Maths cambridge textbook year 11 chapter 7 PDF

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This is the Maths Cambridge textbook for year 11 chapter 7. this chapter covers all types of maths...

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7

Classifying and representing data

Syllabus topic — S1.1 Classifying and representing data This topic involves the planning and management of data collection, the classification of the data and then the representation of the data in tables and graphs.

Outcomes • • • • • • • • • • • •

Describe the distinguishing features of a population and a sample. Investigate data collection methods for both samples and populations. Classify data as numerical or categorical. Class categorical data as nominal or ordinal. Classify numerical data as discrete or continuous. Organise and display data in dot plots and stem-and-leaf plots. Create frequency tables to organise ungrouped and grouped data. Calculate the cumulative frequency from a grouped frequency table. Draw frequency and cumulative frequency graphs. Organise and display data in Pareto charts. Construct and interpret tables and graphs related to motor vehicles and water. Calculate the fuel consumption and running costs of a vehicle.

Digital Resources for this chapter In the Interactive Textbook: • Videos • Literacy worksheet • Quick Quiz • Widgets • Spreadsheets • Study guide

• Solutions (enabled by teacher)

In the Online Teaching Suite: • Teaching Program • Tests

• Teaching Notes

• Review Quiz

Knowledge check In the Interactive Textbook you can take a test of prior knowledge required for this chapter, and depending on your score you may be directed to revision from the previous years’ work. ISBN 978-1-108-43463-8 © Powers 2018 Mathematics Standard Year 11 Photocopying is restricted under law and this material must not be transferred to another party

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Chapter 7 Classif ying and representing dat a

7A

7A Data collection Data collection involves deciding what data to collect, locating it and collecting it. Data comes from either primary or secondary sources. • Primary sources – interviewing people, conducting questionnaires or observing a system in operation • Secondary sources – data collected or created by someone else, such as information gathered from newspapers, books and the internet It is important that procedures are in place to ensure the collection of data is accurate, up-to-date, relevant and secure. If the data collected comes from unreliable sources or is inaccurate, the information gained from it will be incorrect. Data collection methods are used for both a census and a sample survey. • A census is a survey of every person in a population. For example, all the students in your school are regarded as the population. A census can be very expensive and time-consuming, if the population is large. • A sample is only part of a population. For example, a sample of the school population is the students in your class. Estimates are made about the population based on the sample. Samples are cheaper than censuses, but are not as accurate. Bias can also be an issue. A sample must be large enough to give a good representation of the population, but small enough to be manageable. There are many different types of sampling including a random sample, stratified sample and systematic sample.

Random sample A random sample occurs when all members of the population have an equal chance of being selected. For example, six students are selected at random from the entire school population. Lotto is another good example of random sampling. A sample of 6 numbers is chosen from 40 numbers. Random samples are simple and easy to use for small populations. However, for large populations, it is possible to miss out on a particular group.

Stratified sample A stratified sample occurs when categories or strata of a population are chosen and then members from each category are randomly selected. For example, one student is selected from each year 7, 8, 9, 10, 11 and 12. Each year group is a category in a stratified sample. Some other common types of categories are age, sex, religion or marital status. A stratified sample is useful when the categories are simple and easy to determine. However, care needs to be taken when selecting categories to avoid any bias in the data.

Systematic sample Systematic sampling occurs when the population is divided into a structured sample size. For example, the students in the school population areput in alphabetical order and the 100th student,200th student,

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7A Dat a collection

300th student, … are selected. A systematic sample is often used by a manufacturer to ensure the machines are working correctly. Here the manufacturer might test a machine every 30minutes or check the 50th item on a production line. Systematic sampling results in a gap between each selection.

Self-selected sample In a self-selected sample, members of the population volunteer themselves. For example, six students from the entire school population offer to complete a questionnaire. A self-selected sample often occurs on the internet. There is the potential for self-selected sampling to be biased as only people who are motivated and have the time volunteer themselves. RANDOM SAMPLE

STRATIFIED SAMPLE

SYSTEMATIC SAMPLE

SELF-SELECTED SAMPLE

Members of the population have an equal chance of being selected.

Categories of a population are chosen. Members then are randomly selected from each category.

Population is divided into a structured sample size. Members are then selected in a certain order from this structure.

Members of the population volunteer themselves.

Example 1: Distinguishing sample types

7A

A retirement village has63 residents, 42 women and 21 men. Decide whether each sample of resident would be random, stratified, systematic or self-selected. a Every seventh resident b Six of the women and three of the men c Nine names picked from a hat containing the names of the residents. d Residents sorted into alphabetical order and each ninth resident selected. e Residents are divided into four age groups (51− 60, 61–70, 71–80, 81–90) and two residents selected from each age group S O LU T I O N:

1 The population has been divided into a structured sample size – 7 th, 14 th, 21st, … 63rd. 2 The population has been divided into categories – women and the men. 3 Sample is taken at random. 4 The population is divided into alphabetical order and a structured sample size taken. 5 The population has been divided into four age group categories and then randomly selected.

