8th grade - njctl

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8th Grade

Data

2015-11-20

www.njctl.org

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Table of Contents

· Two Variable Data

· Determining the Prediction Equation

· Two-Way Table

· Line of Best Fit

· Glossary

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Two Variable Data

Return toTable ofContents

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Two Variable Data is also called

Bivariate Data

With bivariate data there are two sets of related data that you want to compare.

Two Variable Data

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Example 1: An ice cream shop keeps track of how much ice cream they sell versus the temperature on that day.

Temperature degrees F

Ice Cream Sales $

57.5 215

61.5 325

53 185

60 332

65 406

72 522

67 412

77 614

74 541

64.5 421

This table shows 10 days of data.

The two variables are:Temperature and Ice Cream Sales.

We can create a scatter plot by plotting the points.

Temperature is the x variableSales is the y variable.

Scatter Plot

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Ten Days of Ice Cream Shop Sales


Ice

Cre

am S

ales

$

Scatter Plot

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What did the scatter plot show us?

Using the Scatter Plot it is easy to see that:

warmer weather leads to more sales.

Scatter Plot

click to reveal

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Scatter Plots are either:

Linear Non-linear

Scatter Plot

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These scatter plot are also non-linear.

Scatter Plot

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If a scatter plot is linear it can be described 3 ways:Negative AssociationPositive Association

No Association

Scatter Plot

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1 What type of scatter plot is shown from the Ice Cream Shop example 1?

A non-linear

B linear, positive association

C linear, negative association

D linear, no association


Ice

Cre

am S

ales

$

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Example 2: Data for 10 students math and science grades are shown in the table. Plot the points to create the scatter plot.

Math Grade

Science Grade

56 6296 9385 8184 8263 60

100 9878 8189 9146 4875 75

Math Grades

Scie

nce

Gra

des

Scatter Plot

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2 What type of scatter plot is shown for the math and science grades from example 2?A non-linear




Math Grades

Scie

nce

Gra

des

Click to reveal solved graph.

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3 What kind of association is shown in the graph?

A non-linear




Time spent studying

Test

Sco

re

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4 What kind of association is shown in the graph ?

A non-linear




Shoe size & Height

shoe size

heig

ht in

in

ches

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5 What association is

shown in this graph?

A non-linear B linear, positive correlation

C linear, negative correlation

D linear, no correlation

Height in inches

Wei

ght i

n P

ound

s

Boy's Height and Weight

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6 Which of the following scenarios would produce a linear scatter plot with a positive correlation?

A Miles driven and money spent on gas

B Number of pets and how many shoes you own

C Work experience and income

D Time spent studying and number of bad grades

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7 Which of the following would have no association if

plotted on a scatter plot?

A Number of toys and calories consumed in a day

B Number of books read and reading scores

C Length of hair and amount of shampoo used

D Person's weight and calories consumed in a day

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What kind of predictions can you make

from looking at the graph?

Predictions

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Number of Hours

Resting Heart Rate

12 616 7810 700 9016 652 854 7514 623 781 878 69

Survey Data

A student wanted to find out if there was a relationshipbetween the number of hours a person exercised in one weekand their resting heart rate. 15 people were surveyed and the table at the right shows the results.

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Plot the results of the survey on

a scatter plot.Number of

HoursResting Heart Rate

12 616 7810 700 9016 652 854 7514 623 781 878 69

Scatter Plot

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Linear Relationship? Association?

Is there a linear relationship?

Is there a positive or negative association?

According to your scatter plot, does a person who exercise generally have a lower resting heart rate than a person that doesn't exercise?

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Hours Math Grade

2 967 754 861 94

0.5 978 702 903 8710 681 946 754 88

Sandy wanted to find out if there was a relationship between the number of hours a student spent browsing the Internet ineach day and their math grades for the marking period.She surveyed several students and the results are shownin the table at the right.

Survey Data

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Look at your results. Is the scatter plot linear or non-linear? Is there a positive or negative association?

What can you say about the math scores as more hours are spent browsing the Internet?


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YearTemperatur

e in F

2,000 30.42,001 30.12,002 37.32,003 26.72,004 24.82,005 30.32,006 38.92,007 37.12,008 34.52,009 27.32,010 31.4

The table shows average temperatures for the month of January in New Jersey from 2000 to 2009.

Is it linear?Is there a positive association,negative association, or neither?


