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100

UNIT 6

DATA MANAGEMENT

MATH 421A

15 HOURS

Revised June 1, 00



101

UNIT 6: Data Management

Previous Knowledge

With the implementation of APEF Mathematics at the Intermediate level, students should be able

to:

- Grade 7- distinguish between biased and unbiased sampling

- select appropriate data collection methods

- construct a histogram

- read and make inferences for data displays

- determine measures of central tendency

- create and solve problems using the numerical definition of probability

- identify all possible outcomes of two independent events

- Grade 8- develop and apply the concept of randomness

- construct and interpret box and whisker plots

- determine the effect of variations in data on the mean, median and mode

- Grade 9- determine probabilities involving dependent and independent events

- determine theoretical probabilities of compound events

Overview:

- sampling techniques and Bias

- measures of Central Tendency and 50% Box Plots

- 90% Box Plots and Applications

- Probability and Applications (Expected Values)



102

SCO: By the end of grade

10 students will be

expected to:

F1 design and conduct

experiments using

statistical methods and

scientific inquiry

F2 demonstrate an under-

standing of concerns

and issues that pertain

to the collection of data

F12 draw inferences about

a population/sample

and any bias that can

be identified

F14 demonstrate an under-

standing of how thesize of a sample

affects the variation in

sample results

G5 develop an

understanding of

sampling variability

Elaborations - Instructional Strategies/Suggestions

Sampling Techniques (8.1)

Invite student groups to explore the following questions:

“If you want to know what percent of high school students on PEI know

the capitals of the Canadian provinces, how would you do this and who

would you ask? Would the results represent the views of the entiregrade 10 population?

Class discussion might touch on these topics:

What does the term “ population” mean?

Is it reasonable to survey the entire population?

If the response is no, then how do we select a representative sample to

be surveyed?

Concept of Bias should be introduced at this point.

Bias is some influence that prevents the sample from being

representative of the entire population.

Challenge student groups to determine possible ways to select a biased

sample.(ex. Sample selected could be only grade 12 Canadian Studies

classes)Invite students to explore ways of selecting an unbiased sample.

Students should read pp.365-367 in Math Power 10.

Probability sampling

< simple random < every member of the population has an equal chance

of being selected.

Ex. All students’ names are put in a hat and 30 are

selected

< systematic < every nth member of a population is selected

Ex: If the school population is 630 and you want to

select a sample of 30 students, 630 ÷ 30 = 21.

Therefore in an alphabetical student list select every

21st student.<stratified < the population is divided into groups, or strata,

from which random samples are taken.

Ex: School is divided into grades and you want 30

people. Randomly pick 10 people from each grade.

<cluster < choose a random sample from one group within a

population.

Ex: School is subdivided by classes. A class is

chosen randomly and all members are selected.

Non-Probability sampling (not random)

<convenience < no thought or effort has been put into selecting the

sample. It is designed to be convenient for the

sampler.

Ex: Samplers survey their friends at the cafeteria

table.



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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Sampling Techniques (8.1)

Journal/Pencil/Paper

A survey result indicates that “ .. most Canadians feel that the

Senate is a waste of tax-payers’ money.” What are some of

the questions you should ask about this survey?

( who was surveyed- was it random across Canada? ; What

age groups were surveyed? ; What socio-economic groups

were surveyed?)

Pencil/Paper

Identify the population you would sample for an opinion on

each topic:

a) minimum driving age

b) student parking spaces

c) fees for athletic teams

d) cafeteria food

Pencil/Paper

You intend to survey the school population to determine

whether the students would attend another dance this month.

Describe a sampling method for each sampling technique:

a) systematic

b) convenience

c) simple random

d) stratified

Presentation

Bring an example of a recent survey in a newspaper or magazine to class and discuss the validity of the survey. Was

there bias in the survey question(s)? What sampling method

do you think was used?

Project

Try to find out what company does the surveys during the

election campaign and ask questions relating to bias and

sampling methods.

Sampling Techniques

Mathpower 10 p.368 # 1,6,11,14,17,

21,24



104


10 students will be

expected to:

F12 draw inferences about

a population/sampleand any bias that can

be identified

G2 design “yes/no” type

questions

F4 construct various

displays of data


Sampling Techniques (cont’d) (8.1)

< Volunteers < members of a population choose to participate in a

survey.

Ex: Interested students volunteer to participate (mail-

in or phone-in surveys fall under this category)

Various Types of Bias (8.2)

< Selection (Sampling) Bias

This is the type of bias created by faulty sample selection this

generally would not happen in probability sampling procedures.

