Chapter 8 Simple Linear Regression

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CHAPTER 8

Draw a scatter plot/diagram to see relationship

between two variables.

Understand and interpret the terms dependentvariable and independentvariable.

Find linear regression model and make predictions.

Study on the strength of the relationship called

correlation analysis.

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In a simple linear relationship, only TWO variablesare involved:

X = independent variable

Y = dependent variable

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Examples:

1. A sociologist wants to find out if increase in crime

rate is due to increase in cost of living.X = cost of livingY = crime rate

2. A fitness instructor wants to find out the relationshipbetween weight loss and the amount of workout time.X = amount of workout timeY = weight

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A plot between the pairs (x, y) values.

To examine relationship between two variables, X

and Y.

Gives general idea whether X is related to Y.

Plots that give a certain pattern means there is arelationship between X and Y.

Plots that have no particular pattern means there isno relationship between X and Y.

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Increasing pattern. As X increases, Y also increases.

Positive linear relationship between X and Y.

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Decreasing pattern. As X increases, Y decreases.

Negative linear relationship between X and Y.

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No particular pattern.

No relationship between X and Y.

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CHAPTER 8

Question:

You are a marketing analyst for Hasbro Toys. You gather

the following data:

Sketch a scatter plot of the data above. 11

Ad (RM) Sales (Units)

1 1

2 1

3 24 2

5 4


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CHAPTER 8

Answer:

1. Is X and Y

related?

2. Positive or

Negative

Relationship?

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01234

0 1 2 3 4 5

Sales, Y

Advertising, X


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A mathematical equation that describes the linearrelationship between X and Y.

Can be used to predict the values of Y from knownvalues of X.

Represents a straight line, so it is of the form y=mx + c,where m is the slope and c is the y-intercept.

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In statistical regression, we write the linear model as

Y = + X +

where = y-intercept

= slope = random error component

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This regression line is usually estimated by using the

paired sample data. The estimated regression line isgiven by

wherea = estimated b = estimated

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bXaY '


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The method used to find the values ofa and b is

slightly different from the familiar method youlearned in algebra.

Uses the concept of Least-Square Method.

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Formula to estimate a and b:

Now we can fit the regression line to the data usingthe values ofa and b. The estimated regression line is

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bn XY X Y

n X X

aY

nb

X

n

( ) ( )( )

( ) ( )

2 2

bXaY '


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CHAPTER 8

Question:

You are an economist for the county cooperative. You

gather the following data.

Find the estimated regression line relating crop yield and

fertilizer.

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Fertilizer (lb.) Yield (lb.)4 3.0

6 5.5

10 6.5

12 9.0


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CHAPTER 8

Answer:

Construct this table first.

Total:

Mean: 8 6

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X Y X XY

4 3.0 16 12

6 5.5 36 33

10 6.5 100 65

12 9.0 144 108

32 24.0 296 218


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CHAPTER 8

Answer:

Using values from the table, estimate a and b.

Therefore, the estimated regression line is

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65.0)32()296(4

)24)(32()218(42

b

8.0)8(65.06 a

XY 65.08.0'


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CHAPTER 8

Answer:

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0246810

0 5 10 15

Yield (Y)

Fertilizer (X)

.8 .65y x


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CHAPTER 8

Answer:

What do a andb in the regression line means?

1. Y-intercept, a = 0.8Average Crop Yield (Y) is expected to be 0.8 lb. when

no Fertilizer (X) is used. X = 0, Y = 0.8

2. Slope, b = 0.65 Crop Yield (Y) is expected to increase by 0.65 lb. for

each 1 lb. increase in Fertilizer (X).

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CHAPTER 8

Question:

A student wants to know the relationship between

number of pages and the price of the book. To analyze

this, he selects a sample of 8 textbooks currently on salein a bookstore.

Develop a regression line to fit the data given.

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CHAPTER 8

Question:

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Book No. of Pages (X) Price (Y)

History 500 84

Algebra 700 75

Geometry 800 99

Physics 600 72

Sociology 400 69Biology 500 81

Statistics 600 63

Nursing 800 93


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CHAPTER 8

Answer: Construct this table first.

Total: 4900 636 3150,000 397,200

Mean: 612.5 79.526

X Y X XY

500 84 250,000 42000

700 75 490,000 52500800 99 640,000 79200

600 72 360,000 43200

400 69 160,000 27600

500 81 250,000 40500600 63 360,000 37800

800 93 640,000 74400


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CHAPTER 8

Answer:

Using values from the table, estimate a and b.

Therefore, the estimated regression line is

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0514.0)4900()3150000(8

)636)(4900()397200(8 2

b

48)5.612(0514.05.79 a

XY 0514.048'


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Now, that we have estimated the regression line, we

can predict Y given any values of X.

This can be found by substituting X into the estimatedregression line,

However, the value of X to insert in the equation must

be within the range of X in the data set.

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bXaY '


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For Example 3, predict the price of the book that has

550 pages.

Thus, if the book is 550 page thick, the price isestimated to be RM76.27

REMEMBER! To predict Y , X must have values within the data set

range.

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27.76)550(0514.048' Y


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Correlation measures the strength of a linearrelationship between two variables.(strong? weak?)

Correlation coefficient tells us about thestrength and direction of a relationship.

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A numerical measure for correlation of thequantitative data is the Pearson correlationcoefficient, r.

The formula is given by

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]][)()([))(()(

2222 YYnXXn

YXXYnr


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0 r 1

Values ofrclose to 1strong positive linear

relationship between X and Y.

Values ofrclose to -1strong negative linear

relationship between X and Y. Values ofrclose to 0 little or no linear relationship

between X and Y.

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CHAPTER 8

Question:

A food analyst wants to know how much a person would

spend on food, given certain amount of income. He

selects a random sample of 7 people with their incomeand food expenditure as shown below.

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Income

(RM 00)

35 49 21 39 15 28 25

Food Expend.

(RM 00)

9 15 7 11 5 8 9


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CHAPTER 8

Question:

(i) Find the estimated regression line for the data.

(ii) How much would a person spend on food if his

income is RM 3000?

(iii) Compute Pearson correlation coefficient, r. Interpretthe r value.

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CHAPTER 8

Answer: Construct this table first.

Total: 212 64 7222 646 2150

Mean: 30.2857 9.1429 36

Income, X Food Exp,

Y

X Y XY

35 9 1225 81 31549 15 2401 225 735

21 7 441 49 147

39 11 1521 121 429

15 5 225 25 7528 8 784 64 224

25 9 625 81 225


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CHAPTER 8

Answer:

(i) Therefore, the estimated regression line is

The slope, b = 0.2642 means the relationship is

positive. That is, people with higher income will

spend more on food.

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2642.0)212()7222(7

)64)(212()2150(72

b

1414.1)2857.30(2642.01429.9 a

XY 2642.01414.1'


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CHAPTER 8

Answer:

(ii) If income is RM3000, that is X=30, then food

expenditure is

So we expect him to spend RM906.74 on food if his

income is RM3000.

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0674.9)30(2642.01414.1' Y


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CHAPTER 8

Answer:

(iii) Pearson correlation coefficient, r

The value r = 0.9587 shows a very strong positiverelationship between income and food expenditure.

When income is high, thefood expenditure also

increases.

9587.0

]646467][)212()7222(7[)64)(212()2150(7

22

r

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Chapter 8 Simple Linear Regression