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1.03 Associations between categorical variables

Worksheet
Explanatory and response variables
1

Identify the explanatory and response variable in each pair given below:

a

Gender, Favourite dog breed

b

Child's eye colour, Parent's eye colour

c

Country of birth, Language spoken at home

d

Favourite ice cream flavour, Birth place

2

In a study on car accidents, the response variable was "type of injury". Give an example of a possible explanatory variable.

3

In a study on professional careers, the response variable was "income bracket". Give an example of a possible explanatory variable.

4

In a study on sporting injuries, the explanatory variable was "type of sport". Give an example of a possible response variable.

5

In a study on houses, it was found that an explanatory variable was the amount of money spent on renovations. Give an example of a possible response variable.

6

In a study on spending habits, the explanatory variable was "type of job". Give an example of a possible response variable.

7

A graph is to display the relationship between the following variables:

  • Fitness level

  • Time spent exercising

Which of these variables should be on the horizontal axis of the graph?

8

Consider the following variables:

  • Ticket sales

  • Revenue from a show

a

State whether the following statements are correct:

i

The revenue made from a show affects ticket sales.

ii

Ticket sales affect the revenue made from a show.

b

Which is the explanatory variable and which is the response variable?

9

For each pairs of variables, state which variable directly affects the other:

a

Arm length (cm), Weight (kg)

b

Temperature (\degreeC ), Distance from the equator (km)

c

Weekly income, Number of friends

d

Time of travel (minutes), Distance covered (m)

e

Age (years), Time spent watching television

f

Time spent working, Wages earned

g

Time spent to finish a novel, Number of pages

h

Marital status, film preference

i

Height (cm), Number of children

j

Hair Length, Gender

k

Number of pets, Time spent caring for pets

l

Time spent training, Athletic performance.

10

The scatter plot shows the relationship between sea temperature and the amount of healthy coral:

a

Which variable is the response variable?

b

Which variable is the explanatory variable?

11

The following graph shows the height of a ball after it is dropped off the side of a building:

a

Which variable is the response variable?

b

Which variable is the explanatory variable?

\text{Time}
\text{Height}
Two-way tables
12

36 students were asked whether or not they were allergic to nuts and dairy. The two-way table is provided below.

Allergic to nutsNot allergic to nuts
Allergic to dairy106
Not allergic to dairy614
a

State the number of students that are allergic to nuts.

b

State the number of students that are allergic to nuts or dairy, or both.

c

State the number of students that are allergic to at most one of the two things.

13

150 tennis players were asked whether they would support equal prize money for the women’s and men’s draw.

a

Complete the given table.

b

How many more players are there in support of equal prize money than against it?

c

Find the percentage of the male tennis players that support equal prize money. Round your answer to one decimal place if necessary.

SupportDo not support
Males35
Females6612
14

This table describes the departures of flights out of an airport for the months of May and June.

a

Calculate the percentage of flights in June that were delayed, correct to one decimal place.

Departed on timeDelayed
May14421
June11640
b

Calculate the fraction of the total number of flights during the two months that departed on time in May.

c

Calculate the percentage of flights in June that were on time, correct to one decimal place.

15

Two motels promoted packages for 1 night and 2 night stays. Results on the number of packages sold are recorded in the table:

a

Find the percentage of the stays that are 2 night stays, correct to two decimal places.

b

Find the fraction of the two night stays that were in Motel X.

c

Find the percentage of the total accommodation that was in Motel X, correct to two decimal places.

One night stayTwo night stays
\text{Motel }X463266
\text{Motel }Y804817
16

A vet has a total of 28 pets visit their practice in a day.

The pets are categorised based on whether they have been vaccinated and whether they have been microchipped as shown in the Venn diagram:

Complete the following table:

MicrochippedNot Microchipped
Vaccinated
Not vaccinated
17

Sophia asked some people in her community whether they were vegetarian or not.

29 responders said they were vegetarian, of which 8 were children. 14 children said they were not vegetarian, and 11 adults said they are not vegetarians.

a

Complete the given two-way table based on the results of Sophia's survey:

b

What proportion of responders are vegetarian?

c

What proportion of adults are vegetarian?

Not vegetarianVegetarian
Children
Adults
18

Glen surveyed all of the students in Year 12 at his school and summarised the results in the following table:

Play netballDo not play netball
\text{Height} \geq 170 \text{ cm} 62138
\text{Height} < 170 \text{ cm} 60140
a

Which variable is the explanatory variable?

b

Create a row percentage frequency table for this data. Round the values to the nearest percentage.