< Response Bias

This bias is created by faulty question or survey construction.

In other words the wording of the question influences the

response. This can occur in all sampling techniques.

Ex: In the question “Is it really fair that young people are not

allowed to drive until they are 16?” the phrase “really fair”shows a bias in the question.

< Non-Response Bias

This bias is created when a large number of people do not

complete a survey.

Ex: Mail out questionnaires commonly have a poor response.

People do not mail them back, therefore, a bias is created

because inferences are made on sketchy results.

Measures of Central Tendency

Generate discussion to see what students’ current knowledge is on

mean, median and mode.

Mean ( ) < The arithmetic average.

Median < The middle number. Once the list is in ascending order,

the median is the middle value. If there is an even

number of values, the median is the average of the

middle two. Half of the data is below the median and

half of the data is above the median.

Ex: 1, 3, 4, 4, 6, 7, 8, 8, 8, 9, 10 (Median = 7)

Ex: 1, 2, 2, 3, 5, 5, 6, 6, 6, 7, 8, 9 (Median = 5.5)

Mode < The most frequently occurring number

Ex: In the first list above the mode = 8 and in the

second list mode = 6.




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Various Types of Bias (8.2)

Journal

In a short paragraph describe in your own words the types of

bias that can occur and give an example of each.

Group ActivityStudy newspapers, magazines, TV commercials, etc. Find as

many statements as possible that you feel are biased. Identify

each one as a response, non-response, or selection bias.

Project

Contact a polling company and ask for copies of the questions

used to survey political party popularity during the last

election. Study the questions for any bias and determine the

method of sampling.


Pencil/paper(See p.112 for explanation on constructing box plots)

Each student in the class picks a number from 1 to 10. Write

the data from the entire class on the board and find the mean,

median and mode. Draw a 50% box plot.

Pencil/Paper/Estimation

A random generator(TI-83) is used to generate 20 numbers

from 1 to 100. Estimate the mean, median and mode from the

data below. Calculate the mean, median and mode and relate

these to your estimates. Draw a 50% box plot.

55 100 91 95 46 75 94 17 19 5372 71 24 75 80 24 98 6 77 19

Pencil/Paper

Listed below are the heights, in centimetres, of 35

competitors in an Olympics event. Examine the data to

determine the spread (range) of the data, where the data was

centred, and if any extreme heights existed. Construct a 50%

box plot on the data below.

190 192 175 180 189 184 184 187 178

175 195 185 183 187 185 182 184 195

180 187 183 185 198 181 185 180 189

185 167 184 188 183 185 189 175

Various Types of Bias

Mathpower 10 p.372 #1-13


Note to teachers:

To use the TI-83 as a random number

generator.

Math <<<< PRB 5:randInt(

generates numbers from 1 to 100 in

groups of 20.



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10 students will be

expected to:

F5 calculate various

statistics using

appropriate

technology, analyze

and interpret displays

and describe the

relationships

G4 interpret and report on

the results obtained

from surveys and

polls, and from

experiments


Measures of Central Tendency (cont’d)

For Box Plots we must look at the data in quarters or quartiles.

Q1 (first quartile) < the first quartile is the mid-value of the first half of

the data (ie. up and not including the median).Ex: 1, 3, 4, 4, 6, 7, 8, 8, 8, 9, 10 (Q1 = 4)

Q3 (third quartile)< the third quartile is the mid-value of the second

half of the data (ie. after the median).

Ex: 1, 3, 4, 4, 6, 7, 8, 8, 8, 9, 10 (Q3 = 8)

Once we have determined the Median and the quartiles we can then

plot this data in a Box Plot. A box plot has 50% of the values inside

the box and the left whisker represents the first quarter of the data and

the right whisker represents the fourth quarter of the data.

Ex: 1, 3, 4, 4, 6, 7, 8, 8, 8, 9, 10

Now we have Q1= 4, Median = 7, and Q3 = 8

For this example we will use a number line from 1 to 10 with a scale of

1.

In general, more valid inferences can be made when the measures of

central tendency are all close together. The more they are dispersed

the less valid the inferences.



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Pencil/Paper

The results of an experiment to determine the effect of

temperature on the speed of sound in air consisted of taking

nine measurements at 100 C and nine taken at 220 C. The data

is displayed below.

a)

draw a 50% box plot for each set of data b) What is the median speed at the lower temperature? At the

higher temperature?

c) Between what two speeds do 50% of the data lie for each

plot.

d) From your results, what do you think is the effect of an

increase in temperature on the speed of sound in air?

