VCE General Mathematics Further
Unit 2 2014
Bivariate Data SAC
Name: _______________________________
Total:
/ 35marks
Conditions:
(
%)
45 min, Casio Calculator and Double sided handwritten Summary Sheet permitted
Section A: Multiple Choice ( 6 x 2 = 12 marks)
1. In which of the scatterplots below would the relationship between the variables be best described
as moderate negative
A
D
2.
c
B
E
The Pearson’s product moment correlation coefficient
𝒓 = 𝟎. 𝟖𝟕𝟎𝟓𝟔
Written as a percentage, the coefficient of determination is closest to:
A
B
C
D
E
75.8%
0.758%
0.871%
87.1%
75%
3. In which one of the following situations would there most likely be a negative correlation between
the two variables
A
B
C
D
E
The number of days above 250 over summer holidays and the number of times a teenager goes swimming
The number of hours spent training for a 2km race and the actual time taken to run the 2 km on the day
Student shoe sizes and their test scores on a Maths test
An adult weight and their waist measurement in cm
A child’s height and their reading age
4. A student doing research believes that there may be a mathematical correlation between
a person’s total income AND the amount they spend on holidays.
She collects some data from a group of adults, draws a scatterplot of the data and then constructs
the least squares regression line.
Which one of these equations could be the line of best fit between these two variables?
( hint: first identify the independent variable and the dependent variable )
5.
+ 0.02 x holiday expenditure
A
annual income = 1000
B
annual income = 1000 − 0.02 x holiday expenditure
C
holiday expenditure = 1000
D
holiday expenditure = 1000 − 0.02 x annual income
E
holiday expenditure = − 0.02 x annual income
+ 0.02 x annual income
Consider this data that compares the height of 6 teenage boys with the heights of their mothers.
Enter this data in your calculator and extract any relevant information
Height of mothers
M
154
159
168
170
175
176
Height of sons
S
164
167
172
180
180
187
Which of these statements can be made about this data?
6.
A
There is a strong negative correlation between these two variables
B
The equation of the line of best fit is 𝑴 = 𝟎. 𝟗𝟓 𝑺 + 𝟏𝟕. 𝟎𝟒
C
89.5% of the variation in S values can be explained by variation in M values
D
94.6% of the variation in S values can be explained by variation in M values
E
When a mother is tall, this will result in her son being tall
For a set of bivariate data, involving the variables x and y, the coefficient of determination 𝒓𝟐 =
The least squares regression line ( line of best fit ) is given by the equation 𝒚 = −𝟎. 𝟐𝟐𝒙 + 𝟖. 𝟕𝟓
The value of Pearson’s coefficient r for this set of data, correct to 2 decimal places, would be
A
B
C
D
E
−0.22
0.80
8.75
−0.80
0.41
𝟎. 𝟔𝟒
Section B: Short Answer Section:
Show all relevant work steps. All answers to 2 decimal places
1.
Consider this table of bivariate data
x
y
a)
.
15
148
18
161
19
125
23
138
25
149
30
90
32
127
36
105
Complete a scatterplot on the grid provided, labelling and scaling each axis carefully.
b)
Find Pearson’s correlation coefficient ( 𝒓 ) for this data
c)
Describe the form, strength and shape of the correlation as shown on your scatterplot.
d)
Using the Coefficient of Determination, write a statement explaining the relationship
between the two variables.
e)
Write the equation of the least squares regression line ( i.e. the line of best fit ) for this data.
f)
Using this equation, predict the y value when 𝑥 = 27
( 2 + 1 + 2 + 2 + 1 + 1 = 9 marks )
2.
A set of data comparing two variables x and y has the following summary statistics
̅ = 𝟔. 𝟑𝟕𝟓 𝒂𝒏𝒅 𝒚
̅ = 𝟏𝟕. 𝟑𝟕𝟓
𝒓 = −𝟎. 𝟗𝟖𝟓𝟓 𝑺𝒙 = 𝟑. 𝟑𝟑𝟓 , 𝑺𝒚 = 𝟓. 𝟖𝟐𝟗 , 𝒙
Find the equation of the least squares regression line ( i.e. the line of best fit )
( 3 marks )
3.
The government statistician declares there is a moderate positive correlation between the cost of
petrol at the bowser (C )and the exchange rate between the Australian dollar and the US dollar ( E ).
eg on a particular day, petrol costs 147.0 cents a litre and one Aussie dollar buys 93.0 US cents
He calculates the line of best fit between the variables has the equation
a)
𝐂 = 𝟎. 𝟔𝟕𝟓 𝐄 + 𝟖𝟒. 𝟐𝟐𝟓
If the exchange rate on a particular day is 98.3 cents US ( to buy $1 Aus ), use this equation
to estimate the cost of petrol on that day.
b) If the exchange rate on another day rises to 103.4 cents US ( to buy $1 Aus ), what might
you expect to pay to fill up an 80 litre tank with petrol?
( 2 + 3 = 5 marks )
4.
Badlands finished on the bottom of the ladder in a country football league in 2014. They appoint a
new coach for 2015.
He decides to analyse some of the team statistics of the 8 teams from 2014 in an attempt to explain
the success of the better teams. He understands that their success is probably based on a wide range
of factors, but he is keen to see if there are any obvious strong correlations between certain team
statistics and their winning records. Two of his findings are shown in the tables below
WINS compared to average tackles per game
Team ladder
in 2014
List 1
Average tackles
per game ( T)
Bullato
Kooribura
Lascelles
Smithtown
5. Whakem South
Mountainvale
Ridgefield
Badlands
85
92
96
78
80
55
66
48
List 2
Number of
wins ( W )
16
14
12
10
8
4
3
1
6.
WINS compared to % of players over 180cm
Team ladder
in 2014
Bullato
Kooribura
Lascelles
Smithtown
Whakem South
Mountainvale
Ridgefield
Badlands
List 3
Percentage of
players over
180cm (P)
52%
58%
24%
44%
36%
32%
38%
36%
List 2
Number of
wins ( W )
16
14
12
10
8
4
3
1
Enter the data into your calculator, setting T as List 1, W as List 2 and P as list 3
a) Comparing the data in Lists 1 and 2, find Pearson’s Correlation Coefficient between the
average number of tackles per game and the number of wins during the season.
Comment on the correlation between these two variables.
b) Comparing data in Lists 3 and 2, find Pearson’s Correlation Coefficient between the
percentage of players over 180cm and the number of wins during the season. Comment
on the correlation in between these two variables.
c) Using the two Correlation Coefficients you have found and the two resulting Coefficients
of Determination, write a statement suggesting to the new coach which of the two team
variables ( average tackles per game or percentage of players over 180cm ) appears to
be more statistically significant when compared to the number of games won. Explain
your answer fully.
( 2 + 2 + 2 = 6 marks )