Buoi 1.2-EMBS6Regre, Simple Lineae Regression,

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Slides by
John
Loucks
St. Edward’s
University

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Chapter 12, Part A
Simple Linear Regression
 Simple Linear Regression Model
 Least Squares Method
 Coefficient of Determination
 Model Assumptions
 Testing for Significance

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 Regression analysis can be used to develop an
equation showing how the variables are related.
 Managerial decisions often are based on the
relationship between two or more variables.
 The variables being used to predict the value of the
dependent variable are called the independent
variables and are denoted by x.
 The variable being predicted is called the dependent
variable and is denoted by y.

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 The relationship between the two variables is
approximated by a straight line.
 Simple linear regression involves one independent
variable and one dependent variable.
 Regression analysis involving two or more
independent variables is called multiple regression.

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Simple Linear Regression Model
y = b0 + b1x +e
where:
b0 and b1 are called parameters of the model,
e is a random variable called the error term.
 The simple linear regression model is:
 The equation that describes how y is related to x and
an error term is called the regression model.

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Simple Linear Regression Equation
 The simple linear regression equation is:
• E(y) is the expected value of y for a given x value.
• b1 is the slope of the regression line.
• b0 is the y intercept of the regression line.
• Graph of the regression equation is a straight line.
E(y) = 0 + 1x

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 Positive Linear Relationship
E(y)
x
Slope b1
is positive
Regression line
Intercept
b0

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 Negative Linear Relationship
E(y)
x
Slope b1
is negative
Regression line
Intercept
b0

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 No Relationship
E(y)
x
Slope b1
is 0
Regression line
Intercept
b0

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Estimated Simple Linear Regression Equation
 The estimated simple linear regression equation
• is the estimated value of y for a given x value.
• b1 is the slope of the line.
• b0 is the y intercept of the line.
• The graph is called the estimated regression line.
^
𝑦 =𝑏0 +𝑏1 𝑥

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Estimation Process
Regression Model
y = b0 + b1x +e
Regression Equation
E(y) = b0 + b1x
Unknown Parameters
b0, b1
Sample Data:
x y
x1 y1
. .
. .
xn yn
b0 and b1
provide estimates of
b0 and b1
Estimated
Regression Equation
Sample Statistics
b0, b1
^
𝑦 =𝑏0 +𝑏1 𝑥

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Least Squares Method
 Least Squares Criterion
where:
yi = observed value of the dependent variable
for the i th observation
= estimated value of the dependent variable
for the i th observation
min

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 Slope for the Estimated Regression Equation
where:
xi = value of independent variable for i th
observation
= mean value for dependent variable
= mean value for independent variable
yi = value of dependent variable for i th
observation
𝑏1=
∑ (𝑥𝑖 − 𝑥 )( 𝑦𝑖 − 𝑦 )
∑ (𝑥𝑖 − 𝑥 )
2

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 y-Intercept for the Estimated Regression Equation
𝑏0= 𝑦−𝑏1 𝑥

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Reed Auto periodically has a special week-long sale.
As part of the advertising campaign Reed runs one or
more television commercials during the weekend
preceding the sale. Data from a sample of 5 previous
sales are shown on the next slide.
 Example: Reed Auto Sales

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 Example: Reed Auto Sales
Number of
TV Ads (x)
Number of
Cars Sold (y)
1
3
2
1
3
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17
27
Sx = 10 Sy = 100
𝑥=2 𝑦=20

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Estimated Regression Equation
 Slope for the Estimated Regression Equation
 y-Intercept for the Estimated Regression Equation
 Estimated Regression Equation
𝑏0= 𝑦− 𝑏1 𝑥=20−5(2)=10
𝑏1 =
∑ (𝑥𝑖 − 𝑥 )( 𝑦𝑖 − 𝑦 )
∑ (𝑥𝑖 − 𝑥 )
2
=
20
4
=5
^
𝑦 =10+5 𝑥

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 Excel Worksheet (showing data)
Estimated Regression Equation
A B C D
1 Week TV Ads Cars Sold
2 1 1 14
3 2 3 24
4 3 2 18
5 4 1 17
6 5 3 27
7

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 Producing a Scatter Diagram
Step 1 Select cells B2:C6
Step 2 Click the Insert tab on the Ribbon
Step 3 In the Charts group, click Insert Scatter (X,Y)
Step 4 When the list of scatter diagram subtypes appears,
Click Scatter (chart in upper left corner)
Using Excel’s Chart Tools for
Scatter Diagram & Estimated Regression Equation

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 Editing a Scatter Diagram
Step 1 Click the Chart Title and replace it with
Reed Auto Sales Estimated Regression Line
Step 3 When the list of chart elements appears:
Click Axis Titles (creates placeholders for
titles)
Click Gridlines (to deselect gridlines option)
Click Trendline
Step 2 Click the Chart Elements button

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 Editing a Scatter Diagram (continued)
Step 4 Click the Horizontal (Category) Axis Title and
replace it with TV Ads
Step 5 Click the Vertical (Value) Axis Title and
replace it with Cars Sold
Step 6 Select the Format Trendline option
Step 7 When the Format Trendline dialog box appears:
Select Display equation on chart
Click the Fill & Line button
In the Dash type box, select Solid
Close the Format Trendline dialog box

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y = 5x + 10
0
5
10
15
20
25
30
0 1 2 3 4
TV Ads
Cars
Sold

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Coefficient of Determination
 Relationship Among SST, SSR, SSE
where:
SST = total sum of squares
SSR = sum of squares due to regression
SSE = sum of squares due to error
SST = SSR + SSE
=

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 The coefficient of determination is:
where:
SSR = sum of squares due to regression
SST = total sum of squares
r2
= SSR/SST

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r2
= SSR/SST = 100/114 = .8772
The regression relationship is very strong; 87.72%
of the variability in the number of cars sold can be
explained by the linear relationship between the
number of TV ads and the number of cars sold.