Pencil/Paper

A survey of weekly television viewing time of 25 female and

26 male teenagers produced the following data.

a) Find the measures of central tendency (mean, median and

mode)

b) What type of sampling technique would you assume wasused?

c) What types of conclusions can you make about the survey?




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10 students will be

expected to:

F4 construct variousdisplays of data

F26 construct, interpret and

apply 90% box plots

F30 organize and display

information in manydifferent ways with

and without

technology


Measures of Central Tendency (cont’d)

Re-doing the previous example using the TI-83:

Stat 1:Edit clear all lists, then enter the data in L1

If the data must be arranged in ascending order

press Stat 2:Sort A(L1) where A is

ascending

To graph a 50% box plot:

2nd Stat Plot 1:Plot 1 and having the following settings

the 4th graph choice doesn’t connect the outliers to the box while the 5 th

choice of graph does. Typically we will be using this 5th choice. It is abox and whiskers plot with outliers.

to graph set the appropriate window dimensions or press zoom 9:zoom

stat

press trace and see the minimum, Q1, the me

dian, Q3 and the maximum

by cursoring across the box plot.



109



Pencil/Paper/Technology

A teacher has the following results in percent in a class test.

76, 43, 56, 74, 96, 89, 55, 66, 49, 80, 85, 93, 95, 77, 96, 70,

98, 46, 78, 55, 76, 95, 95, 96, 52, 98, 73, 95, 81, 96, 59, 94,

44, 92, 96. Sort the data in ascending order. And draw a 50%

box and whiskers plot.

Solution

Enter the data in the TI-83. Sort the data Stat 2:Sort A(L1).

Graph the data on the TI-83. To see the graph, set the window

dimensions by pressing zoom 9:stat

To see the mean, minimum, Q1, median, Q3 and the maximum

press Stat <<<< Calc 1:1-var Stats enter and scrolling downLLooking at the sorted data determine the mode.

What inferences can be made from the graph?

Half the class has a mark over the median 80, and 1/4 over Q3

95 . Because the median is 80 and we see a short upper

whisker then a lot of the class is very high. The lower whisker

is long which means that there are a few really low students

dragging the mean down.

Note to teacher: If the box is really short then the middle 50%

have marks very close together. If the box is long then there isa large range of marks in the middle 50% of students.

Communication/Journal.

Make inferences about the following box plot.

The median is skewed around 85% with a short upper whisker

and therefore a lot of marks there. The range of the upper half

is very small thus the upper half of the class have marks veryclose together. The lower half have a greater range and thus a

greater dispersion of marks. Marks in upper half are high

because median is 85%.




110

# marked 8 9 10 11 12 13 14 15 16 17 18 19

Frequency 1 2 6 2 14 11 21 17 15 8 2 1


10 students will be

expected to:

F26 construct, interpret and

apply 90% box plots

Elaboration - Instructional Strategies/Suggestions

90% Box Plots

Binomial Population < A population that has two possible outcomes.

In other words, in response to a question the

answer is either YES or NO.

Ex: Toss of a coinEx: Did you pass your test?

Ex: Are you a band student?

90% Box Plots combine results of many small samples of the

population. These box plots then allow us to make inferences on the

population as a whole or backwards from population to sample. The

box plots given are for sample sizes 20, 40, and 100.

Ex: In a school of 1000 students, a sample of 20 students is surveyed.

This procedure is repeated 100 times and each time the 20 students are

randomly chosen. (Not necessarily the same students). This gives us

the data to create a 90% Box Plot for sample size 20.

In the above example, assume the population is known be 70% enrolled

in the English Program and 30% in French Immersion. When

conducting a survey (as explained in the above paragraph) the following

data is obtained and placed in a frequency table.

In a 90% Box Plot, 10% of the values are contained in the two whiskers

together. Out of 100 trials, 10% would be 10. In the table above we

need to count frequencies from both ends until we are as close to 10 as

possible. Working our way in from both sides, the closest we get to 10

is 12 which is obtained when using the first three columns on the left

and the last two columns on the right. The rest of the values are

contained in the box.

Now would be a good time to show the students the entire 90% Box

Plot for sample size 20 table and let them realize that all this work has

generated only 1 of the box plots in this table. So instead of doing all

this work from now on use the tables provided.

In order to do the worthwhile tasks you will need to be able to read the

box plot tables. Instructions are given in Addison-Wesley 10 text p. 548

and 556.