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MonthTemperatur

e in F

1 35.42 38.83 49.84 52.85 65.36 70.27 78.28 759 67

10 5711 4912 40.8

The table shows average temperature by month for New Jersey. Month 1 = January, Month 2 = February, etc.Make a scatter plot using the data from the table.Is the graph linear? Is there an association?


Ans

wer

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Shoe Size v. Girl's Height

Shoe Size

Hei

ght i

n In

ches

8 What association is shown in this graph?A non-linear B linear, positive association

C linear, negative association D linear, no association

Shoe Size

Girl's Height in

Inches5 55

5.5 54

8 64

7.5 65

9 70

6 52

7.5 63

8 66

Ans

wer

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Poll 10 girls and 10 boys from your class on their heights and shoe size. Make a scatter plot for your observations.

Girls Height (in inches)

Shoe Size

Boys Height (in inches)

Shoe Size

PollTe

ache

r Not

es

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Wake Up Time

How Long to Get Ready

Survey your classmates and to find out what time they wake up on a school day and how long it takes them to get ready. Make a scatter plot of your results.

Is there an association with the time a student wakes up and how long it takes them to get ready?

Survey

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Line of Best Fit


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Line of Best Fit

Bivariate data plotted on a scatter plot shows us negative or positive association (correlation).

A line of best fit, or trend line, can help us predict outcomes using the data that you already have.It is drawn on a scatter plot that best fits the data points.

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Line of Best Fit

Notice that the points form a linear like pattern. To draw a line of best fit, use two points so that the line is as close as possible to the data points.

Our line is drawn so that it fits as close as possible to the data points. This line was drawn through (35,82) and (50,90).

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Time spent studying

Test

Sco

re

Line of Best Fit

Predict the test score of someone who spends 52 minutes studying.Predict the test score of someone who spends 75 minutes studying.

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Shoe size & Height

heig

ht in

inch

es

shoe size

Draw a line of best fit, or trend line, on this graph.

Predict the height of a person who wears a size 8 shoe.Predict the shoe size of a person who is 50 inches tall.

Line of Best Fit

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9 Consider the scatter graph to answer the following: Which 2 points would give the best line of fit?

X Y3 9

4.5 8

5 76 58 49 310 1

B

CD

A

A A and D

B B and C

C C and D

D there is nopattern

Ans

wer

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10 Consider the scatter graph to answer the following: Which 2 points would give the best line of fit?

X Y5 26 47 38 4

9 4.5

9 510 3

A

CB D

A A and D B B and C C C and D D there is no

pattern

Ans

wer

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11 Which two points would you pick to draw

the line of best fit?

D

X Y

2 96

7 75

4 86

1 94

0.5 97

8 70

2 90

3 87

10 68

1 94

6 75

4 88

B

C

AA A and B B B and C C C and D D A and D

Ans

wer

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Shoe Size

Girl's Height

in Inches

5 555.5 548 64

7.5 659 706 52

7.5 638 66


Shoe Size

Hei

ght i

n In

ches

A

B

C

D

12 Which two points

would you use to

draw the line of

best fit?

A A and D B C and D C B and D

Ans

wer

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13 A scatter plot is shown on the coordinate plane.

Which of these most closely approximates the line of best fit for the data in the scatter plot?

A

B

C

D

From PARCC EOY sample test non-calculator #15

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Line of Best Fit

Using the scatter plot you created for shoe size v. girls' heights and shoe size v. boys' heights, determine line of best fit that goes through each of these scatter plots.

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Determining the Prediction Equation


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Line of Best FitThe points form a linear like pattern, so use two of the points to draw a line of best fit.

Our line is drawn so that it fits as close as possible to the data points. This line was drawn through (35,82) and (50,90).

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Use the two points that formed the line to write an equation for the line.

Find m Find b

Prediction Equation

This equation is called the Prediction Equation.The slope also shows that a student's score will increase by 8 for every 15 minutes of studying they do.

Where S is the score for t minutes of studying.

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Prediction Equations can be used to predict other related values.

If a person studies 15 minutes, what would be the predicted score?

This is an extrapolation, because the time was outside the range of the original times.

Prediction Equation

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If a person studies 42 minutes, what would be the predicted score?

This is an interpolation, because the time was inside the range of the original times.

Prediction Equation

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Interpolations are more accurate because they are within the set.

