 1.04 Associations between numerical variables

Worksheet
Correlation
1

State whether the following graphs show a positive correlation, a negative correlation, or neither:

a
b
c
d
2

State whether the following graphs show a linear or non-linear correlation:

a
b
c
d
e
3

Identify the type of correlation in the following scatter plots, as either:

• Strong negative correlation

• Weak negative correlation

• No correlation

• Weak positive correlation

• Strong positive correlation

a
b
c
d
e
4

Identify the type of relationship represented by the following scatter plots, as either:

• Positive linear

• Negative linear

• No relationship

a
b
c
5

Identify the type of relationship in the following scatter plots as either:

• Strong positive linear

• No relationship

• Weak positive linear

• Weak negative linear

• Strong negative linear

a
b
6

For each pair of variables, state whether their relationship would have a positive or negative correlation:

a

Time spent studying and exam performance.

b

Rainfall and traffic accidents.

c

Height and weight.

d

Temperature and ice cream sales.

7

Describe the relationship between the variables in the following scatter plots:

a
b
8

Yvonne has a high income but does not like her job. Which dot in the graph would represent her?

9

Scientists conducted a study where each person was asked to read a paragraph and then recount as much information as they can remember. They found that the longer the paragraph, the less information each person could retain.

Is the correlation between the length of the paragraph and the information retained, positive or negative?

10

A database compare the income per capita of a country and the child mortality rate of that country. A sample of the data is shown in the table.

If a scatter plot was created from the entire database, describe the relationship you would expect it to have in terms of strength, direction and shape.

11

After a mathematics exam, Lachlan commented that he left out Question 10 because it was not relevant to the topics that were assessed, but that the other questions were fine. The teacher felt that Question 10 and Question 7 were assessing the same skills, so she decided to look at the results of students across all 8 classes. She plotted the percentage who had gotten Question 10 correct against the percentage who had gotten Question 7 correct as shown below:

a

According to the graph, who is correct: Lachlan or the teacher?

b

Let's assume that Lachlan performed exceptionally well in every other question. If a whole class of students had the same result as Lachlan, in which region would this class’s results appear on the scatter plot?

Scatter plots
12

The following table has data results from an experiment:

a

Construct a scatter plot for this data.

b

Describe the shape of the correlation between the data points as linear or non-linear.

13

A study was conducted to find the relationship between the age at which a child first speaks and their their aptitude test results when they are teenagers. The data is shown in the table below:

a

Construct a scatter plot for this data.

b

Describe the correlation between the variables in terms of strength, direction and shape.

14

A student was performing an experiment to study the relationship between the current and voltage through a resistor. He noted his results in the following table:

a

Construct a scatter plot for this data.

b

Describe the correlation between the variables in terms of shape and direction.

15

The following table shows the time taken to finish a lap on a race track for several average speeds:

a

Create a scatter plot to display this data.

b

Describe the type of correlation between the variables in terms of direction and shape.

16

A group of 10 students sat two tests and wanted to see if there was a relationship between their score in test 1 compared to test 2. Their scores are recorded in the table below.

a

Create a scatter plot for this data, in which the variable on the x axis is Test 1, and the variable on the y axis is Test 2.

b

Describe the correlation between the two sets of scores.

17

The following table shows the number of traffic accidents associated with a sample of drivers in different age groups:

a

Construct a scatter plot for this data.

b

Describe the correlation between a person's age and the number of accidents.

c

Is there an outlier in this data set?

18

The following table shows the average IQ of a random group of people against their height:

a

Construct a scatter plot to represent this data.

b

Describe the relationship between IQ and height.

c

How tall is the person who appears to be an outlier?

19

The marks of 12 students in Maths and Sport were recorded in the following table:

a

Construct a scatter plot for the students' marks in Maths vs their marks in Sport.

b

Describe the correlation between students' marks in Maths and marks in Sport.

c

Which student's scores appear to represent an outlier?

