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6. Statistics
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Middle Years

6.04 Correlation coefficient

Lesson
Worksheet
Practice
Worksheet
Pearson's correlation coefficient
1

Describe the type of linear relationship between the variables in the following scatter plots:

a
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y
b
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c
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y
d
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e
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2

Estimate the value of the correlation coefficient for the data in the following scatter plots:

a
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x
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y
b
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x
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c
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d
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y
e
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3

Describe the linear relationship between the variables with the following correlation coefficients:

a

0.96

b

0.66

c

0.36

d

- 0.06

e

- 0.34

f

- 0.66

4

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.

5

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

6

Describe the type of correlation the following correlation coefficients indicate:

a

r = 1

b

r = 0

c

r = -1

7

For each of the following graphs, write down an estimate of the correlation coefficient:

a
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x
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y
b
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x
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y
c
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y
d
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y
8

For each of the following pairs of relationships:

i

Determine whether the two linear relationships have the same direction.

ii

State the relationship that has the stronger correlation.

a

The linear relationship between a set of data for variables x and y has a correlation coefficient of 0.3. The linear relationship between a set of data for variables x and z has a correlation coefficient of 0.9.

b

The linear relationship between a set of data for variables x and y has a correlation coefficient of - 0.9. The linear relationship between a set of data for variables y and z has a correlation coefficient of - 0.5.

c

The linear relationship between a set of data for variables x and y has a correlation coefficient of 0.8. The linear relationship between a set of data for variables s and t has a correlation coefficient of - 0.5.

d

The linear relationship between a set of data for variables x and y has a correlation coefficient of 0.3. The linear relationship between a set of data for variables x and t has a correlation coefficient of - 0.9.

9

A researcher plotted the life expectancy of a group of men against the number of cigarettes they smoke a day. The results were recorded and the correlation coefficient r was found to be - 0.88.

Describe the correlation between the life expectancy of a man and the number of cigarettes smoked per day.

10

A researcher was evaluating the relationship between the number of years in education a person completes and the number of pets they own. The results were recorded and correlation coefficient r was found to be - 0.3.

Describe the correlation between a person's years of education and the number of pets they own.

11

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.

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t
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\text{Height}
12

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.

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\text{Age}
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\text{Happiness}
13

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.

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x
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y
Correlation vs causation
14

Determine whether the following describe a relationship that is correlated but not causal:

a

The sales of ice cream and increase in temperature.

b

The number of hours worked and how much money is made for a given person.

c

The amount of showers had in a day and the amount of the water bill.

d

The amount of rainfall received, and level of water in a lake.

e

The larger the dimensions of a rectangular verandah, the more area.

f

The season of the year and the number of water related injuries.

g

Increase in temperature, and the level of mercury in a thermometer.

h

The number of students shouting in class and the number of detentions received.

15

For each of the following data examples, determine if there is a causal relationship between the variables:

a

The number of times a coin lands on heads and the likelihood that it lands on heads on the next flip.

b

The amount of weight training a person does and their strength.

16

Determine whether the following describe a causal relationship and not just a correlation:

a

An individual's decision to work in construction and his diagnosis of skin cancer.

b

The number of minutes spent exercising and the amount of calories burned.

c

A decrease in temperature and the increase in attendance at an ice skating rink.

d

As a child's weight increases, so does her vocabulary.

17

Determine whether the following are examples of variables with no correlation:

a

The age of a child and their shoe size.

b

The age of a child and their height.

c

The age of a child and the number of pets owned.

d

The age of a child and the amount of adjectives learned.

18

The table shows the number of fans sold at a store during days of various temperatures:

\text{Temperature (\degree C)}68101214161820
\text{Number of fans sold}1213141718192123
a

For this data, will r be greater than zero, less than zero or equal to zero?

b

Is there a causal relationship between the variables? Explain your answer.

19

A study found a strong positive association between the temperature and the number of beach drownings.

a

Does this mean that the temperature causes people to drown? Explain your answer.

b

Is the strong correlation found a coincidence? Explain your answer.

20

Many trees lose their leaves in winter. Does this mean that cold temperatures cause the leaves to fall?

21

A study found a strong correlation between the approximate number of pirates out at sea and the average world temperature.

a

Does this mean that the number of pirates out at sea has an impact on world temperature?

b

Is the strong correlation found a coincidence? Explain your answer.

c

If there is correlation between two variables, is there causation?

22

Scatter plots for two sets of data are shown below:

Set A

x
y

Set B

x
y
a

Which data set has the strongest linear correlation between the variables?

b

Which data set appears to have a non-linear relationship? Explain your answer.

c

Why does data set B have the weakest correlation between its variables?

d

State whether the following pairs of variables could be represented by data set B:

i

Marks in an English examination and distance travelled from home to school.

ii

Cost of cars and cost of petrol.

iii

Distance travelled in a car and the cost of a driver’s license.

iv

Height and weight of students at school.

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