Mdm4ub Unit 1 Lesson 01 - Advanced Functions Grade 12 Ontario Math Course

MDM4U-B

1UNITUNIT

1Statistics

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Mathematics of Data Management MDM4U-B Unit 1 Introduction 1

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Table of Contents

Unit 1: Statistics

Lesson 1: Importance of Data

Lesson 2: Collecting Data: Sampling

Lesson 3: Analysis of One-variable Data

Lesson 4: Analysis of Two-variable Data

Lesson 5: Use and Misuse of Statistics

Suggested Answers to Support Questions

Unit 2: Combinatorics

Lesson 6: Organized Counting and Permutations

Lesson 7: Combinations

Lesson 8: Combinatorics Problems Involving Repetition and Overlaps

Lesson 9: Pascal’s Triangle and Its Applications

Lesson 10: Mid-course Review


Unit 3: Probability

Lesson 11: Introduction to Probability

Lesson 12: Applying Probability I: Organized Counting and Mutually Exclusive Events

Lesson 13: Applying Probability II: Independent and Dependent Events

Lesson 14: Probability Distributions for Discrete Random Variables

Lesson 15: Binomial and Hypergeometric Probability Distributions


Unit 4: Probability Distribution

Lesson 16: Probability Distributions for Continuous Random Variables

Lesson 17: The Normal Distribution

Lesson 18: Applications of the Normal Distribution

Lesson 19: Designing and Carrying Out an Investigation

Lesson 20: Course Review


Copyright © 2009 The Ontario Educational Communications Authority. All rights reserved.www.ilc.org

Unit 1 Introduction Mathematics of Data Management MDM4U-B2

Introduction “The weather forecast says that there is a 40% chance of rain tomorrow.”

“A new cereal claims that it will reduce your chance of getting heart disease by 10%, compared to another leading brand.”

“Surveys show that two out of three people prefer the taste of coffee over tea.”

You are exposed to statements like these dozens of times every day. But what do they mean? Are they accurate or even helpful? How can you tell? You need a way to judge the reliability of all this data and assess the validity of their conclusions. Statistics can help you to do this.

Statistics is the science of gathering, organizing, and analyzing various types of data. Statistical analyses are used everywhere. For example, a political party might want to know what proportion of voters plans to vote for them in an upcoming election, or a medical researcher might want to know how diet and cancer are linked.

With the growth of the Internet, there has been an explosion of interest in data management and statistical analysis. Data sets can now be easily linked together to create massive global data banks on almost every aspect of life. Statistical analysts are hired to mine this information in order to look for useful patterns.

As you work your way through the problem-solving activities in this unit, you will come to understand the importance of data, how it is collected, and the ways in which statistical analysis can be used to inform or mislead.

Overall ExpectationsAfter completing this unit, you will be able to

• demonstrateanunderstandingoftheroleofdatainstatisticalstudiesandthevariability inherent in data and distinguish different types of data

• describethecharacteristicsofagoodsample,somesamplingtechniques,andprinciples of primary data collection, and collect and organize data to solve a problem

• analyze,interpret,andassessthevalidityofstatisticalconclusionsusingone-andtwo-variable data

• interpretandanalyzegraphsandmakeconclusionsbasedonthegraphs

• interpretstatisticspresentedinthemediaandotherpublicsourcesandidentifythebiases that can occur while collecting data

• seehowdatamanagementisusedinavarietyofcareers

If you have not yet registered on the ILC website, please do so now. Follow the instructions in the Course Introduction, in the “Materials” section, under “Computer Access.”

Copyright © 2009 The Ontario Educational Communications Authority. All rights reserved. www.ilc.org

MDM4U-B

Importance of Data

1

Mathematics of Data Management MDM4U-B Lesson 1 1

IntroductionHistorians sometimes distinguish major periods in human history based on their dominant technological developments. You’ve probably heard of the Stone Age, the Bronze Age, the Iron Age, and the Industrial Age. Many people say that we now live in the Information Age. Each of us has more information available at our fingertips today than was even imaginable a few decades ago. All human knowledge is now available to be searched by anyone anywhere, using a few keystrokes, and more data is added every second.