20

A study was conducted to compare running times in various outdoor temperatures. The table below lists the time taken to sprint 400 metres by runners in different temperatures:

a

Construct a scatter plot for this data.

b

How many runners were tested in the study?

c

Describe the correlation between temperature and sprint time as positive or negative.

d

Describe the correlation between temperature and sprint time as strong or weak.

e

Which pair of temperature and sprint time represents an outlier?

21

A researcher is studying the relationship between the number of passers-by near an emergency situation and the time taken until help is offered. The results are shown in the table below:

a

Construct a scatter plot to represent the data in the table.

b

Comment on the relationship between the number of passers-by and the time until assistance is offered by a passer-by to a person in an emergency.

c

Describe the correlation between the number of passers-by and the time until assistance is offered by a passer-by to a person in an emergency.

22

Scientists were looking for a relationship between the number of hours of sleep we receive and the effect it has on our motor and processing skills. Some people were asked to sleep for different amounts of time, and were all asked to undergo the same driving test in which their reaction time was measured. The results are shown in the table:

a

Construct a scatter plot for this data.

b

Describe the relationship between the amount of sleep and reaction time.

23

To determine whether the presence of sharks in a coastal region is influenced by cage diving, the number of nearby cage diving operations and the number of nearby shark sightings was recorded each month over several months. The results are shown in the table:

a

Construct a scatter plot for this data.

b

According to the data, is it possible to determine whether cage diving operations encourage more sharks to come near the shoreline?

c

Describe the relationship between the number of shark sightings and the number of cage diving operations.

24

The market price of bananas varies throughout the year. Each month, a consumer group compared the average quantity of bananas supplied by each producer to the average market price (per unit). The results are shown in the table:

a

Construct a scatter plot for this data.

b

Describe the relationship between the supply quantity and the market price of bananas.

c

What needs to happen for a supplier to receive a high price per banana?

Pearson's correlation coefficient
25

Describe the type of correlation the following correlation coefficients indicate:

a
r = 1
b
r = 0
c
r = -1
26

A pair of data sets have a correlation coefficient of \dfrac{1}{10} while a second pair of data sets have a correlation coefficient of \dfrac{3}{5}. Which pair of data sets have the stronger correlation?

27

If the explanatory variable increases, describe the effect on the response variable for the following studies:

a

A study found that the correlation coefficient between heights of women and probability of being turned down for a promotion was found to be - 0.90.

b

A study found that the correlation coefficient between population of a city and number of speeding fines recorded was found to be 0.83.

c

A study found that the correlation coefficient between length of hair and length of fingernails was found to be 0.07.

d

A study found that the correlation coefficient between number of bylaws a council has about dog breeding and number of dogs available for adoption at the local shelter was found to be 0.55.

28

For each of the following graphs, write down an appropriate value for the correlation coefficient:

a
b
c
d
29

The scatter diagram shows data of the height of a ball kicked into the air as a function of time:

a

Which type of model is appropriate for the data, Linear or Non-linear?

b

Write down a possible value of Pearson’s correlation coefficient, r, for this set of data.

30

The scatter diagram shows data of a person's level of happiness as a function of their age:

a

Which type of model is appropriate for the data, linear or non-linear?

b

Write down a possible value of Pearson’s correlation coefficient, r, for this set of data.

31

The scatter diagram shows data of the height of an object after it is pushed off a rooftop as a function of time:

a

Which type of model is appropriate for the data, linear or quadratic?

b

Write down a possible value of Pearson’s correlation coefficient, r, for this set of data.

32

A climate scientist wishes to investigate whether there is a relationship between the altitude of a city and the average maximum temperature of the city. Data was collected and is shown in the table below:

a

State the explanatory variable in this problem.

b

Construct a scatter plot for this data.

c

Describe the correlation between the two variables.

d

From the following values, select the value that is most likely to be the correlation coefficient for this data.

• 0.75
• -1
• 0.6
• -0.8
e

Can the scientist conclude that the altitude of a city causes the average maximum temperature? Explain your answer.

Outcomes

ACMGM052

construct a scatterplot to identify patterns in the data suggesting the presence of an association

ACMGM053

describe an association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak)

ACMGM056

use a scatterplot to identify the nature of the relationship between variables