Contents
12.1 Two-way tables12.2 Probabilities and two-way tables 12.3 Application of contingency tables Chapter review
Probability
and two-way
tables
Syllabus subject matter
Syllabusguide chapter 12 Exploring and understanding data
■ Interpretation and use of probability as a measure of chance in a range of practical and theoretical situations
■ Interpretation in context of row and column
percentages for a contingency table (two-way table of frequencies)
■ Misuse of probabilities, including
misinterpretation of row and column percentages in contingency tables
■ Use of areas in histograms to estimate
probabilities
12.1
Two-way tables
When we have information about a group that could be presented as two different frequency tables, we can use a two-way table to show more relationships between the data. This is particularly valuable for categorical data, or data that can be grouped into a small number of categories.
The marginal frequencies of a two-way table are normally placed on the right and the bottom of the table. They are the same as the frequencies that would be obtained from a one-way table of each characteristic. The simplest two-way tables have only two categories for each characteristic.
Two-way tables
A two-way table (contingency table) shows data concerning two different characteristics of the same population. One characteristic has frequencies shown in columns and the other has frequencies shown in rows. This is also called
cross-tabulation. Each cell of the table shows the frequency for the particular combination
of the characteristics shown at the side and top of the table.
It is usual to add frequencies across and down the table to obtain marginal frequencies.
!
A market researcher interviewed 50 people about their preferences of ice-cream flavours. Of the 30 women, 20 said they preferred chocolate, while 14 of the men preferred other flavours.
a Construct a two-way table of this information. b How many men preferred chocolate?
Solution
a Show gender vertically and flavour preference horizontally.
Fill in the known information. 20 women preferred chocolate.
14 men preferred other flavours.
There were 30 women altogether. There were 50 people altogether.
Chocolate Other flavour
Male 14
Female 20 30
50
Example
1
Continuous or discrete data with a large number of different scores are grouped into a small number of classes. These are often categorised as ‘low’, ‘medium’ and ‘high’.
Complete the missing cells. There must be 10 ‘Other female’ to add across to 30.
The ‘Other’ total down must be 24.
There must be 20 men to add down to 50.
Complete the remaining cells using the side totals.
Check that all totals for the marginal frequencies on the right and on the bottom agree.
b Read from the completed table. 6 men preferred chocolate flavour.
Chocolate Other flavour
Male 6 14 20
Female 20 10 30
26 24 50
The heights of some Year 12 students and the numbers of computers in their houses are shown below.
Adam 185 cm, 3 Ann 168 cm, 4 Bonnie 157 cm, 2
Charmaine 172 cm, 5 Chris 194 cm, 1 Colin 165 cm, 3 David 178 cm, 2 Elena 176 cm, 1 Harry 169 cm, 0 Helga 158 cm, 3 Janita 162 cm, 4 Lee 183 cm, 2 Mary 164 cm, 1 Michael 167 cm, 2 Mirza 179 cm, 2 Pam 160 cm, 2 Peter 177 cm, 1 Phoong 166 cm, 3 Roberto 185 cm, 4 Robyn 156 cm, 3 Sam 164 cm, 4 Stefan 172 cm, 3 Thomas 167 cm, 2 Xing 155 cm, 1
a Make a two-way table of the information with the height in classes. b Is there any noticeable pattern?
Solution
a Classify height into three classes:
Short: less than 166 cm Average: 166–175 cm Tall: over 175 cm. Put height across and number of computers down. Fill in the table. It is more accurate to do tally marks first, as shown.
b The only noticeable pattern is that there seem to be more tall people with 1 or
2 computers, but this is probably a fluke.
Height (cm)
Short 166
Average 166–175
Tall 175
0 0 1 0 1
1 2 0 3 5
2 2 2 3 7
3 3 2 1 6
4 2 1 1 4
5 0 1 0 1
9 7 8 24
Computers
Height (cm)
Short 166
Average 166–175
Tall 175
0 |
1 | | | | |
2 || | | | | |
3 | | | | | |
4 | | | |
5 |
Computers
Where data is classified in several ways, a multicolumn histogram may be drawn. Such histograms can be constructed from two-way tables, or used to construct two-way tables.