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a Systematic sample b Stratified sample c Random sample d Systematic sample e Stratified sample

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7A

Chapter 7 Classif ying and representing dat a

Exercise 7A

LEVEL 1

1 State whether a census or a sample is the most appropriate way to collect this data. a Information on the shopping experience of people in the city b John collecting the heights of his best friends c The travelling habits of the Jones family towork d Australians watching the grand final e Number of people eating toast for breakfast f Length of time every AAA battery lasts g Number of people entering a gym between 5 p.m. and6 p.m. h Holly collecting the length of time students inher class spend on the internet i The world’s reaction to climate change j Shop manager’s reaction to a drop in sales.

Example 1

2 State whether the sample is random, stratified, systematic or self-selected. a A police officer breathalyses every tenth person. b Each person is given a raffle ticket and the tickets are drawn out of a hat. c Each person leaving a bus was given a survey form. We used the data from the people who sent it back. d A business has 240 married and 120 unmarried employees. A sample was chosen to include 10 of the married and 5 of the unmarried employees. e Students were sorted into alphabetical order and each third student selected. f Individuals were randomly selected using their tax file number. g Every 12th jogger was selected from an alphabetical list. h Ten cards were randomly selected from a normal deck of cards. i Ten girls and ten boys were randomly selected from a concert audience. j Ten people who arrive at a shopping centre each day completed the survey.

3 Michael uses a random sample to survey 10% of the local community. In the local community there are 810 males and 920 females. How many people does Michael need to survey?

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7A Dat a collection

4 Amelia plans to conduct a random sample to survey the netball players inthe local association. There are 2850 players in the local association and she plans to survey 171 players. What percentage of thepopulation is her sample? 5 Paige uses a stratified sample to survey 5% of her school population. At the school there are 80 teachers and 1160 students. a How many teachers should complete the survey? b How many students should complete the survey? c How many people should complete the survey? 6 Tyler uses a stratified sample to survey 25% of the members of his swimming club. He uses their sex as a category and selects a random group of female and male swimmers. There are 88 female swimmers and 112 male swimmers in his club. a How many swimmers are in the entire population? b How many female swimmers are in the sample? c How many male swimmers are in the sample? 7 Osman uses a stratified sample to survey 7.5% of his chat room friends. He uses marital status as a category and selects a random group of married and unmarried friends. There are200 married and 240 unmarried friends in his chat room. a How many friends are in the entire population? b How many married friends are in the sample? c How many unmarried friends are in the sample? 8 Taylia uses a stratified sample to survey20% of the senior students from her school. There are 205Year 11 students and 180 Year 12 students. How many students should Taylia choose from Year 12? 9 Ming uses a stratified sample to survey12 1 % of the junior students from his school. There are 2 88 Year 7, 120 Year 8, 104 Year 9 and 128 Year 10 students. a How many students are in the entire population? b How many students should Ming choose in following years? i Year 7 ii Year 8 iii Year 9 iv Year 10

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LEVEL 2

10 A survey was conducted in a school on whether Australia should remain a constitutional monarchy or become a republic. The results are shown below. Male

Female

Total

Republic

51

79

130

Monarchy

23

47

70

Total

74

126

200

a How many males surveyed did not think Australia should change to a republic? b What percentage of people favour changing to a republic? c This survey is not a good random sample of all Australians. Why? 11 A sample of 30 students is taken from a primary school that has an enrolment of 420 students from kindergarten to Year 6. The sampling is designed so that the proportion of each year of the sample matches the population. There are 4 students from Year in the sample. How many Year students are there in the school population? 12 A store has 400 employees of which 208 are female and 192 are male. The store intends to survey 25 of its employees. A stratified survey is to be conducted. a How many females should be surveyed? b How many males should be surveyed? LEVEL 3

13 Identify any possible issues with each of the following survey questions. a Do you like the government’s new policy? Yes/No b Alan is a lazy boss who should be forced to pay his diligent workers more money. Agree or disagree? 14 Kayla surveyed a group of 15 people at theTamworth country music festival on their music preferences. She used this data to draw conclusions for the entire population of NSW. a Do you think her conclusions will be accurate? Give a reason. b What would be a more appropriate method of sampling music preferences?

