chapter 11 simple linear regression analysis ( 线性回归分析 )
DESCRIPTION
Business Statistics in Practice. Chapter 11 Simple Linear Regression Analysis ( 线性回归分析 ). Two quantitative variables Correlation. A relationship between two variables. Explanatory (Independent)Variable. Response (Dependent)Variable. y. x. Hours of Training. Number of Accidents. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 11
Simple Linear Regression Analysis (线性回归分析 )
Business Statistics in Practice
Two quantitative variablesCorrelation
What type of relationship exists between the two variables and is the correlation significant?
x y
Cigarettes smoked per dayScore on SATHeight
Hours of Training
Explanatory(Independent)Variable
Response(Dependent)Variable
A relationship between two variables.
Number of AccidentsShoe Size Height
Lung CapacityGrade Point Average IQ
Correlation(相关 ) vs. Regression(回归 )
A scatter diagram (散点图 ) can be used to show the relationship between two variables
Correlation (相关 ) analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the
relationship No causal effect (因果效应 ) is implied with correlation
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PriceIndex(000s)
8.07.57.57.37.27.27.17.17.06.26.25.1
Relationship between Market Index and Stock Price
50
60
70
80
90
100
5 6 7 8 9 10
Index
Pric
e
Example The twelve days of stock prices and the overall market index on each day are given as follows:
The Coefficient of CorrelationCoefficient of Correlation (r).It is a measure of the strength of the relationship (linear) between two variables
It can range from -1.00 to 1.00.
2
1
( )n
xx ii
ss x x
2
1
( )n
yy ii
ss y y
xy
xx yy
ssr
ss ss
1
( )( )n
xy i ii
ss x x y y
Negative values indicate an inverse relationship and positive values indicate a direct relationship.
Values of -1.00 or 1.00 indicate perfect and strong correlation.
Values close to 0.0 indicate weak correlation.
The Coefficient of CorrelationCoefficient of Correlation (r).
Different Values of the CorrelationCoefficient
Introduction to Regression Analysis
Regression analysis is used to: Predict the value of a dependent variable based on the
value of at least one independent variable Explain the impact of changes in an independent
variable on the dependent variable
Dependent variable: the variable we wish to predict or explain
Independent variable: the variable used to explain the dependent variable
Simple Linear Regression Model
Only one independent variable, X Relationship between X and Y is
described by a linear function Changes in Y are assumed to be caused
by changes in X
Types of Relationships
Y
X
Y
X
Y
Y
X
X
Linear relationships Curvilinear relationships
Types of Relationships
Y
X
Y
X
Y
Y
X
X
Strong relationships Weak relationships
(continued)
Types of Relationships
Y
X
Y
X
No relationship(continued)
ii10i εXββY Linear component
Simple Linear Regression Model
Population Y intercept
Population SlopeCoefficient
Random Error term
Dependent Variable
Independent Variable
Random Error component
(continued)
Random Error for this Xi value
Y
X
Observed Value of Y for Xi
Predicted Value of Y for Xi
ii10i εXββY
Xi
Slope = β1
Intercept = β0
εi
Simple Linear Regression Model
i10i XbbY
The simple linear regression equation provides an estimate of the population regression line
Simple Linear Regression Equation (Prediction Line)
Estimate of the regression
intercept
Estimate of the regression slope
Estimated (or predicted) Y value for observation i
Value of X for observation i
The individual random error terms ei have a mean of zero
Least Squares Method (最小二乘方法 )
b0 and b1 are obtained by finding the values of b0
and b1 that minimize the sum of the squared
differences between Y and :
2i10i
2ii ))Xb(b(Ymin)Y(Ymin
Y
b0 is the estimated average value of Y when the value of X is zero
b1 is the estimated change in the average value of Y as a result of a one-unit change in X
Interpretation of the Slope(斜率 ) and the Intercept(截
距 )
Estimation/prediction equation
Least squares point estimate of the slope 1
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ySSnx
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b
iiyy
iiixx
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xx
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The Least Square Point Estimates
Least squares point estimate of the y-intercept 0
nx
xny
yxbyb ii 10
Example 11.1Example 11.1 The House Price Case
A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)
A random sample of 10 houses is selected Dependent variable (Y) = house price in $1000s Independent variable (X) = square feet
050
100150200250300350400450
0 500 1000 1500 2000 2500 3000
Square Feet
Hou
se P
rice
($10
00s)
Graphical Presentation
House price model: scatter plot
050
100150200250300350400450
0 500 1000 1500 2000 2500 3000
Square Feet
Hou
se P
rice
($10
00s)
Graphical Presentation
House price model: scatter plot and regression line
feet) (square 0.10977 98.24833 price house
Slope = 0.10977
Intercept = 98.248
Interpretation of the Intercept, b0
b0 is the estimated average value of Y when the value of X is zero (if X = 0 is in the range of observed X values) Here, no houses had 0 square feet, so b0 = 98.24833
just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet
feet) (square 0.10977 98.24833 price house
Interpretation of the Slope Coefficient, b1
b1 measures the estimated change in the average value of Y as a result of a one-unit change in X Here, b1 = .10977 tells us that the average value of a
house increases by .10977($1000) = $109.77, on average, for each additional one square foot of size
feet) (square 0.10977 98.24833 price house
317.85
0)0.1098(200 98.25
(sq.ft.) 0.1098 98.25 price house
Predict the price for a house with 2000 square feet:
The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850
Predictions using Regression Analysis
Model Assumptions1. Mean of Zero
At any given value of x, the population of potential error term values has a mean equal to zero
2. Constant Variance AssumptionAt any given value of x, the population of potential error term values has a variance that does not depend on the value of x
3. Normality AssumptionAt any given value of x, the population of potential error term values has a normal distribution
4. Independence AssumptionAny one value of the error term is statistically independent of any other value of
Example 11.2: Fuel Consumption Case
Week
Average Hourly Temperature x (deg F)
Weekly Fuel Consumption y (MMcf)
1 28.0 12.42 28.0 11.73 32.5 12.44 39.0 10.85 45.9 9.46 57.8 9.57 58.1 8.08 62.5 7.5
Example 11.2: Excel Output of Regression on Fuel Consumption Data
What Does r2 Mean?
The coefficient of determination, r2, is the proportion of the total variation in the n observed values of the dependent variable that is explained by the simple linear regression model
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variationTotal variation2
Explainedr
Mechanics of the F Test
T test: To test H0: β1= 0 versus Ha: β1 0 at the level of significance
Test statistics based on F
Reject H0 if F(model) > F or p-value < F is based on 1 numerator and n-2 denominator
degrees of freedom
2)-)/(n variationed(Unexplain variationExplained
F
Chapter Summary Discussed correlation -- measuring the strength of
the association Introduced types of regression models Reviewed assumptions of regression and correlation Discussed determining the simple linear regression
equation Addressed prediction of individual values Model diagnostic