1 Try to categorise subject choices of
students in your class as ‘mainly science’, ‘mainly arts’, ‘mainly commerce’ or ‘other’. Choose some career orientations for next year, such as ‘TAFE’,
‘apprentice’, ‘university’, ‘employment’.
2 Gather information from your class and
draw up a two-way table. You may want to extend your data gathering to another class if it can be arranged by your teacher.
3 Can you make any conclusions from your
table?
4 Discuss the difficulties of classifying the
information in a meaningful way.
Investigation
Subject choices and careers
The histogram below shows the results of a survey of preferred car colours. Use the histogram to construct a two-way table of preferred colours by gender.
Solution
Since the columns show both gender and colours, the table can be constructed with gender across and colour down the table. Each frequency can be read directly from the histogram.
Car colour preference
White 0 5 10
Red Orange 20
15
Number
25 30
Male Female
Yellow Green Blue Purple Black
Male Female
White 21 25 46
Red 9 7 16
Orange 2 5 7
Yellow 15 16 31
Green 6 7 13
Blue 11 12 23
Purple 3 1 4
Black 8 2 10
75 75 150
Exercise 12.1
Two-way tables
1 A two-way table was constructed as shown
at right, but only partially completed.
a How many women were there? b How many singles were there?
2 The occupations of a group of workers were
classified as shown in the table. How many of each of the following were there?
a Blue-collar workers b Female blue-collar workers c Women
d White-collar workers e People
3 Construct a two-way table from the following histogram.
4 Construct a multicolumn histogram from this
two-way table showing the numbers of people from city and country areas who prefer to drive a sedan, station wagon, 4WD or ute.
Modelling and problem solving
5 At a bus-stop at Tarragindi, 30 people got onto a banana bus. All were going to Griffith
Uni or Garden City. Just 4 of the men and 8 of the women were going to Griffith Uni. The remaining 7 men were going to Garden City.
a Construct a two-way table of this information. b How many women were going to Garden City?
6 In a survey, 60 Year 12 students were asked about their preferences in watching football.
Of the 25 girls, 8 preferred to watch AFL, 12 preferred rugby and the others preferred soccer. Of the boys, 20 watched rugby and 6 watched soccer.
a Construct a two-way table of this information. b How many boys preferred to watch AFL?
Male Female
Married 11 28
Single 5 7
Male Female
Blue-collar 16 23
White-collar 28
Service 32 54
64
Source: Choice, September 2001
Reported car breakdowns by make and age
Holden 0
100 200
Ford 400
300
Number
500
1–4 years 5–7 years
Toyota Mitsubishi
8–11 years
City Country
Sedan 40 30
Station wagon 10 12
4WD 15 20
Ute 5 15
Additional exercise
12.2
Probabilities and two-way tables
When we are calculating percentages in order to find probabilities, it is very important to make the basis of the percentage clear. That is, the total that is used in the denominator of the percentage should be clear.When percentages are calculated from values in two-way tables, there are three possibilities: a value can be calculated as a percentage of the total of the row on the right side, the total of the column at the bottom, or the grand total of the whole table.
7 Voters were asked about their feelings about a republic, as well as their voting intentions
for the next election. Of the Labor voters, 28 were in favour of a republic, 17 were against and 4 didn’t care. Of the Liberal/National voters, 25 were in favour, 27 were against and 3 didn’t care. Of those who favoured minor parties, 8 were in favour, 6 were against and 7 didn’t care.
a Construct a two-way table of this information. b How many Labor voters were interviewed? c How many people favoured a republic?
8 Some people in Cairns gave information about their ages and the numbers of brothers and
sisters they had. The following information was collected, where the age is given first and then the number of siblings.