111


90% Box Plots (population to sample, sample to

population)

Group Activity/Paper/Pencil

Divide class into groups and have each group create a 90%

box plot based on a different percent of marked items.

Note to teachers: To generate the data using the TI-83 for a

situation where 80% of the school population is enrolled in

the English Program:

Math <<<< over to Prb 7:randBin( Random binomial)

(Sample size, probability, number of samples). In this samplethe sample size is 20, the probability is 80% and this is

repeated 100 times

80% of 20 = 16 so we would expect out of every 20 people

surveyed 16 would be in the English Program. This program

generates 100 numbers with this restriction but taking into

account the fact that there is some uncertainty in the sampling

process. In the first 20 people you survey it might happen that

most (or very few) of them are in the English Program so that

you may not have exactly 16 out of 20 in the English

Program. If enough groups of 20 students are surveyed the

average should move closer to 16.

For the following problems and those in the SuggestedResources use the Box Plot tables at the end of this unit.

Pencil/Paper

20% of the school population take Canadian Studies. In a

random sample of 20 students, what range of students might

be taking Canadian Studies.

Pencil/Paper

If 34% of the student population regularly attends school

dances, is it likely that a random sample of 40 students would

contain 20 students who attend dances.

Pencil/Paper In a random sample of 20 grade 10 students 7 said they have a

driver’s license. Make an inference about the percent of grade

10 students who have a driver’s license. ( ex. Math 10 p.556)

90% Box Plots

see worksheet at end of unit

Activity

Estimating the size of a wildlife

population Math 10 p.560

Instructions for this activity are at the

back of the unit.

Math 10 p.561 # 1, 3-5

Problem Solving Strategies

Math Power 10 p.397 #1,3,6



112


10 students will be

expected to:

G10 find probability givenvarious conditions


Probability (p.374)

A simple way of introducing students to the study of probability is to do

an activity like the following:

Each card has a letter written on it

if the cards were placed in a hat, what is the chance (or probability) that

you will draw (assume that after each draw the cards are replaced):

a) a vowel

b) a consonant

c) an E

d) an X

Now challenge the students to come up with a definition of probability.

Probability < The ratio of the number of favourable outcomes to thetotal number of possible outcomes.

P(outcome) is the probability of getting that outcome. For example,

when rolling a die P(3) is the probability of rolling a 3 which equals .

Using a deck of 52 cards a person draws a jack.

a) What are the chances of drawing a second jack if the first jack has

been replaced? ( ) This is an example an independent event . An

independent event is when each event has an equal chance of

occurring.

b) What are the chances of drawing a second jack if the first jack was

not replaced. ( ) This is an example of a dependent event .

Expected Values (8.3)

Have students play the game as described in example 2 p. 381. Students

need to keep track of the number of rolls needed to win. The table

included on p.110 at the end of this unit is to help students record their

result with this activity.

When students have completed the activity, record the number of rolls it

took each student to win and then find the class mean (experimental

solution).

Now go through the solution to the example to calculate the expected

value of each roll. Use the expected value to find the number of rolls

expected to win (theoretical solution).



113


Probability (p.374)

Pencil/Paper

A jack is drawn from a deck of 52 cards.

a) What is the probability of drawing a second jack from the

deck if the first jack is replaced?

b) What is the probability of drawing a second jack from the

deck if the first jack is not replaced?

Journal

How do independent and dependent events differ?

Expected Values (8.3)

Pencil/Paper

In a contest at a local coffee/donut store the prizes are as

shown. What is the expected value for this contest?

= .94

If you spend more than $.94 at the store then you will spend more than you win on average.

Pencil/Paper

At the Old Home Week Exhibition there is a game of chance

where you toss 2 coins. If both come up heads you will win

$4. If only one comes up heads you will win $1. If neither

comes up heads they pay you nothing. It costs $2 to play this

game. Complete the table below to determine the expected

value for this game. Should you play this game

Probability

Mathpower 10

Scrabble p.362#1, 2d,g, i-k

Rock, Scissors, Paper all

p.374 #1 do any three

p.375 #3 a-c use chart p.381

Expected Values

Mathpower 10 p.382 # 2-6,9,10,13

Math 10 p.575 # 1-6

Journal

Design a game where you will raise

money for the school council during

the winter carnival. (Make sure you

don’t lose money for the school but

still give participants a reasonable

chance of winning.