19

Members of a gym were asked what kind of training they do. Each of them only did one kind of training. The table shows the results:

CardioWeight
Male1535
Female228
a

Which variable is the explanatory variable?

b

Create a row percentage frequency table for this data. Round the values to the nearest percentage.

c

Does there appear to be an association between the type of training and the gender of gym members? Explain your answer.

d

Does a person’s gender cause them to choose a certain type of training? Explain your answer.

20

An online streaming service was interested in the viewing habits of high school students and whether there was any association between year group and preferred genre. 25 students were chosen at random from each year group at a particular school and asked the following questions:

  • Question 1: What is your favourite genre? Cartoons, Reality TV, Movies or Documentaries

  • Question 2: Which year group are you in?

a

Which variable, genre or year group, represents the explanatory variable in this problem?

b

The results of the survey are recorded in the table below.

CartoonsReality TVMoviesDocumentariesTotal
Year 71553A25
Year 81354325
Year 91145525
Year 109B6725
Year 11646925
Year 1213111025
TotalC243536D

Find the values of:

i
A
ii
B
iii
C
iv
D
c

Calculate the percentage of:

i

Year 7 students who watch cartoons.

ii

Year 12 students who watch cartoons.

d

Comment on the percentage of Year 7 and Year 12 students who watch cartoons.

Stacked column graphs
21

170 people were surveyed about their music preference. The results have been recorded in the table below:

Music PreferenceMaleFemaleTotal
\text{Rock and Roll}241943
\text{Classical}81523
\text{Pop}171734
\text{Rap}628
\text{Country and Western}172441
\text{R and B}6915
\text{Punk}426
\text{Total}8288170
a

State the explanatory variable for this data set.

b

Which of the following 100\% stacked column charts should be used to look for an association between the variables?

A
B
c

Does the stacked column chart suggest that there is an association between music preference and gender? Explain your answer.

22

A group of year 12 students surveyed their class and recorded the hair colour and eye colour for each student. The results are displayed in the 100\% stacked column chart shown:

a

State the explanatory variable for this chart.

b

Does the chart suggest an association between eye colour and hair colour? Explain your answer.

c

Can we say that having blue eyes causes a high chance of having blonde hair? Explain your answer.

23

The local library constructed divided bar graphs showing the last six months of book loans by category:

Assume that the total number of loans was exactly the same for each month.

a

In which month did subscribers borrow the most business books?

b

Which month was the best month for children's books?

c

State the response variable for this data.

d

Does the divided bar graph suggest that there is an association between the variables? Explain your answer.

24

This stacked column chart shows the fitness levels for various categories of smokers:

a

State the response variable for this chart.

b

Does there appear to be an association between fitness level and category of smoker?

c

Describe this association.

25

25 students were chosen at random from each year group at a particular school and asked the following questions:

  • Question 1: What is your favourite genre? Cartoons, Reality TV, Movies or Documentaries

  • Question 2: Which year group are you in?

The results are displayed in the divided column graph below:

Using the graph above or another method, describe the association between:

a

Cartoons and Year Group.

b

Documentaries and Year Group.

c

Genre and Year Group

26

Weekend workers at a fast food store were surveyed as to which pathway they followed at school (ATAR or Non-ATAR) and the type of phone (IPhone, Samsung or Other). The results are displayed in the column graph below.

a

Complete the frequency table, for the above data:

iPhoneSamsungOtherTotal
ATAR25
non-ATAR27
b

A 100\% stacked column graph is to be created that can be used to help determine if there is a relationship between the variables. Which variable (School Pathway or Type of Phone) should be plotted on the horizontal x axis?

c

Choose the statement that best describes the association between School Pathway and Type of Phone.

A

It appears that there is an association. Students following an non-ATAR pathway are more likely to have an iPhone.

B

There appears to be no association between the variables. The numbers in each category of phone are similar.

C

It appears that there is an association. Students following an ATAR pathway are more likely to have an iPhone.

D

It appears that there is an association. Students following an ATAR pathway are more likely to have a Samsung phone.

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Outcomes

ACMGM049

construct two-way frequency tables and determine the associated row and column sums and percentages

ACMGM050

use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association

ACMGM051

describe an association in terms of differences observed in percentages across categories in a systematic and concise manner, and interpret this in the context of the data

ACMGM055

identify the response variable and the explanatory variable

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