The farther points are away from the data set the less reliable the prediction.

Using the same prediction equation, consider:

If a person studies 120 minutes, what will be their score?

What is wrong with this prediction?

Prediction Equation

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If a student got an 80 on the test, What would be the predicted length of their study time?

The student studied about 31 minutes.

Prediction Equation

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14 Consider the scatter graph to answer the following: What is the slope of the line of best fit going through A and D?

X Y3 95 76 58 49 3

10 1

A

D(9, 3)

(3, 9)A

B

CD

Ans

wer

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15 Consider the scatter graph to answer the following: What is the y-intercept of the line of best fit going through A and D?

X Y3 94.5 8

5 76 58 49 310 1

A

D

(9, 3)

(3, 9)A 9

B 10

C 11

D 12

Ans

wer

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16 Consider the scatter graph to answer the following: The equation for our line is y = -1x + 12. What would the prediction be if x = 7? Is this an interpolation or extrapolation?

X Y3 94.5 8

5 76 58 49 310 1

A

D

A 5, interpolation

B 5, extrapolation

C 6, interpolation

D 6, extrapolation

Ans

wer

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X Y3 94.5 8

5 76 58 49 310 1

A

D

A -4, interpolation

B -4, extrapolation

C -2, interpolation

D -2, extrapolation

Ans

wer

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X Y3 9

4.5 8

5 76 58 49 310 1

A

D

A 1, interpolation

B 1, extrapolation

C 2, interpolation

D 2, extrapolation

Ans

wer

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19 In the previous questions, we began by using the table at the right. Which of the predicted values: (7,5) or (14, -2) will be more accurate and why?

A

B

C

D

X Y3 9

4.5 8

5 76 58 49 310 1

(7,5); it is an interpolation.

(7,5); there already is a 5 and a 7 in the table

(14, -2) it is an extrapolation

(14, -2); the line is going down and will become negative

Ans

wer

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20 What is the slope of this best

fit line that goes through

A and C?

A

B

C

D

X Y3 6

2 5

5 9

4 8

1 3

6 10

7 12

9 14

C

A

Ans

wer

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21 What is the y-intercept of

the line of best fit that goes

through A and C?

A

BC

D

C

A

X Y3 6

2 5

5 9

4 8

1 3

6 10

7 12

9 14

Ans

wer

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X Y3 6

2 5

5 9

4 8

1 3

6 10

7 12

9 14

22 The equation for the line of best fit

is . What would the prediction be if

y = 4.5? Is this an interpolation or

extrapolation?

A 8, interpolation

B 8, extrapolation

C 6.5, interpolation

D 6.5, extrapolation

Ans

wer

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X Y3 6

2 5

5 9

4 8

1 3

6 10

7 12

9 14

23 The equation for the line of best fit

is . What would the prediction be if

y = 8? Is this an interpolation or

extrapolation?

A

B

C

D

interpolationextrapolation

interpolation

extrapolation

Ans

wer

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Shoe Size

Girl's Height

in Inches

5 55

5.5 54

8 64

7.5 65

9 70

6 52

7.5 63

8 66


Shoe Size

Hei

ght i

n In

ches

Prediction EquationCalculate the prediction equation using the two labeled points.

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Shoe Size

Girl's Height

in Inches

5 55

5.5 54

8 64

7.5 65

9 70

6 52

7.5 63

8 66


Shoe Size

Hei

ght i

n In

ches

24 What is the slope of the

prediction equation

for this graph?

Ans

wer

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Shoe Size

Girl's Height

in Inches

5 55

5.5 54

8 64

7.5 65

9 70

6 52

7.5 63

8 66


Shoe Size

Hei

ght i

n In

ches

25 A girl with a size 7 shoe

and height of 56 inches

will be an interpolation.

True

False

Ans

wer

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Shoe Size

Girl's Height

in Inches

5 55

5.5 54

8 64

7.5 65

9 70

6 52

7.5 63

8 66


Shoe Size

Hei

ght i

n In

ches



will be an interpolation.

True

False

Ans

wer

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Shoe Size

Girl's Height

in Inches

5 55

5.5 54

8 64

7.5 65

9 70

6 52

7.5 63

8 66


Shoe Size

Hei

ght i

n In

ches

27 What will the height be

of a girl with a size

8.5?

