1.
Statements I, II, III, IV and V represent descriptions of the correlation between two variables.
I
II
III
IV
V
High positive linear correlation
Low positive linear correlation
No correlation
Low negative linear correlation
High negative linear correlation
Which statement best represents the relationship between the two variables shown in each of the
scatter diagrams below.
(a)
y
10
y
10
8
8
6
6
4
4
2
2
0
(c)
(b)
2
4
6
8
10 x
y
10
0
(d)
8
6
6
4
4
2
2
2
4
6
8
10 x
4
6
8
10 x
2
4
6
8
10 x
y
10
8
0
2
0
Answers:
(a) …………………………………………
(b) …………………………………………
(c) …………………………………………
(d) …………………………………………
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
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
(a)
Find the product moment (Pearson’s) correlation coefficient (r).
(b)
Use your result from part (a) to comment on the statement:
‘Those who do well in Mathematics also do well in English.’
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:
Student
High School grade, x
University grade, 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
(a)
Find the correlation coefficient r, giving your answer to two decimal places.
(b)
Describe the correlation between the high school grades and the university grades.
(c)
Find the equation of the regression line for y on x in the form y = ax + b.
2
4.
Eight students in Mr. O’Neil’s Physical Education class did pushups and situps. Their results
are shown in the following table.
Student
1
2
3
4
5
6
7
8
number of pushups (x)
24
18
32
51
35
42
45
25
number of situps (y)
32
28
38
40
30
52
48
52
The graph below shows the results for the first seven students.
y
60
50
number
of 40
situps
(y) 30
20
10
O
10
20
30
40
50
number of pushups (x)
60
x
(a)
Plot the results for the eighth student on the graph.
(b)
If x = 34 and y = 40 , draw a line of best fit on the graph.
(c)
A student can do 60 pushups. How many situps can the student be expected to do?
3
5.
Several candy bars were purchased and the following table shows the weight and the cost of
each bar.
Weight (g)
Cost (Euros)
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
(a)
Find the correlation coefficient, r, giving your answer correct to 3 decimal places.
(b)
Describe the correlation between the weight of a candy bar and its cost.
(c)
Calculate the equation of the regression line for y on x.
(d)
Use your equation to estimate the cost of a candy bar weighing 109 g.
4
6.
The sketches below represent scatter diagrams for the way in which variables x, y and z change
over time, t, in a given chemical experiment. They are labelled 1 , 2 and 3 .
x
y
1
(a)
× ×
× ×
×× ×× ×
××
× × ×× ×
time t
z
2
× × ×
×
×
× ×
× × ×
×
×× ×
× ×
time t
××
× ×
× ×
××
××
× ××
×
××
3
time t
State which of the diagrams indicate that the pair of variables
(i)
is not correlated;
(ii) shows strong linear correlation.
(b) A student is given a piece of paper with five numbers written on it. She is told that three
of these numbers are the product moment correlation coefficients for the three pairs of
variables shown above. The five numbers are
0.9,
–0.85,
–0.20,
0.04,
1.60
(i)
For each sketch above state which of these five numbers is the most appropriate
value for the correlation coefficient.
(ii)
For the two remaining numbers, state why you reject them for this experiment.
5
7.
It is thought that the breaststroke time for 200 m depends on the length of the arm of the
swimmer.
Eight students swim 200 m breaststroke. Their times (y) in seconds and arm lengths (x) in cm
are shown in the table below.
Length of arm,
x cm
Breaststroke,
y seconds
(a)
1
2
3
4
5
6
7
8
79
74
72
70
77
73
64
69
135.1
135.7
139.3
141.0
132.8
137.0
152.9
144.0
Calculate the mean and standard deviation of x and y.
(b) Find the correlation coefficient, r.
(c)
Comment on your value for r.
(d) Calculate the equation of the regression line of y on x.
(e)
Using your regression line, estimate how many seconds it will take a student with an arm
length of 75 cm to swim the 200 m breaststroke.
6
8.
Tania wishes to see whether there is any correlation between a person’s age and the number of
objects on a tray which could be remembered after looking at them for a certain time.
She obtains the following table of results.
(a)
Age (x years)
15
21
36
40
44
55
Number of objects
remembered (y)
17
20
15
16
17
12
Use your graphic display calculator to find the equation of the regression line
of y on x.
(b) Use your equation to estimate the number of objects remembered by a person aged 28
years.
(c)
Use your graphic display calculator to find the correlation coefficient r.
(d) Comment on your value for r.
9.
The following table shows the cost in AUD of seven paperback books chosen at random,
together with the number of pages in each book.
Book
1
2
3
4
5
6
Number of pages (x)
Cost (y AUD)
(a)
7
50
120
200
330
400
450
630
6.00
5.40
7.20
4.60
7.60
5.80
5.20
Plot these pairs of values on a scatter diagram. Use a scale of 1 cm to represent 50 pages
on the horizontal axis and 1 cm to represent 1 AUD on the vertical axis.
(b) Write down the linear correlation coefficient, r, for the data.
(c)
Stephen wishes to buy a paperback book which has 350 pages in it. He plans to draw a
line of best fit to determine the price. State whether or not this is an appropriate method in
this case and justify your answer.
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