Alanna 28, 2 Allan 55, 1 Andrew 18, 1 Anita 29, 0 Ann-Marie 41, 2 Astrid 43, 1 Bjorn 16, 0 Bruce 24, 2 Carlo 35, 3 Chris 53, 2 Denise 28, 3 Duke 37, 3 Elaine 35, 0 Eleni 34, 1 Jenny 19, 0 Jodie 28, 6 Joy 33, 2 Kandai 34, 2 Kayo 16, 1 Ken 19, 3 Louise 22, 2 Lynne 17, 4 Michael 17, 1 Miguel 22, 2 Paul 34, 2 Peter 35, 0 Rebecca 46, 5 Sandra 58, 4 Shaia 23, 6 Shayne 16, 0 Tom 16, 4 Trudy 44, 0 Wendy 32, 5
a Construct a two-way table of this information using suitable categories for ages. b How many young people had no siblings?
c Is there any noticeable pattern?
What is the implied total for each of the following calculations?
a The percentage of dog-owners who also like cats b The percentage of pet-owners who have a dog c The probability that a cat-owner also has a dog
Solution
a ‘Percentage of’ shows the total. The total is the number of dog-owners.
b ‘Percentage of’ shows the total. The total is the number of pet-owners.
c ‘Probability that’ shows the total. The total is the number of cat-owners.
Two-way probabilities
Probabilities may be calculated as percentages from two-way tables:
• Percentages calculated by row are based on the marginal frequencies on the right sides of the rows.
• Percentages calculated by column are based on the marginal frequencies at the bottoms of the columns.
• Percentages calculated by table total are based on the total of the whole table at the bottom right corner.
!
The table shows the preferences of people for ice-cream flavours. Use the table to find:
a the percentage of females who like
chocolate flavour
b the probability of being a female chocolate lover c the probability that a chocolate lover is female.
Solution
a This is calculated by row, from the total
of females.
Percentage= × 100%
≈ 67%
Write the answer. About 67% of females like chocolate flavour.
b This is calculated by total, from the table
total.
Probability= × 100%
= 40%
Write the answer. The probability of being a female chocolate lover is 40%.
c This is calculated by column, from the total of chocolate lovers.
Probability= × 100%
≈ 77%
Write the answer. The probability that a chocolate lover is female is about 77%.
Chocolate Other
flavour
Male 6 14 20
Female 20 10 30
26 24 50
20 30
---20 50
---20 26
From Example 5, it is clear that is it vital to make sure that the total from which the percentage is calculated is made clear.
In Chapter 6, you saw that probabilities calculated from frequency tables would be unreliable when based on very small frequencies. In the same way, probabilities calculated from two-way tables may be unreliable when based on very small frequencies. The rule of five is applied to two-way tables to ensure that probabilities and expected frequencies can be used with some confidence.
The rule of five is useful when deciding how to categorise data so that meaningful results can be obtained.
Rule of 5
Probabilities calculated using frequencies less than 5 are regarded as unreliable. Cells in two-way tables should be combined to ensure that individual frequencies are greater than 5 before relative frequencies are calculated.
Similarly, expected frequencies less than 5 should be regarded as unreliable.
!
The data below shows the results of a Queensland survey of support for daylight saving. Five people from each of various towns were asked for their opinions.
Roma Y N N N N Ipswich Y Y Y N N Gold Coast Y Y Y Y N Brisbane Y Y Y Y N Cairns Y Y N N N Mt Isa Y N N N N Bamaga N N N N N Nambour Y Y Y Y N Rockhampton Y Y N N N Show this information as a two-way table.
Solution
There are not enough results to classify by each location. Ipswich, the Gold Coast, Brisbane and
Nambour are all in the south-east corner, while the other locations are scattered. Classify in this way and complete the table.
Each cell satisfies the rule of five.
Townsville
Cunnamulla
Rockhampton
Brisbane Charleville
Ipswich Mount Isa
Roma
Gold Coast Nambour
Cairns Bamaga
Cooktown
Daylight saving
For Against
South-east corner 15 5 20
Other locations 6 19 25
21 24 45
When several classifications of data are possible, the choice of classification will usually be influenced by knowledge about the survey question. In Example 6, the locations could have been classified as coastal and inland, but ‘south-east corner’ and ‘other locations’ are used because this reflects a difference in population concentration as well as sunrise/sunset times.
