商务统计学英文课件 (5)

Two variable model Y
(continued)
Yˆ b0 b1X1 b2X2
Slope for variable X1
X2
Slope for variable X2
X1
Example: 2 Independent Variables
n A distributor of frozen dessert pies wants to evaluate factors thought to influence demand
n Dependent variable: Pie sales (units per week) n Independent variables: Price (in $)
Advertising ($100’s)
n Data are collected for 15 weeks
Pie Sales Example
Total
14 56493
The Multiple Regression Equation
Sales 306.526 - 24.975(Price) 74.131(Advertising)
where Sales is in number of pies per week Price is in $ Advertising is in $100’s.
In this chapter we will use Excel or Minitab to obtain the regression slope coefficients and other regression summary measures.
Multiple Regression Equation
Business Statistics: A First Course
Fifth Edition
Multiple Regression
Learning Objectives
In this chapter, you learn: n How to develop a multiple regression model n How to interpret the regression coefficients n How to determine which independent variables to
Prediction interval for an individual Y value, given these X values
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Predictions in Minitab
Predicted Yˆ value
Predicted Values for New Observations
Confidence interval for the mean value of Y, given these X values
Sales 306.526 - 24.975(Price) 74.131(Advertising)
ANOVA Regression Residual Total
Intercept Price Advertising
df 2
12 14
SS 29460.027 27033.306 56493.333
r2 SSR regression sum of squares
SST
total sum of squares
Multiple Coefficient of Determination In Excel
Regression Statistics
Multiple R
0.72213
R Square
The regression equation is Sales = 307 - 25.0 Price + 74.1 Advertising
Predictor Coef SE Coef T P
Constant 306.50 114.30 2.68 0.020
Price
-24.98 10.83 -2.31 0.040
Advertising ($100s) 3.3 3.3 3.0 4.5 3.0 4.0 3.0 3.7 3.5 4.0 3.5 3.2 4.0 3.5 2.7
Multiple regression equation:
Sales = b0 + b1 (Price) + b2 (Advertising)
Using The Equation to Make Predictions
Predict sales for a week in which the selling price is $5.50 and advertising is $350:
Sales 306.526 - 24.975(Price) 74.131(Advertising) 306.526 - 24.975 (5.50) 74.131(3.5) 428.62
Check the “confidence and prediction interval estimates” box
Predictions in PHStat
(continued)
Input values
Predicted Y value
Confidence interval for the mean value of Y, given these X values
Excel Multiple Regression Output
Regression Statistics
Multiple R
0.72213
R Square Adjusted R Square Standard Error
0.52148 0.44172 47.46341
Observations
15
Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Pie Sales 350 460 350 430 350 380 430 470 450 490 340 300 440 450 300
Price ($) 5.50 7.50 8.00 8.00 6.80 7.50 4.50 6.40 7.00 5.00 7.20 7.90 5.90 5.00 7.00
Predicted sales is 428.62 pies
Note that Advertising is in $100’s, so $350 means that X2 = 3.5
Predictions in Excel using PHStat
n PHStat | regression | multiple regression …
Advertising 74.13 25.97 2.85 0.014
S = 47.4634 R-Sq = 52.1% R-Sq(adj) = 44.2%
Analysis of Variance
Source
DF SS MS F P
Regression 2 29460 14730 6.54 0.012
Residual Error 12 27033 2253
0.52148
Adjusted R Square
0.44172
Standard Error
47.46341
Observations
15
r2 SSR 29460.0 .52148 SST 56493.3
52.1% of the variation in pie sales is explained by the variation in price and advertising
b2 = 74.131: sales will increase, on average, by 74.131 pies per week for each $100 increase in advertising, net of the effects of changes due to price
New
Obs Fit SE Fit 95% CI
95% PI
1 428.6 17.2 (391.1, 466.1) (318.6, 538.6)
Values of Predictors for New Observations New Obs Price Advertising
1 5.50 3.50
Multiple regression equation with k independent variables:
Estimated (or predicted) value of Y
Estimated intercept
Estimated slope coefficients
Yˆ i b0 b1X1i b 2X 2i b k X ki
ANOVA Regression Residual Total
df 2
12 14
SS 29460.027 27033.306 56493.333
MS 14730.013
2252.776
F
Significance F
6.53861
0.01201
Input values
Prediction interval for an individual Y value, given these X values
Coefficient of Multiple Determination
n Reports the proportion of total variation in Y explained by all X variables taken together
MS 14730.013
2252.776
F
Significance F
6.53861
0.01201
Coefficients 306.52619 -24.97509 74.13096
Standard Error 114.25389 10.83213 25.96732
t Stat 2.68285 -2.30565 2.85478
Y-intercept
Population slopes
Random Error
Yi β 0 β1X1i β 2 X 2i β k X ki ε i
Multiple Regression Equation
The coefficients of the multiple regression model are estimated using sample data
include in the regression model n How to determine which independent variables are more
important in predicting a dependent variable n How to use categorical variables in a regression model
The Multiple Regression Model
Idea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (Xi)
Multiple Regression Model with k Independent Variables:
b1 = -24.975: sales will decrease, on average, by 24.975 pies per week for each $1 increase in selling price, net of the effects of changes due to advertising
P-value 0.01993 0.03979 0.01449
Lower 95% 57.58835 -48.57626 17.55303
Байду номын сангаас
Upper 95% 555.46404 -1.37392 130.70888
Minitab Multiple Regression Output
Sales 306.526 - 24.975(Pri ce) 74.131(Adv ertising)