Data defines our modern culture. Data is now a resource that is increasingly being “mined” by statistical analysts to reveal patterns that enrich our lives or a company’s bottom line. Data is everywhere. For example, data is collected on employment numbers, stock-market fluctuations, the box-office success of the latest blockbuster movie, and how rapidly the earth’s ecosystems are changing.

But what is data? How do you know what it means? Are you being informed or misled by data, and how can you tell the difference? The ability to assess the accuracy and relevancy of data is one of the most important skills to possess in the Information Age.

In this lesson, you will begin your study of data analysis and statistics by exploring the concept of data and its importance. You will learn to recognize the main types of data and understand that variability is inherent in data. You will be able to distinguish between data involving one variable and that involving more than one variable, and how this difference affects the way the data might be analyzed.

Planning Your Study You may find this time grid helpful in planning when and how you will work through this lesson.

Suggested Timing for This Lesson (Hours)

Data and Statistics ¼Types of Data ¾Sources of Data ¾One Variable or Many? ¼Variability in Data and Sampling ½Depicting Data ½Key Questions ¾


Lesson 1 Mathematics of Data Management MDM4U-B2

What You Will Learn After completing this lesson, you will be able to

• identifyanddescribetheroleofdatainstatisticalstudies

• distinguishandexplainreasonswhyvariabilityisinherentindata

• identifydifferenttypesofstatisticaldata



Data and Statistics You are bombarded with data and statistics every day. Data appears on your television and computer screens, in advertisements, on radio news reports, in newspapers and magazines, and on websites. You have to deal with streams of data at work, and then again when you get home. Sports reports are full of data on player statistics and game scores. Business news is full of economic forecasts, such as detailed data on stock price fluctuations and business balance sheets. Health news stories are full of data on risk factors and the benefits of new treatments. Weather reports consist of data on expected temperatures and precipitation. Even entertainment news is increasingly full of statistics on the box-office success of recent hit movies and the salaries that some movie stars make.

You are often urged to take action on the data by purchasing a product or service, choosing a particular course of action, or embracing a particular point of view. Data is used to both inform and manipulate you. But how can you avoid being manipulated? How can you tell whether the data is reliable and the statistics presented represent true patterns?

This course will provide you with the knowledge and skills to determine for yourself whether data seems reliable and whether the patterns depicted and the predictions made from them are valid. In this first lesson, you will learn what data is, understand why it varies, and recognize how to identify its source and reliability. Later in this unit, you will learn how to use statistical tools to find patterns in data and make reliable predictions. Perhaps most importantly, you will learn how to recognize when data is being presented in a false or misleading way.

Importance of Statistics Many situations exist where obtaining data and drawing conclusions from it are of interest. Some examples were mentioned in the introduction to this lesson. Here are some more examples of applications of statistical studies:

• Asmallbusinessownerwantstopredicttheeffectofraisingpricesonsales.

• Acommunitygroupwantstoknowifaproposednewlandfillneartheirtownwillincrease water pollution in their wells.

• Theownerofaminor-leaguehockeyteamwantstopredictattendancepatternsfortheupcoming season.

• TheGovernmentofOntarioisinterestedinknowingthenumberofOntarianswhostart a college program two years after high-school graduation.

For governments, collecting data and analyzing patterns on their population is a particularly important step in helping to create new policies. To run a country as big as Canada, you can imagine that there are many kinds of data and statistics to collect. TheGovernmentofCanadahasanagencycalledStatisticsCanadathatcollectsdata



and provides many statistics about the country. Here is a quote from the current head of Statistics Canada:

“Our data serve a very useful role in the functioning of our country, allowing Canadians to make informed decisions and governments of all levels to develop appropriate policies. We take this role very seriously indeed.” — Munir A. Sheikh, Chief Statistician of Canada (http://www.statcan.gc.ca/about-apercu/chief-chef-eng.htm)

As part of this course, you are encouraged to visit Statistics Canada’s website and explore the daily stories (“The Daily”) found under the “News” link.

S u p p o r t Q u e s t i o n

Do not send your answers in for evaluation.