Exercise 12.2
Probabilities and two-way tables
1 For each of the following, state the total for the percentage that is to be calculated. a The percentage interviewed who favour daylight saving and are women b The percentage of women who favour daylight saving
c The percentage of those in favour of daylight saving who are women
d The percentage of teenagers who own pets e The percentage of pet-owners who are teenagers
2 The table below shows the results of a survey of employees’ preferences for the length of
their lunch-break (with appropriate changes to finishing time). Use the table to find the percentage of:
a workshop workers who prefer hour b office workers who prefer hour
c those who prefer hour who are office workers
d all workers who prefer hour and are office
workers
e on-site workers who prefer 1 hour.
3 The table below shows the results of quality testing samples of toys produced by different
production lines in a factory. Use the table to find the percentage of:
a failures that are from line A b line A that fail quality control c total production from line A
d good production that is from line A e line C that fail quality control.
Modelling and application
4 The data below shows the results of a survey of support for oil exploration on the Great
Barrier Reef. People in different states were asked for their opinions.
NSW: Y Y Y N N N N N N Vic: Y Y N N N N N N N Qld: Y Y Y Y N N N N N N N NT: Y Y Y N N N N SA: Y Y N N N N N N WA: Y Y Y Y N N N N N
Tas: Y N N N N ACT: Y N N N N
Show the information as a two-way table, grouping by distance from the Reef.
5 The data below shows the results of a survey of the number of takeaway meals people have
eaten in the last week. People were asked for their age and the number of takeaway meals eaten. The age is given first in each case:
35, 2 55, 1 58, 2 51, 0 39, 1 52, 1 26, 0 52, 2 23, 2 59, 0 24, 3 24, 4 48, 0 41, 4 17, 1 19, 1 32, 3 26, 4 26, 4 59, 3 39, 2 21, 1 65, 2 59, 0 46, 2 41, 1 22, 3 46, 3 56, 3 25, 2 24, 2 26, 4 38, 3 59, 2 17, 7 46, 3 32, 5 31, 4 43, 1 17, 3 Show the information as a two-way table with suitable age grouping.
Lunch-break
hour hour 1 hour
Workshop 14 20 12
On-site 20 14 6
Office 6 8 13
1 2
-- 34 --1
2 ---1 2 ---1
2
---1 2
---Quality control
Pass Fail
Line A 35 6
Line B 42 8
12.3
Application of contingency tables
In medicine, contingency tables are used in research to determine whether some medical conditions may be influenced by other factors. A classic case is the research on the connection between smoking and lung cancer. In this case, the percentages of smokers and non-smokers who contracted lung cancer were tabulated. The result was that the probability of a smoker contracting lung cancer was found to be much higher than the probability of a non-smoker. Such studies do not show that there is a cause and effect relationship, but they do indicate where research should be conducted. In the case of smoking, a chemical was found in tobacco smoke that was shown to actually cause lung cancer.Application of contingency tables
Patterns in contingency tables may be found by comparing marginal percentages with percentages calculated by row or by column. Variation of the percentages indicates a possible conclusion.
Partial tables can sometimes be completed using totals across rows or down columns.
!
A study of the deaths of 10 000 men found that 2500 of them were smokers. The number of men who died of lung cancer was 715, and 630 of these were smokers.
a Construct a contingency table for smokers and lung cancer deaths.
b Calculate the probabilities of dying of lung cancer for smokers and non-smokers. c Compare the risks of dying of lung cancer for smokers and non-smokers.
Solution
a Make rows for smoking and columns
for lung cancer.
Fill in the known values.
Use the totals to complete the table.
b Since we want the probabilities for
smokers and non-smokers, percentages should be calculated by row.
× 100% = 25.2%, etc.
Write the answers. The probabilities of dying from lung cancer are 25.2% for smokers and 1.1% for non-smokers.