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# of rolls Sum Points Total

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20



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Estimating the size of a Wildlife Population(bi-nomial population:tagged or not tagged)

To estimate the number of animals in a species, wildlife biologists use a capture - recapture sampling

technique. To simulate this process popcorn can be used. Have approximately 100 kernels in each zip-

lock bag where 10 in each bag has been spray painted black. ( see Math 10 p.560 for detailed

instructions)

Students do not know how many popcorn are in the bag that they have. Don’t allow

them to count them yet, that is done in step 8.

1. Place the unmarked popcorn(natural colour) in a styro-foam cup

º this represents the population at large

2. Count the number of marked popcorn (black)

º this is the number captured and released

3. Place the marked popcorn in the cup and mix the popcorn up.

º this represents the release of the captured into the wild where they mix with the

rest of the population

4. Pick 40 popcorn from the cup(don’t look - this is the random sample)

º this is the recapture

5. Count the number of marked popcorn

º this represents the number of marked items in the sample

6. Use the chart (sample size 40) to determine the percentage range of marked items in the population

For example, if there were 6 marked popcorn kernels, then by using the table we would get a

percentage range of 8% to 26%.

7. Use the steps below to estimate the size of the entire population (the total number of kernels in the bag)

Using the 8% to 26% range. We know that 10 kernels are marked so the total population could range

from 38 to 125.

.08n = 10 .26n = 10

n = 125 n = 38

Therefore there is a 90% probability that there are between 38 and 125 popcorn (marked and unmarked)

in your bag.

8. Count the total number of popcorn in your bag. Does your prediction fall in an acceptable range?



116

# marked 8 9 10 11 12 13 14 15 16 17 18 19

Frequency 1 2 6 2 14 11 21 17 15 8 2 1

# marked 8 9 10 11 12 13 14 15 16 17 18 19

Frequency 1 2 6 2 14 11 21 17 15 8 2 1

Construction of box plots

If we look at 50% box plots then 50% of the data (values) are contained in the box and the remaining

50% are contained in the two whiskers combined.

For our example, 50% of 100 trials is 50. In the frequency table below (from p.106) we must try to get

the two whiskers adding to as close to 50 as possible (can’t be less than 50)

If we work inward from the outside columns in the table we see this development;

Column 1 2 3 4 5 6 7 8 9 10 11 12

Combining the values of columns 1 and 12 we get a value = 2

Adding to the above total columns 2 and 11 we get = 6



Now as we approach 50 (the total we want) we will probably only be able to add one extra column at a

time

Adding to the above total column 5 we get = 51

If we had chosen to add to the above total column 8 we would have gotten = 54

So we can see that the best result comes from adding column 5 last to get a total of 51.

Column 1 2 3 4 5 6 7 8 9 10 11 12

The same procedure of working from outside to inside is used for 90% box plots.



117

90% Box Plot Problems

Population to Sample (given the % of population, find # possible in a sample)

1) 30% of students at Three Oaks take Physics. In a random sample of 20 students, estimate how many

students could possibly be taking Physics.

2) At a certain school, 80% of the students take History. In a random sample of 40 students, estimate how

many students might be taking History.

3)In the town of Montague 18% of people speak two languages. In a random sample of 100 residents,

estimate how many people might speak two languages.

4) If 28% of 16 year-old people smoke, is it possible that a random sample of 40 people would contain 19

smokers?

5) The probability (chances) of correctly answering a true/false question is 50%. If you guess the

answers, can you correctly guess 24 out of 40 questions correctly, 90% of the time?

6) The probability of guessing a multiple choice question (each question has 5 possible answers) is 20%.

If you guess the answers, can you guess 15 out of 40 questions correctly 90% of the time?

Sample to Population (given the # possible in a sample, find the % of the population)

1) 8 out of 40 randomly selected grade 10 students say that they have a part-time job. Make an inference

about the percent of grade 10 students that have a part-time job.

2) In Westisle High School a survey showed that 12 out of 20 randomly selected students come from a

farm home. Use the box-plots to estimate the percent of students in Westisle who come from a farm

background.

3) Bluefield has 900 students. A survey showed that 26 out of 40 students were bussed to school:

a) make an inference about the percent of students who go to school by bus.

b) use the answer from (a) to estimate how many students are bussed.

Project/Presentation

4) Design a one question yes/no survey about a topic of your choice. Conduct your survey with a randomsample of 40 people. Use the results to make an inference about the percent of people who would

answer yes on the survey question. Explain how you chose your random sample. Which method of

sampling did you use? How were you able to eliminate bias in your question?



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