Ans

wer

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Shoe Size

Girl's Height

in Inches

5 55

5.5 54

8 64

7.5 65

9 70

6 52

7.5 63

8 66


Shoe Size

Hei

ght i

n In

ches



will be an extrapolation.

True

False

Ans

wer

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Shoe Size

Girl's Height

in Inches

5 55

5.5 54

8 64

7.5 65

9 70

6 52

7.5 63

8 66


Shoe Size

Hei

ght i

n In

ches

29 Using the prediction

equation, what will the

height be of a girl

who has a size

10 shoe?

Ans

wer

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Prediction Equation

Using the scatter plot you created for the shoe size v. girls' heights and shoe size v. boys' heights from your class, determine the prediction equation for each graph.

Using the equation, how tall is a girl that wears a 9.5 size shoe?

How tall is a boy that wears a 6.5 shoe?

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Two-Way Tables


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Take a Bicycle to School

Do Not Take a Bicycle to School Total

Take the Bus to School 5 7 12

Do Not Take the Bus

to School6 12 18

Total 11 19 30

Two-Way TablesWe can also organize data gathered in a two-way table.

Two-way tables display information as it pertainsto two different categories.

Here is an example of a two-way table:

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Two-Way Tables

What does the two-way table show us?

The table below shows information gathered from 30 students. They were asked if they took a bus or a bicycle to school.




Do Not Take the Bus

to School6 12 18

Total 11 19 30

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Do Not Take the Bus

to School6 12 18

Total 11 19 30

As you can see from the table, some students take the bus,other students ride their bicycles, take the bus or ride a bicycle to school. Several students do not take a bus nor ride their bicycles to school.

Two-Way Tables

Let's answer some questions using the data from the table.

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30 From this table, how many students take the bus

or ride their bicycle to school?




Do Not Take the Bus

to School6 12 18

Total 11 19 30

Ans

wer

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31 How many students take the bus, but do not

ride their bicycles to school?




Do Not Take the Bus

to School6 12 18

Total 11 19 30

Ans

wer

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32 How many students do not take the bus to school?




Do Not Take the Bus

to School6 12 18

Total 11 19 30

Ans

wer

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33 How many students ride their bicycles to school,

but do not take the bus?




Do Not Take the Bus

to School6 12 18

Total 11 19 30

Ans

wer

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Henry surveyed students from several classes to find out if they did chores and received an allowance. 65 students did chores. Of those 65 students, 49 received an allowance. There were 26 students that did not do chores and did not receive an allowance. 10 students that did not do chores, but received an allowance.

Set up your table, and label the categories.

Allowance No Allowance Total

Chores

No Chores

Total

Two-Way Tables

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Two-Way Tables


Chores 65

No Chores

Total

65 students did chores. Where would you write that number?

Notice that the "Chores" and "No Chores" categories are in the rows, and the "Allowance" and "No Allowance" categories are in the columns.

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Two-Way Tables


Chores 49 65

No Chores

Total

Of those 65 students, 49 received an allowance. Where would you write the 49?

Look at the "Chores" category, then "Allowance" since the 49 students who did chores received an allowance.

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Two-Way Tables


Chores 49 65

No Chores 26

Total

There were 26 students that did not do chores and did not receive an allowance.

Look at the "No Chores" category and "No Allowance" category.

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Two-Way Tables


Chores 49 65

No Chores 10 26

Total

10 students that did not do chores, but received an allowance.

Look for the "No Chores" category then "Allowance" category.

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Chores 49 65 - 49 = 16 65No Chores 10 26 10 + 26 = 36

Total 49 + 10 = 59 16 + 26 = 4265 + 36 = 101 or

59 + 42= 101

If you did your math correctly, the total row and column should be the same.

Two-Way TablesThis is the table filled using the information that was given. Although some of the cells are not filled, you can easily find the rest of the information with simple math.

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Chores 49 16 65No Chores 10 26 36

Total 59 42 101

Two-Way Tables

Here is the final table. Now you can answer some questions using the data.

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34 How many students took this survey?



Total 59 42 101

Ans

wer

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35 How many students do chores, but do not receive an allowance?



Total 59 42 101

Ans

wer

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36 How many students do not do chores, but still receive an allowance?



Total 59 42 101

Ans

wer

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Laptop Computer

No Laptop Computer Total

Desktop Computer

No Desktop Computer

Total

Survey your class to find out if each student has a laptop computer and/or desktop computer at home.