Lung cancer
Yes No
Smoker 630 2 500
Non-smoker
715 10 000
Lung cancer
Yes No
Smoker 630 1870 2 500
Non-smoker 85 7415 7 500
715 9285 10 000
Lung cancer
Yes No
Smoker 25.2% 74.8% 100%
Non-smoker 1.1% 98.9% 100%
7.2% 92.8% 100%
630 2500
In Example 7, the row percentages were quite different from the marginal percentages at the bottoms of the columns. This suggests that there are differences in the conditions that affect the results. In some cases, percentages for part of a contingency table may be given, so we need to use the percentages by table total to complete the table.
In pathology testing, a false positive is an incorrect positive result. Similarly, a false
negative is an incorrect negative result. Because of their use as diagnostic tools, pathology
tests are set up so that false positives far outweigh false negatives. This means that doctors can be fairly sure of negative results from these tests, but need to consider all the diagnostic evidence when a pathology test is positive. This is even more marked for rare diseases.
c Compare the probabilities. 25.2% ÷ 1.1% ≈ 23
Write the answer. Based on this study, smokers have nearly 23 times the chance of dying from lung cancer as non-smokers.
Teacher notes
About 2% of people suffer from diabetes at some time, and the test is about 99% accurate.
a Set up a contingency table for test results and disease presence, assuming that the test
accuracy is not affected by the presence of the disease.
b Find the probability that a positive result is false. c Find the probability that a negative result is false.
Solution
a Make rows for results and columns for
diabetes.
Fill in the known value for diabetes. Fill in the ‘No’ value using the table total.
Identify the ‘correct result’ cells.
Fill in these cells.
Use column totals to complete the ‘incorrect result’ cells.
Find the row totals.
b In this case, we want percentages by row.
Recalculate from the row totals.
× 100% ≈ 67%
× 100% ≈ 0.02%, etc.
Calculate the ‘correct result’ probabilities. True positive= 99% of 2%
= 0.99 × 2% =1.98%
True negative= 99% of 98%
= 0.99 × 98% =97.02%
Diabetes
Yes No
Positive
Negative
2% 98% 100%
T
est
Diabetes
Yes No
Positive 1.98% 0.98% 2.96%
Negative 0.02% 97.02% 97.04%
2.00% 98.00% 100.00%
T
est
Diabetes
Yes No
Positive 67.00% 33.00% 100%
Negative 0.02% 99.98% 100%
2.00% 98.00% 100%
T
est
1.98 2.96
---0.02 97.04
Contingency tables can be used to calculate expected frequencies. However, expected frequencies of less than 5 should be regarded as unreliable. In these cases it is probably more correct to say ‘less than 5’ than to give a definite value.
Row and column frequencies and percentages from contingency tables are often
misinterpreted. You should be able to understand and correct misinterpretations of this kind. A positive result is false when you don’t have diabetes.
Write the answer from the table. The probability that a positive result is false is about 33%.
c A negative result is false when you do have diabetes.
Use the percentage by row. The probability that a negative result is false is about 0.02%.
The contingency table for preference of ice-cream flavour and gender, by table total, is shown at right.
a How many from 35 men would you
expect to prefer chocolate flavour?
b From 35 of the same mixture of men
and women, how many male chocolate lovers would you expect?
Solution
a We need the percentages by row for this
calculation, so re-do the table.
× 100% = 30%, etc.
Find the expected number. Expected frequency = 30% of 35
= 0.3 × 35 = 10.5
Write the answer. We would expect about 10 or 11 men to prefer chocolate flavour.
b This is out of the entire group, so we
should use table total percentages.
Expected frequency= 12% of 35
= 0.12 × 35 = 4.2
This is below the rule of 5 number. From 35 people, less than 5 male chocolate lovers would be expected.
Chocolate Other flavour
Male 12% 28% 40%
Female 40% 20% 60%
52% 48% 100%
Chocolate Other flavour
Male 30% 70% 100%
Female 67% 33% 100%
52% 48% 100%
12 40
---Example
9
The contingency table at right was used by a student to conclude that ‘30% of women would vote for paid maternity leave’. Explain why the conclusion is incorrect and find the correct result.