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Using Excel to Compute the
 Adding r 2
Value to Scatter Diagram
Step 2 When the Format Trendline dialog box appears:
Select Display R-squared on chart
Close the Format Trendline dialog box
Step 1 Right-click on the trendline and select the
Format Trendline option

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 Adding r 2
Value to Scatter Diagram
y = 5x + 10
R
2
= 0.8772
0
5
10
15
20
25
30
0 1 2 3 4
TV Ads
Cars
Sold
Using Excel to Compute the

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Sample Correlation Coefficient
𝑟𝑥𝑦=(signof 𝑏1)√Coefficient of Determination
𝑟𝑥𝑦=(signof 𝑏1)√𝑟2
where:
b1 = the slope of the estimated regression
equation
^
𝑦 =𝑏0 +𝑏1 𝑥

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The sign of b1 in the equation
Sample Correlation Coefficient
rxy = +.9366
𝑟𝑥𝑦=(signof 𝑏1)√𝑟2
𝑟𝑥𝑦=+√.8772

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Assumptions About the Error Term e
1. The error  is a random variable with mean of zero.
2. The variance of  , denoted by  2
, is the same for
all values of the independent variable.
3. The values of  are independent.
4. The error  is a normally distributed random
variable.

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Testing for Significance
To test for a significant regression relationship, we
must conduct a hypothesis test to determine whether
the value of b1 is zero.
Two tests are commonly used:
t Test and F Test
Both the t test and F test require an estimate of s 2
,
the variance of e in the regression model.

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 An Estimate of s 2
where:
s2
= MSE = SSE/(n - 2)
The mean square error (MSE) provides the estimate
of s 2
, and the notation s2
is also used.
SSE=

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 An Estimate of s
• To estimate s, we take the square root of s2
.
• The resulting s is called the standard error of
the estimate.
s =

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Testing for Significance: t Test
 Hypotheses
 Test Statistic
where
H0: b1 = 0
Ha: b1 ≠ 0
𝑡=
𝑏1
𝑠𝑏1
𝑠𝑏1
=
𝑠
√∑ (𝑥𝑖 − 𝑥 )2

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 Rejection Rule
where:
t is based on a t distribution
with n - 2 degrees of freedom
Reject H0 if p-value < a
or t < -tor t > t

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1. Determine the hypotheses.
2. Specify the level of significance.
3. Select the test statistic.
a = .05
4. State the rejection rule. Reject H0 if p-value < .05
or |t| > 3.182 (with
3 degrees of freedom)
𝑡=
𝑏1
𝑠𝑏1
H0: b1 = 0
Ha: b1 ≠ 0

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5. Compute the value of the test statistic.
6. Determine whether to reject H0.
t = 4.541 provides an area of .01 in the upper
tail. Hence, the p-value is less than .02. (Also,
t = 4.63 > 3.182.) We can reject H0.
=

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Confidence Interval for 1
 H0 is rejected if the hypothesized value of 1 is not
included in the confidence interval for 1.
 We can use a 95% confidence interval for 1 to test
the hypotheses just used in the t test.

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 The form of a confidence interval for 1 is:
where ta/2 is the t value providing an area
of a/2 in the upper tail of a t distribution
with n - 2 degrees of freedom
b1 is the
point
estimator
𝑏1 ± 𝑡𝞪 / 2 𝑠𝑏1
is the
margin
of error
𝑡𝞪/ 2 𝑠𝑏1

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Reject H0 if 0 is not included in
the confidence interval for 1.
0 is not included in the confidence interval.
Reject H0
or 1.56 to 8.44
 Rejection Rule
 95% Confidence Interval for 1
 Conclusion
= 5 +/- 3.182(1.08) = 5 +/- 3.44
𝑏1 ± 𝑡𝞪 / 2 𝑠𝑏1

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 Hypotheses
 Test Statistic
Testing for Significance: F Test
F = MSR/MSE
H0: b1 = 0
Ha: b1 ≠ 0

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 Rejection Rule
where:
F is based on an F distribution with
1 degree of freedom in the numerator and
n - 2 degrees of freedom in the denominator
Reject H0 if
p-value < a
or F > F

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1. Determine the hypotheses.
2. Specify the level of significance.
3. Select the test statistic.
a = .05
4. State the rejection rule. Reject H0 if p-value < .05
or F > 10.13 (with 1 d.f.
in numerator and
3 d.f. in denominator)
F = MSR/MSE
H0: b1 = 0
Ha: b1 ≠ 0

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5. Compute the value of the test statistic.
6. Determine whether to reject H0.
F = 17.44 provides an area of .025 in the
upper tail. Thus, the p-value corresponding to F
= 21.43 is less than .025. Hence, we reject H0.
F = MSR/MSE = 100/4.667 = 21.43
The statistical evidence is sufficient to conclude
that we have a significant relationship between the
number of TV ads aired and the number of cars sold.

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Some Cautions about the
Interpretation of Significance Tests
 Just because we are able to reject H0: b1 = 0 and
demonstrate statistical significance does not enable
us to conclude that there is a linear relationship
between x and y.
 Rejecting H0: b1 = 0 and concluding that the
relationship between x and y is significant does
not enable us to conclude that a cause-and-effect
relationship is present between x and y.

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End of Chapter 12, Part A

Buoi 1.2-EMBS6Regre, Simple Lineae Regression,

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