1. Provide three examples of applications of statistical studies in the health sciences.

(Example: Study on the role of smoking in lung function reduction)

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___________________________________________________________________

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There are Suggested Answers to Support Questions at the end of this unit.

Types of Data Data can be broadly defined as facts or information. Data can be collected on almost anything. The object you are collecting data about is usually called a variable. A variable is a term used to describe any characteristic that you are studying or observing. It’s called a variable because the specific value of the object you are observing can vary. For example, in a study of birth weights of 100 newborns, the weight at birth is called a variable because the birth weight of babies will vary across the study; each baby will be born at a slightly different weight. The data in the study would be the 100 individual birth weights.

In almost every case, the data collected is associated with a variable under study. That’s why people often use the terms “variable” and “data” to describe the same thing. For example, if they say that a variable is non-numerical, it means that the data it represents is not based on numbers.



Examples of data (and variables):

• Theweightofaperson

• Theweightofasalmon

• Theageofapinetree

• Thecountryinwhichapersonwasborn

• Thepoliticalpartyforwhomapersonisplanningtovote

• Aperson’sfavouritetypeofmusic

• Whetherornotapersonsubscribestoanewspaper

• Theamountofrainfallinamonth

• Theamountoftimesinceaperson’slastdentalvisit

• Thenumberofshoesapersonowns

• Whetherapersonrentsorownshisorherhome

Because data (and variables) can be of so many different types, it is helpful to categorize them. The data’s type will determine how it is analyzed statistically because there are different statistical methods optimized for each type. You will explore these methods later in this course.

Quantitative versus Qualitative VariablesData most often refers to information that can be expressed using numbers. This is called quantitative or numerical data. The age of a person is an example of quantitative data (or a quantitative variable). But not all data involves numbers. The name of the country in which someone was born is an example of data that is not a number. This type of data is called qualitative or non-numerical data. Qualitative data and variables can also be called categorical when they describe the category to which an individual belongs. For example, if a shirt comes in three sizes (small, medium, and large), and if someone wears a medium size, then they are in the medium category for that shirt. The variable of shirt size in this case would be called a categorical variable because its values fall into categories.

Examples of quantitative data:

• Theheightofaprofessionalbasketballplayerisaquantitative(ornumerical)variable

• Theageorweightofaperson

• Thepopulationofacity

• Thetimeittakestotraveltowork



Examples of qualitative data:

• Thenameoftheuniversityaprofessionalbasketballplayerwenttoisaqualitative(orcategorical) variable

• Whethersomeoneismaleorfemale

• Whethersomeoneisleft-orright-handed

• Aperson’sfavouritecolour

• Thetypeofcarsomeonewantstobuy

Discrete versus Continuous Variables You can divide quantitative or numerical variables into two types: discrete or continuous. There are some quantitative variables where the value can only be a whole number because fractions or decimals would not make sense. These are called discrete variables. These variables tend to be based on data recorded as counts.

Examples of discrete variables:

• Thenumberofpetsapersonowns(forexample,youcan’thavetwo-and-a-halfpets)

• Thenumberofchildrenapersonhas

• Thenumberoftimesapersonvisitedthedentistinthepastthreeyears

• Thepositioninwhichapersonfinishedarace

There is another type of quantitative or numerical variable where decimals or fractions are possible. These variables are measured rather than counted and are referred to as continuous variables. They are not restricted to jumping up and down between whole number values—they can take any value in between.

Examples of continuous variables:

• Height

• Weight

• Speed

• Aperson’sbodytemperature

• Thetimetakentofinisharace

Here is an example to contrast the two types:

The height of a hockey player is a continuous variable because it is measured. Meanwhile, the number of goals a hockey player scores in a season is discrete because these can only be expressed as whole numbers. A hockey player can be six-and-a-half feet tall but he can’t score six-and-a-half goals.



S u p p o r t Q u e s t i o n s


2. Identify whether each of the following variables is quantitative or qualitative. If quantitative, categorize it as either discrete or continuous.