Make a two-way table showing your results.

Two-Way Tables

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Relative Frequency

Using two-way tables, we can calculate relative frequencies.

Relative frequencies are ratios that compares the value of a certain category to the subtotal in that category.

As you have previously learned, the frequency is the quantity of just how many of a certain event occurs.Relative frequency is how many compared to the subtotal. The relative frequency is written as a fraction or decimal.

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Relative Frequency

Example: There are 12 girls in a class of 20 students.

The frequency of number of girls in a class is 12.The relative frequency of the number of girls in the class is or 0.60.

What is the frequency of girls in your class? What is the relative frequency?

What is the frequency of boys in your class? What is the relative frequency?

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Do Not Take the Bus

to School6 12 18

Total 11 19 30

Relative Frequency

Calculate the relative frequency for the two-way table from earlier by row and then by column.

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Take the Bus to School

0.42 + 0.58 = 1.00

Do Not Take the Bus

to School

0.33 + 0.67 = 1.00

Total0.37 + 0.63 = 1.00

For this cell, the relative frequency of students taking a bicycle to school or the bus to school is divided by the total number of students that take the bus to school.

Relative Frequency

By row:

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Do Not Take the Bus

to School

Total 1.00 1.00 1.00

For relative frequency by column, the number of students that take a bicycle to school or take a bus to school is divided by the number of students that take a bicycle to school.

Relative Frequency

By column:

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0.42 + 0.58 = 1.00

Do Not Take the Bus

to School

0.33 + 0.67 = 1.00

Total0.37 + 0.63 = 1.00

Let's answer some questions using the relative frequencies.

By row:

What is the relative frequency of students that take a bicycle to school and also take a bus to all students taking a bus to school?

Relative Frequency

Ans

wer

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0.42 + 0.58 = 1.00

Do Not Take the Bus

to School

0.33 + 0.67 = 1.00

Total0.37 + 0.63 = 1.00

By row:

What is the relative frequency of students that do not take a bicycle to school and do not take a bus to all students that do not take a bus to school?

Relative FrequencyA

nsw

er

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0.42 + 0.58 = 1.00

Do Not Take the Bus

to School

0.33 + 0.67 = 1.00

Total0.37 + 0.63 = 1.00

By row:

37 What is the relative frequency of students that take a bicycle to school but do not take a bus to the total number of students that do not take the bus?

Ans

wer

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0.42 + 0.58 = 1.00

Do Not Take the Bus

to School

0.33 + 0.67 = 1.00

Total0.37 + 0.63 = 1.00

By row:

38 What is the relative frequency of the students that do not take a bicycle to school, but do take the bus to the all the students that take the bus to school?

Ans

wer

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39 By Column:What is the relative frequency of students that take a bicycle to school and also take a bus to school, to the total number of students that take a bicycle to school?

By column: Take a Bicycle to School



Do Not Take the Bus

to School

Total 1.00 1.00 1.00

Ans

wer

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Do Not Take the Bus

to School

Total 1.00 1.00 1.00

40 What is the relative frequency of students that do not take a bicycle to school and do not take the school bus to the total number of students that do not take a bicycle to school?

Ans

wer

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Do Not Take the Bus

to School

Total 1.00 1.00 1.00

41 What is the relative frequency of students that take a bicycle to school, but do not take the bus to all students that take a bicycle to school?

Ans

wer

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Total 59 42 101

Use the following two-way table to calculate the relative frequencies by row.

Relative Frequency By Row


Chores

No Chores

Total

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Chores 1.00

No Chores 1.00

Total 1.00

For example, does there seem to be a relationship between whether or not a student receives an allowance compared to whether or not a student does chores?

By row:

Why do we calculate relative frequencies? We can use relative frequencies to determine if there is an association between the two categories.

Relative Frequency

Approximately 0.75 or 75% of students that receive an allowance do chores, and out of those that do chores only 0.25 or 25% of students receive no allowance.

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Total 59 42 101

Use the following two-way table to calculate the relative frequencies by column.

Relative Frequency By Column


Chores

No Chores

Total

Is there a relationship between students that do chores to the amount of students that receive an allowance?

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Cat No Cat TotalDog

No DogTotal

Construct a two-way table using the following information.