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7B Classification of data

7B Classification of data There are many types of data that can be collected. For example, if you ask six friends how many pets they own, they might give you the following data: 1, 0, 2, 4, 1, 15. However not all data are numbers. For example, if you also record the gender of each of your friends, you get the following data: male, male, female, female, female, male. Data is divided into two broad classifications: categorical and numerical.

Categorical data Categorical data represents characteristics such as a person’s gender, marital status, address or music they like. Categorical data can take on numerical values (such as ‘0’ for unsatisfactory and ‘1’ for satisfactory), but those numbers don’t have mathematical meaning. Categorical data is further classified as nominal or ordinal. • Nominal data uses a name or label that does not indicate order. For example, a student’s gender could be classified as an ‘F” for female and an ‘M’ for male. • Ordinal data uses a name or label that does indicate order. For example, the quality of work could be classified as an ‘A’ for excellent, ‘B’ for good and ‘C’ for satisfactory. It shows a sequence A, B and C. Categorical data has no quantity or amount associated with each category.

Numerical data Numerical data indicates a quantity and is used to perform calculations. For example, if we asked each student in the class their height, we would expect to get a variety of answers. However each answer is a number. Numerical data is further classified as discrete or continuous. • Discrete data is data that can only take exact numerical values. For example, the number of sisters will give rise to numbers such as 0, 1 or 2. Counting a quantity often results in discrete data. • Continuous data is data that can take any numerical value (depending on the degree of accuracy). For example, a student’s height will give rise to numbers such as 171.2 cm and 173.5 cm. Measuring a quantity often results in continuous data. CLASSIFICATION OF DATA 1 Categorical data – data is classified by the name of the category it belongs to. a Nominal data – name does not indicate order. b Ordinal data – name does indicate order. 2 Numerical data – data indicates a quantity and is used to perform calculations. a Discrete data – data can only take exact numerical values. b Continuous data – data can take any numerical value. ISBN 978-1-108-43463-8 © Powers 2018 Mathematics Standard Year 11 Photocopying is restricted under law and this material must not be transferred to another party

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Chapter 7 Classif ying and representing dat a

Example 2: Classifying data as categorical or numerical

7B

Classify the data from these situations as categorical or numerical. a The heart rate of a group of personal trainers b The most watched television show in Australia c The number of people living in Smith Ave d The reasons for people travelling to work by train S O LU T I O N:

1 The heart rate, such as 70 beats per minute, can be measured and results in a number. 2 A television show, such as the news, does not result in a number. 3 The number of people living in Smith Ave, such as27, can be counted and results in a number. 4 The reason for travelling to work by train, such asit is cheaper, does not result in a number.

a The heart rate is numerical data. b A television show is categorical data. c The number of people living in Smith Ave is numerical data. d The reasons for travelling to work is categorical data.

Example 3: Classifying data as a nominal or ordinal

7B

Classify the following numerical data as nominal or ordinal. a School year level b Internet use at home S O LU T I O N:

1 Year level such as Year 11 indicates order but has no mathematical meaning. 2 Internet use such as email is a label that does not indicate any order.

a Year level is ordinal data. b How the internet is used at home is nominal data.

Example 4: Classifying data as a discrete or continuous

7B

Classify the following numerical data as discrete or continuous. a The number of pets in your family b The perimeter of the school S O LU T I O N:

1 The number of pets can be counted and is exact. 2 The perimeter of the school is a measurement of distance and assumes a value.

a The number of pets is discrete data. b The perimeter of the school is continuous data.

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7B Classification of data

Exercise 7B Example 2

1 Classify the data from these situations as categorical or numerical. a The favourite colours of Jenny’s friends b The number of people travelling in a car c The weight of each student in the year in kilograms d People rating their doctor on personal service (high, medium or low) e The number of students in each class f The IQ of a group of students g Responses to a survey question (agree or disagree) h A person’s lucky number i A female’s favourite mobile phone j The distance from Sydney to Wollongong k The cost of bread at the supermarket l The community’s preferred leader m The number of computers in the school

Example 4

2 Classify the following numerical data as discrete or continuous. a The price paid for a can of soft drink b The number of people at a concert c The time between trains d The number of pages in the newspaper e The amount of water used in the past month f The number of people in your immediate family g The numbers drawn in this week’s lotto h The length of the cricket pitch i The distance measured for the long jump at the world championships j The score achieved from a quiz consisting of 10 questions k The height of the tallest person in the world

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LEVEL 1

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Chapter 7 Classif ying and representing dat a

3 State whether the following is categorical, discrete or continuous data. a The heights of members of a football team b The distance to drive to the train station c The different types of ice creams d The quality of food in a restaurant e The eye colour of a group of people f The number of pets in a household g The time to swim 50 metres h The number of goals scored in the first match of the season i Toda...