Quantitative or qualitative

Discrete or continuous

a) Height of a bicycle

b) Age of a cat to the nearest year

c) Volume of juice in a can

d) Names of countries a person has visited

e) Letter grade in a course

f) Eye colour

g) How someone feels about the current government

h) Whether someone drinks coffee, tea, both, or neither in the morning

i) The number of cars you can see from your window

j) Whether you have ever gone fishing



3. For each of the following questions, identify whether the variable is discrete or continuous.

Discrete or continuous

a) How often do you ride a bike in a week?b) How many buttons are on your shirt?c) When is the best time to go to the store to avoid line-ups? d) How much water is in a glass?

Sources of Data Regardless of its type, the reliability and usefulness of data will depend on its source. As you know, stories that you hear second- or third-hand from friends are often not completely true. The same goes for the reliability of data. The meaning of data will depend on its context and what it was collected for.

If you have acquired the data first-hand, you know its source and you know what it was collected for because you obtained it yourself. You know about the potential biases in the data and whether or not it is complete. But if you get the data second-hand from another source, say, from a friend who did his or her own study or from a newspaper or web article that summarizes their study’s results, then you are getting the data filtered through the bias of that source. You can’t be as sure of its reliability any more. Maybe the source left something out or interpreted something incorrectly.

Because this is such an important issue in data analysis, there are terms used to describe the source of the data.

Primary versus Secondary Data Primary data is data that you collect yourself. For example, data that you collect for a study you are conducting on the eating habits of your co-workers is primary data. Secondary data is data collected by someone else. When your employer uses your study to change their menu in the company cafeteria, your data is secondary to them. Here is another example: You conduct an interview with your neighbour. Information from this interview is primary data to you. If you write a book with that primary data and someone then uses that book as a reference to write a report, then the data is considered secondary to the person who wrote the report.

The benefit of primary data is that you know all about it. The problem with secondary data is that often, the reliability, accuracy, and integrity of the data is uncertain. Who collected it? Is it biased? How old is it? Can the data be verified, or does it have to be taken on faith? All of these questions can be difficult to answer if you did not collect the data yourself.



But in many cases, you don’t have the time, resources, or ability to collect all of the data yourself. For example, in your study of cafeteria food, you may need a list from the food vendor showing the calorie count of each dish. You can’t easily collect that information yourself, first-hand, because it requires the ability to carry out some chemical procedures you may not have access to. So, how can you determine the reliability of this secondary data? That’s where the source of the data becomes important.

Primary versus Secondary Sources Primary sources are those that created the original data. They have a direct knowledge of the situation. The actual source can either be the people who created the data themselves or it can be the documents they created. For example, a research paper published by a team of medical researchers studying heart disease and diet is a primary source of information on this topic. The data is still secondary to you because you did not collect it yourself, but at least you are able to read the original study and see how the researchers collected the data and analyzed the results. If you can’t collect the data for yourself, then going to the primary source is the next best option.

Secondary sources are ones that report on the results of other studies. For example, secondary sources can be news reports, encyclopedias (like Wikipedia), textbooks, websites, and magazines. In the heart disease study example above, a website reporting on the results of the study would be considered a secondary source to you. Secondary sources are the least reliable sources of information because they usually shorten and edit the original information, thus creating potential biases.

However, some secondary sources are better than others. For example, academic institutions tend to present original information in a less biased way. Textbooks and some encyclopedias can be good as well. The worst secondary sources tend to be those advocating particular opinions, especially Internet “blogs.” It can be hard to determine whether a bias exists in a secondary source, so it is always good to try and confirm the information with another source.

Can you think of a reliable source of secondary data that was mentioned earlier in this lesson?

If your answer is Statistics Canada, then you are correct.





4. Categorize each of the following data sources as primary or secondary.

Primary or secondary

a)AGovernmentofOntariowebsiteshowingunemploymentfigures for the province

b)AGovernmentofOntariowebsitepresentingtheresultsofa survey conducted by businesses on consumer spending patterns

c) A Wikipedia article on nuclear power

d) A journal article published by researchers studying the effect of a new drug on Alzheimer’s disease

One Variable or Many? In some situations, the data you are interested in involves only one variable. For example, if you want to know the proportion of Canadians who are colour-blind, your data will involve a single variable: is an individual colour-blind or not? At other times, your data may involve more than one variable, for example, if you want to know whether colour-blindness is more common among men than women. In that case, you are collecting data on two variables: sex, and whether a person is colour-blind or not. Some studies, particularly in the environmental or medical fields, can involve dozens of variables.

