2.8
3.0
3.5
Total
835
30.4
Given that sx = 8.96, sy = 0.610 and sxy = 4.16, find the correlation coefficient r, giving your answer
to two decimal places.
(2)
(b)
Describe the correlation between the high school grades and the university grades.
(2)
(c)
Find the equation of the regression line for y on x in the form y = ax + b.
(2)
(Total 6 marks)
IB Questionbank Mathematical Studies 3rd edition
3
4.
Several candy bars were purchased and the following table shows the weight and the cost of each bar.
Weight (g)
Cost (Euros)
(a)
Yummy
Chox
Marz
Twin
Chunx
Lite
BigC
Bite
60
85
80
65
95
50
100
45
1.10
1.50
1.40
1.20
1.80
1.00
1.70
0.90
Given that sx = 19.2, sy = 0.307 and sxy = 5.81, find the correlation coefficient, r, giving your
answer correct to 3 decimal places.
(2)
(b) Describe the correlation between the weight of a candy bar and its cost.
(1)
(c)
Calculate the equation of the regression line for y on x.
(3)
(d)
Use your equation to estimate the cost of a candy bar weighing 109 g.
(2)
(Total 8 marks)
IB Questionbank Mathematical Studies 3rd edition
4
5.
A number of employees at a factory were given x additional training sessions each. They were then timed
on how long (y seconds) it took them to complete a task. The results are shown in the scatter diagram
below. A list of descriptive statistics is also given.
14
time taken (seconds)
12
10
8
6
4
2
0
2
4
6
8
10
n = 9,
sum of x values:  x = 54,
sum of y values:  y = 81,
mean of x values: x = 6,
mean of y values: y = 9,
standard deviation of x: sx = 1.94,
standard deviation of y: sy = 2.35,
covariance: sxy = –3.77.
(a)
Determine the product-moment correlation coefficient (r) for this data.
(2)
(b)
What is the nature of the relationship between the amount of additional training and the time taken
(2)
n
(c)
Calculate
 (x
i 1
i
– x )( yi – y ) given that the covariance sxy = –3.77.
(1)
(d) (i)
(ii)
Determine the equation of the linear regression line for y on x.
Find the expected time to complete the task for an employee who only attended three
(4)
(Total 9 marks)
IB Questionbank Mathematical Studies 3rd edition
5
Solutions
1.
(a)
(i)
Sx = 11.2
36.7
r=
(M2)
11.2  3.5
= 0.936 (3 s.f.)
(A1)
(A1)
OR
Sx = 11.6
36.7
r=
11.6  3.5
= 0.904 (3 s.f.)
(A1)
(M2)
(A1)
(ii) The correlation coefficient suggests a strong positive correlation
between the two variables.
(R1)
_
(b)
(c)
_
Sxy
(
x

x
)
( Sx) 2
36.7
y – 10.6 =
(x – 30.4)
11.2 2
y = 0.293x + 1.69 (or y = 0.293x + 1.71) Allow ft from (a) (i))
5
y– y
(i)
(ii)
(M1)
(A2)
y = 0.293 × 33 + 1.69
= 11.359
= 11 hours
(M1)
8 = 0.293x + 1.69
x = 21.54
= 22 years
(M1)
3
(A1)
(A1)
4
[12]
2.
(a)
For applying r =
S xy
S x S y 
or any correct formulae.
For: Sx = 4.0034568…  4
(M2)
Sy = 13.992456…  14 (M2)
r = 0.6399706…  0.64 (2 d.p.) (A1)
(M1)
6
Note: Follow through with candidate’s Sx + Sy.
Accept solutions that use the unbiased estimates for the population
standard deviations.
(b)
This indicates that there is a degree of positive correlation between scores in Mathematics and
scores in English.
(R1)
Note: Follow through using candidate’s v,2
and  52% (v) from table.
Therefore those who do well in Mathematics are likely to do well in English also. (Or equivalent
statements.)
(R1)
2
[8]
IB Questionbank Mathematical Studies 3rd edition
6
3.
(a)
r=
S xy
SxSy
4.16
(8.96)(0.610)
= 0.76
=
(M1)
(A1)
2
(b)
There is a fairly strong positive correlation between high school
(A1) (A1)
2
Note: Award (A1) for strong (or fairly strong) or high, (A1) for positive.
(c)
y– y 
_
S xy
S x2
_
( x  x)
4.16
(x – 83.5)
8.96 2
y = 0.052x – 1.29 (3 s.f.)
Note: Award (C2) for correct answer (from calculator).
y – 3.04 =
(M1)
(A1)
2
[6]
4.
(a)
r=
S xy
(S x S y )

5.81
(19.2  0.307)
= 0.986
(M1)
(A1)
2
1
Note: Award (G2) for 0.985 from GDC.
(b)
Strong, positive correlation
(A1)
(c)
y = 0.182 + 0.0158x
(G3)
OR
5.81
(x – 72.5)
19.2 2
y = 0.0158x + 0.182
y – 1.325 =
(d)
y = 0.0158 × 109 + 0.182
= 1.90 euros.
(M1)(A1)
(A1)
3
(M1)
(A1)
2
[8]
5.
(a)
r= 
3.77
= –0.827
1.94  2.35
or (G2)
(b)
(M1)(A1)
2
moderate/strong (allow approximately linear)
(A1)
negative
(A1)
2
Note: Comments such as: number of sessions increases as time decreases
can be awarded (A1)(A0) but “inversely proportional” receives no
marks.
IB Questionbank Mathematical Studies 3rd edition
7
n
(c)
 (x
i
 x)( y i  y ) = –3.77 × 9 = –33.9
(A1)
1
i 1
(d)
(i)
(y – 9) = 
3.77
(x – 6)
1.94 2
y = –x + 15
(ii)
–3 + 15 = 12 seconds
(M1)
(A1)
or (G2)
(M1)(A1)
or (G2)
4
[9]
IB Questionbank Mathematical Studies 3rd edition
8
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