Paid maternity leave
Yes No
Men 15% 35% 50%
Women 30% 20% 50%
45% 55% 100%
Solution
The table is shown with percentages by table total. The 30% actually shows that 30% of people are women in favour of maternity leave.
To find the required value, recalculate by row.
Write the explanation and correct value. The student has confused percentage by row with percentage by table total. The correct conclusion is that 60% of women would vote for paid maternity leave.
Paid maternity leave
Yes No
Men 30% 70% 100%
Women 60% 40% 100%
45% 55% 100%
Modelling and problem solving
1 A study of causes of death in Australia found that, from 54 000 deaths, 9400 resulted from
heart attack. Of those who died from heart attack, 3020 were regarded as obese. A total of 5840 of the 54 000 studied were considered to be obese.
a Construct a contingency table for heart attack and obesity.
b Calculate the probabilities of dying of heart attack for obese and non-obese people. c Compare the risks of dying of heart attack for obese and other people.
2 The HIV test is about 95% accurate, and about 0.1% of the population is believed to have
the virus.
a Set up a contingency table for test results and HIV, assuming that the test accuracy is not
affected by the presence of the disease.
b Find the probability that a positive result is false. c Find the probability that a negative result is false.
3 An over-the-counter pregnancy test kit is claimed to have an accuracy of 90%. About 30%
of women who buy and use the kit are actually pregnant. Use a contingency table to find the probability that:
a a positive result is false b a negative result is false.
4 The contingency table at right shows the results of a check of the payment of accounts by
married and single women at a fitness club.
a Copy and complete the table.
b What is the probability that an account
is paid up and belongs to a single woman?
c What is the probability that an account in arrears belongs to a single woman?
d From 80 married women members, how many would you expect to be in arrears? e From 80 accounts in arrears, how many would you expect to be from married women?
Paid up In arrears
Married 4% 66%
Single
12% 100%
Application of contingency tables
Exercise 12.3
Additional exercise5 The table below shows the percentages of people interviewed about TV-watching
preferences, by age.
a Use the table to find how many people from a similar group of 150 would: i prefer to watch drama
ii be over 50
iii be under 30 and prefer to watch drama. b Find also:
i the expected number who would prefer to watch sport from 50 people aged 41–50 ii the expected number from 90 current affairs enthusiasts who would be aged 31–40. 6 A national survey of attitudes of fans to football codes grouped results into the following
categories:
Australian-born AFL fans: 27% Overseas-born AFL fans: 4% Australian-born rugby fans: 37% Overseas-born rugby fans: 5% Australian-born soccer fans: 18% Overseas-born soccer fans: 9%
A student concluded that:
a soccer fans were twice as likely to be Australian-born as overseas-born b 9% of overseas-born football fans were likely to be fans of soccer c 27% of AFL fans were Australian-born.
Use a contingency table to check these conclusions and find the correct results for any that are incorrect.
Age
30 31–40 41–50 50
Current affairs 12 10 11 14 47
Sport 10 7 8 6 31
Drama 1 3 5 13 22
23 20 24 33 100
Chapter summary
Chapter
Review
Communication and justification
1 Explain what is meant by a contingency table. 2 What are marginal frequencies?
3 If the marginal frequencies of a 2 × 2 two-way table are known, how many cells must be known in order to complete the table?
4 For a two-way table, what is the difference between percentages calculated by row
and percentages calculated by column?
5 What is the purpose of the rule of 5?
6 The row percentages for one row of a particular two-way table are substantially different
from the marginal percentages at the bottom of the table. What does this suggest?
7 What is meant by a false positive in pathology testing?
Knowledge and procedures
8 This partial two-way table shows the numbers
of students in a first-year economics course at the University of Queensland:
a How many males were from the
northside?
b How many students were from the southside?
9 Construct a two-way table from the following histogram showing the caffeine and
herbal contents of some popular ‘energy’ drinks.