Example: One-variable studies

Suppose that Rachel designs seats on airplanes. She wonders whether people have been getting taller over the years and so will need more leg room on the new planes. She obtains the heights of a representative sample of adult Canadians measured last year, as well as the heights of adults measured 40 years ago. Rachel may be interested in two different populations, but if the only characteristic she measures is height, then this data only involves one variable.

Example: Two-variable studies

There are many examples in statistics where more than one variable is observed and you are looking for possible relationships between the variables. For instance:



• Therelationshipbetweenthelevelofpollutioninacityandtheaveragelifespanoftheresidents in that city

• TherelationshipbetweentheproportionofInternetsubscribersinaneighbourhoodand the voting patterns of that neighbourhood

• Therelationshipbetweenhavingapetandsomebody’semotionalorphysicalhealth

The number of variables collected affects the way that the data is analyzed. In Lesson 3, you will explore data analysis using one variable, and in Lesson 4, you will examine ways to analyze the interrelationship between two variables.

Variability in Data and Sampling In general, there is variability in data. Variability can come from two sources.

The first source of variability is a consequence of the simple fact that individuals are different. This variability is inherent in the population—you can’t avoid it. For example, not everyone will answer a survey question in exactly the same way. So, if you ask a group of people the same question you will get a variety of answers. The group of people you interview is called your sample. Usually, you want your sample to represent the opinions of the larger population in which you are actually interested. Selecting a group within your study population that represents the pattern in the entire population is called taking a representative sample. You will learn more about sampling in the next lesson.

Example of inherent variability:

Imagine that Aisha and Peter are both interested in knowing what proportion of Torontonians subscribes to a newspaper. Neither Aisha nor Peter has the time or the resources to interview all 2.5 million residents of the city. Therefore, they will interview a small group of residents that they hope is representative of the whole population. This smaller group is referred to as a sample, and the group about which data is being collected is the population. Typically, the sampling method will involve some degree of randomness to reduce the amount of bias. However, even if Aisha and Peter both use proper statistical methods, it is unlikely that they will get exactly the same proportion of newspaper subscribers in their two samples, due to chance differences in their choice of residents.

The second source of variability is the limitations of humans and the measuring instruments used to collect the data. These limitations can be reduced by using better technology, but they can’t be eliminated completely.

Example of measurement variability:

Suppose Navdip is helping to coach the junior track team, and the coach wants to collect data on the time it takes each team member to run around the track. The coach and Navdip both use stopwatches to collect the data. Do you think that Navdip and the coach will get the same data? Most likely they will get slightly different values. This difference is due both to human limitation (for example, it is unlikely that Navdip and the coach will start and stop their watches at exactly the same instant) and the limited accuracy of the stopwatches.





5. For each of the following situations, determine

i) the population

ii) the variable being researched

iii) whether the variable is quantitative or qualitative

iv) if quantitative, whether the data is continuous or discrete

a) You are hired by a restaurant to determine how often each customer visits the restaurant each week.

_______________________________________________________________

_______________________________________________________________

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b) The transit commission hires you to record the time between buses at a specific bus stop.

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

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c) You conduct a study on whether residents in city A have more disposable income than those in city B.

_______________________________________________________________

_______________________________________________________________

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d) You survey members of your community for their opinion on the proposed name of the new neighbourhood community centre.

_______________________________________________________________

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e) You collect data on the number of cars parked on your street at the top of each hour.

_______________________________________________________________

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f) You are a marine biologist studying the biodiversity in Long Lake by identifying the species of fish in the lake caught by anglers.

_______________________________________________________________

_______________________________________________________________

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_______________________________________________________________

Depicting DataOnce data has been collected, it needs to be presented effectively to communicate the patterns it contains—otherwise there is no use collecting it. In later lessons, you will learn more about the ways to depict data and statistical results. Here is a brief overview of the topic.