Kelly found that 49 people had dogs in her school. Out of the 49 people, 30 people had cats. 50 people had cats in her school.22 people had neither cats nor dogs at home.

Two-way Table

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Cat No Cat TotalDog

No DogTotal

Cat No Cat TotalDog

No DogTotal

By row:

By column:

Relative Frequency

Using the two-way table, calculate the relative frequencies by column and by row.

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Cat No Cat TotalDog

No DogTotal

42 What is the relative frequency of the people who have a cat and a dog at home to the number of people that have cats?

Cat No Cat Total

Dog 30 19 49

No Dog 20 22 42

Total 50 41 91

Ans

wer

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Cat No Cat TotalDog

No DogTotal

43 What is the relative frequency of the people who have a dog and a cat to the number of people that have a dog?

Ans

wer

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Cat No Cat TotalDog

No DogTotal

44 What is the relative frequency of the people who have no cat, but have a dog to the number of people that have no cats?

Ans

wer

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45 The table shows the results of a random survey of students in grade 7 and grade 8. Every student surveyed gave a response. Each student was asked if he or she exercised less than 5 hours last week or 5 or more hours last week. Based on the results of the survey, which statements are true? Select each correct statement.

A More grade 8 students were surveyed than grade 7 students.

B A total of 221 students were surveyed.

C Less than 50% of the grade 8 students surveyed exercised 5 or more hours last week.

D More than 50% of the students surveyed exercised less than 5 hours last week.

E A total of 107 grade 7 students were surveyed.

From PARCC EOY sample test calculator #3

Slide 106 / 122

Survey your classmates to find out if they play sports and/or play an instrument. Construct a two-way table displaying the results. (Write "yes" or "no") Then calculate the relative frequencies by row and by column.

Is there a relationship between the number of students that play sports vs. the number of students that play an instrument?

Construct a Two-way Table

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Glossary


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Bivariate DataTwo sets of related data that is being

compared. Data of two variables.(Two-Variable Data)

Variables:1. Temperature 2. Sales

Variables:1. Shoe Size

Variables:1. Hours 2. Math Grade

Bivariate Data

1 variable

Univariate Data

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(53,180)

(77,610)

range =

610 - 180

If it is 50o outside, what would

be the predicted ice cream sales? y = 17x - 721 y = 17(50) - 721 y = 851 - 721 y = 129

$129

$129 < $180


be the predicted ice cream sales?

y = 17x - 721 y = 17(90) - 721 y = 1,530 - 721 y = 809

$809

$809 > $610

Extrapolation

A data point that is outside the range of data.

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FrequencyThe quantity of just how many of a

certain even occurs.

The frequency of

kids who do not take

the bus to

school is 18.

The frequency of

kids who take the

bus to school is 12.

The frequency of

kids who ride their

bikes to school is 11.

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y = 17x - 721 y = 17(70) - 721 y = 1,190 - 721 y = 469

$469

$180 < $469 < $610



$350

$180 < $350 < $610

y = 17x - 721 y = 17(63) - 721 y = 1,071 - 721 y = 350

Interpolation

A data point that is inside the range of data.

(53,180)

(77,610)

range =

$610 -

$180

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Linear

A graph that is represented by a straight line.

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Line of Best FitA line on a graph showing the general

direction that a group of points seem to be heading. Trend Line.

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Negative Association

A correlation of points that is linear with a negative slope.

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No Association

A correlation of points that is linear with a slope of zero. A horizontal line graph.

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Non-Linear

A graph that is not represented by a straight line. A curved line.

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Positive Association

A correlation of points that is linear with a positive slope.

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y = mx+b

y = 17x - 721 (53,180)

(73,520)

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Prediction Equation


Ice

Cre

am S

ales

$



y = 17x - 721 y = 17(70) - 721 y = 1,190 - 721 y = 469

$469

An equation that is created using the line of best fit. A line that can predict

outcomes using the given data.

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Relative Frequency

The relative frequency of

students who only take the bus to

the total bus riders is

0.58.


students who only ride their bikes to

the total bike riders is 0.33.


students who only ride their bikes to the total students

is 0.37.

Ratios that compares the value of a certain category to the subtotal in that category.

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A graph of plotted points that show the relationship between two sets of data.

Scatter Plot

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Two-Way Table

A table that displays information as it pertains to two different categories.

School Bus vs. Bicycle

Allowance vs. Chores

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8th grade - njctl

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