10 For each of the following, state the total for the probability or percentage that is to be
calculated.
a The percentage of the crowd at a Marconi–Brisbane game who are Strikers fans b The probability that a randomly chosen teenager from Kalkadoon SHS has a pet dog 11 The table below shows the results of a survey about the preferences of some Year 12
students for Gamecube, Playstation 2 and X-Box. Use the table to find the percentage of:
a females who prefer Gamecube b students who prefer Gamecube and
are female
c Gamecube fans who are female
d males who prefer X-Box
e X-Box enthusiasts who are male.
Ex 12.1 Ex 12.1 Ex 12.1 Ex 12.2 Ex 12.2 Ex 12.3 Ex 12.3 Ex 12.1
Male Female
Southside 210
Northside 650
540 630
Home
Ex 12.1
Contents of energy drinks
None 0 1 2
Low Medium 4
3
Number
5 Herbs
No herbs
High
Caffeine content
Ex 12.2
Ex 12.2
Male Female
Gamecube 28 18
Playstation 2 17 24
Modelling and problem solving
12 In a nursing home, 70 out of 240 women aged over 65 suffered from osteoporosis. From
answers to a questionnaire about their lifestyles between the ages of 20 and 60, it was found that 60 of the women had had very active lifestyles, including 10 of those with osteoporosis. The lifestyles of the remainder were classified as sedentary.
a Construct a contingency table for this information.
b How many of those with sedentary lifestyles developed osteoporosis?
c Find the probabilities of developing osteoporosis for those with very active and sedentary lifestyles.
d Compare the probabilities of developing osteoporosis for women with very active
and sedentary lifestyles.
13 The information below shows the results of a survey of the amount of homework that
high-school students say they do each night. The year is given first, followed by the time in minutes.
11, 110 9, 40 8, 50 9, 30 12, 140 9, 20 11, 140 8, 10 9, 70 8, 10
9, 30 11, 30 10, 10 8, 50 10, 80 11, 150 9, 80 10, 20 12, 70 12, 70
10, 70 11, 110 8, 30 10, 90 8, 50 12, 80 11, 50 9, 80 11, 120 12, 180
8, 20 11, 70 9, 20 8, 10 11, 120 12, 20 11, 80 11, 40 12, 30 8, 30
11, 140 10, 40 12, 130 9, 90 9, 90 12, 160 9, 30 8, 50 11, 150 10, 30
12, 120 8, 20 10, 60 12, 120 10, 30 12, 110 10, 90 12, 60 8, 10 12, 100
8, 20 9, 70 12, 180 8, 30 9, 80 8, 40 9, 80 12, 40 12, 90 8, 30
11, 40 9, 60 10, 30 12, 150 9, 70 12, 60 11, 150 10, 30 9, 70 11, 40
a Construct a two-way table of this information using suitable categories for the time. b How many Year 8s spent a lot of time on homework?
c What percentage of Year 12s spent little time on homework?
14 A test for a rare blood disorder has an accuracy of 98% and 0.4% of the population have
the disease. Use a contingency table to find the probability that:
a a positive result is false b a negative result is false.
15 Girls outnumbered boys 60% to 40% at a large Brisbane secondary school. Of the girls,
3% were classified as very tall, while 7% of the boys were classified as very tall. Construct a contingency table to find:
a the probability that a student selected at random is very tall b the probability that a very tall student from the school is male
c the expected number of very tall male students from 3 busloads of students (180 students) from the school.
16 A survey of hair-washing habits found the following:
40% of those surveyed were boys.
10% of the boys surveyed used conditioner after shampooing their hair. 70% of the girls surveyed used conditioner after shampooing their hair.
A student concluded that this meant that the probability that a student used conditioner was 80%.
After the teacher said that he should think again, the student said that the probability must be the average of the boys’ and girls’ usages, so it would be 40%. Find and explain the correct probability.
Ex 12.1, 3
Ex 12.2, 3
Ex 12.3 Ex 12.3