There are several ways to represent data. Two of the most common are tables and graphs.

Tables Tables are used to organize data. For example, the following table represents the income distribution of a city in Ontario.

Annual income Percentage

Under $10 000 22%

$10 000 to $25 000 29%

$25 000 to $50 000 30%

$50 000 to $100 000 16%

Over $100 000 3%



Notice that the organization of this table made it easy to see the different categories of income and their proportions relative to each other. The data was not presented randomly, but was ordered in sequence of income, from top to bottom. It also presented the income categories on the left-hand side and the percentages on the right-hand side, since the annual income categories are what you want to find first, and since we also tend to read from left to right.

Graphs Anotherwaytorepresentdataisthroughgraphs.Graphsareusedtodisplaydatasothatthe patterns in the data are easier to see, and you can draw conclusions more easily from these patterns. There are dozens of different graph types, each one suited to highlighting a particular kind of pattern in a particular type of data. One of the most common graph types is a pie graph. It is very good at showing patterns in the relative sizes of categories.

The following pie graph displays the same data from the previous table.

Income Distribution

Under $10K22%

Over $100K3%

$25K to $50K30%

$10K to $25K29%

$50K to $100K16%

Income Distribution

Figure 1.1: Income distribution in Ontario

Notice how this graph quickly lets you see that the largest group is the one with the income range of $25–$50K, but that the $10–$25K group is nearly as big. You also get a sense of how relatively small the group is in the over-$100K income range. Your brain’s ability to detect patterns in an image is much better than its ability to see patterns in a list of numbers. But sometimes graphs can also be used to mislead you. You will explore a number of ways to represent data graphically in Lessons 3 and 4, and learn how to spot misleading uses of graphics in Lesson 5.



Key Concepts Data: a collection of organized information, usually the result of an observation, measurement, or experiment. Data may consist of numbers, words, or images.

Variable: any characteristic of the population being studied or observed

Quantitative data: data that is represented using numbers; also called numerical data

Numerical data (or variable): data (or variable) that is represented using numbers

Qualitative data: data that is not represented using numbers; also called non-numerical data; may sometimes also be called categorical data

Non-numerical data (or variable): data (or variable) that is not represented using numbers

Categorical data (or variable): data (or variable) that describes to what category an individual belongs

Discrete data (or variable): data (or variable) that is the result of a count; its values can only be whole numbers

Continuous data (or variable): data (or variable) that is measured; it can take any value

Primary data: data collected by the researcher from the source itself

Secondary data: data collected by people other than the researcher

Sample: the part of the population you actually observed or measured. You want the patterns seen in the sample to be similar to the patterns you would find in the population.

Population: the entire group that you want to know about in a study

Representative sample: a group within your study population that represents the pattern in the entire population



K e y Q u e s t i o n s

Save your answers to the Key Questions. When you have completed the unit, submit them to ILC for marking.

Total: 19 marks

1. Identify the following variables as quantitative or qualitative. (5 marks: 1 mark for each statement)

a) The rate of change of a population (1 mark)

b) A letter grade in a course (1 mark)

c) A person’s favourite kind of music (1 mark)

d) The number of calories per 100 grams of a food item (1 mark)

e) A person’s monthly salary (1 mark)

2. Analyze each statement below and then, for each one, identify

i) the population

ii) the variable

iii) whether the variable is discrete or continuous

(12 marks: 3 marks for each statement)

a) You are surveying the traffic exiting a highway by counting the number of cars exiting the highway at a specific point every 15 minutes (3 marks)

b) Survey asking the age of students in a class (3 marks)

c) A teacher surveying the height of students in a class (3 marks)

d) A survey of the annual salary of working Canadians (3 marks)

3. Describe two examples of applications of statistical studies in politics. (2 marks: 1 mark for each example)

Now go on to Lesson 2. Do not submit your coursework to ILC until you have completed Unit 1 (Lessons 1 to 5).


Mdm4ub Unit 1 Lesson 01 - Advanced Functions Grade 12 Ontario Math Course

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