Statistics 8th Edition Jia Junping Chapter 12 summary of knowledge points of multiple linear regression and answers to exercises after class

Catalog

One 、 Knowledge framework

Two 、 Exercises

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One 、 Knowledge framework

Two 、 Exercises

1 According to the table 12-2 The data for Excel Regression , And the regression results are discussed , Calculation x1＝200,x2＝7 when y The predicted value of .

Explain ： from Excel Output regression results , As shown in the table .

Regression results

variance analysis

So the regression equation is ：y＝25.0287－0.04971x1＋1.928169x2.
among β1＝－0.04971 Express , stay x2 Under the same conditions ,x1 Every increase 1 A unit of ,y Average decline 0.04971 A unit of ;β2＝1.928169 Express , stay x1 Under the same conditions ,x2 Every increase 1 A unit of ,y Average increase 1.928169 A unit of . Determination factor R2＝21.09%, It means that the variation of the dependent variable can be y And x1 and x2 The proportion explained by the linear relationship between is 21.09%. Because this proportion is very low , It shows that the fitting degree of the regression equation is very poor . Estimate the standard error se＝13.34122, The prediction error is also large . The ANOVA table shows ,Significance F＝0.436485＞α＝0.05, indicate y And x1 and x2 The linear relationship between them is not significant . For regression coefficient test P All values are greater than α＝0.05, The two regression coefficients are not significant . When x1＝200,x2＝7 when ,y The predicted value of is ：y＝25.0287－0.04971×200＋1.928169×7＝28.58.

2 The management of an electric appliance sales company thinks that , Monthly sales revenue is a function of advertising expenses , And I want to estimate the monthly sales revenue through advertising expenses .

requirement ：
（1） Take the TV advertising cost as the independent variable , Monthly sales revenue as the dependent variable , Establish the regression equation of estimation .
（2） The independent variables are TV advertising expense and newspaper advertising expense , Monthly sales revenue as the dependent variable , Establish the regression equation of estimation .
（3） Above （1） and （2） The established regression equation of estimation , Is the coefficient of TV advertising cost the same ？ Explain the regression coefficients respectively .
（4） According to the question （2） The established regression equation of estimation , In the total variation of sales revenue , What proportion is explained by the estimated regression equation ？
（5） According to the question （2） The established regression equation of estimation , Test whether the regression coefficient is significant （α＝0.05）.

Explain ：（1） Take the TV advertising cost as the independent variable , Monthly sales revenue as the dependent variable , from Excel The output regression results are shown in the table .

Regression results

Analysis of variance

So the estimated regression equation is ：y＝88.63768＋1.603865x1.
（2） The independent variables are TV advertising expense and newspaper advertising expense , Monthly sales revenue as the dependent variable , from Excel The output regression results are shown .

Regression results

Analysis of variance

So the estimated regression equation is ：y＝83.23009＋2.290184x1＋1.300989x2.
（3）（1） and （2） In the established regression equation of estimation , The coefficients of TV advertising expenses are different .
stay （1） In the regression equation of , Regression coefficient β1＝1.603865 Express ： Every increase in TV advertising expenses 1 Ten thousand yuan , Monthly sales revenue increased on average 1.603865 Ten thousand yuan ; stay （2） In the regression equation of , Regression coefficient β1＝2.290184 Express ： Under the condition of constant newspaper advertising expenses , Every increase in TV advertising expenses 1 Ten thousand yuan , Monthly sales revenue increased on average 2.290184 Ten thousand yuan .
（4） problem （2） in ,R2＝91.9036%,Ra2＝88.665%, It shows that in the total variation of sales revenue , The proportion explained by the estimated regression equation is 88.665%.
（5） problem （2） in ,β1 Of P value 0.000653,β2 Of P value ＝0.009761, All less than α＝0.05, Therefore, the two regression coefficients are significant .

3 The data of early rice harvest, spring rainfall and spring temperature obtained by a farm through experiments are shown in the table .

requirement ：
（1） Try to determine the binary linear regression equation of early rice harvest to spring rainfall and spring temperature .
（2） Explain the practical significance of the regression coefficient .
（3） In your judgment , Whether there is multicollinearity in the model ？

Explain ：（1） from Excel The output regression results are shown in the table .

　 Regression results

Analysis of variance

Therefore, the binary linear regression equation of early rice harvest to spring rainfall and spring temperature is ：

y＝－0.5910＋22.3865x1＋327.6717x2
（2） Regression coefficient β1＝22.3865 Express , Under the condition of constant temperature , Every increase in rainfall 1mm, The wheat harvest increased on average 22.3865kg/hm2; Regression coefficient β2＝327.6717 Express , Under the condition of constant rainfall , Every time the temperature increases 1℃, The wheat harvest increased on average 327.6717kg/hm2.
（3） From the relationship between rainfall, temperature and harvest , There is a strong relationship between the two variables and the harvest , And from the table 12-16 From the data of , There is also a strong relationship between temperature and rainfall , therefore , There may be multicollinearity in the model .

4 A real estate appraisal company wants to estimate the selling price of real estate in a city （y） And the appraised value of the property （x1）、 Property valuation （x2） And usable area （x3） Build a model , In order to make a reasonable prediction of the sales price . So , It collected 20 Real estate appraisal data of residential buildings , As shown in the table .

use Excel Regression , Answer the following question ：
（1） Write the estimated multiple regression equation .
（2） What is the proportion explained by the estimated regression equation in the total variation of sales price ？
（3） Test whether the linear relationship of the regression equation is significant （α＝0.05）.
（4） Test whether the regression coefficients are significant （α＝0.05）.

Explain ：（1） from Excel The output regression results are shown in the table .

Regression results

Analysis of variance

Therefore, the estimated multiple regression equation is ：y＝148.7005＋0.8147x1＋0.8210x2＋0.1350x3
（2） In multiple linear regression analysis , The adjusted judgment coefficient should be used to measure the goodness of fit of the regression model , Adjusted judgment coefficient Ra2＝87.83%, It indicates that the total variation of sales price , The proportion explained by the estimated regression equation is 87.83%.
（3） because Significance F＝3.88E－08＜α＝0.05, Therefore, the linear relationship of the regression equation is significant .
（4）β1 Of P value ＝0.1311＞α＝0.05, No significant ;β2 Of P value ＝0.0013≤α＝0.05, remarkable ;β3 Of P value ＝0.0571＞α＝0.05, No significant .

5 surface 12-21 It's random 15 Relevant data of similar products sold in large shopping malls （ Company ： element ）.

requirement ：
（1） Calculation y And x1、y And x2 The correlation coefficient between , Is there any evidence that the sales price and the purchase price 、 There is a linear relationship between selling price and selling expense ？
（2） Based on the above results , Do you think it is effective to predict the sales price by using the purchase price and selling expenses ？
（3） use Excel Regression , And test whether the linear relationship of the model is significant （α＝0.05）.
（4） Explain the judgment coefficient R2, Conclusions and problems （2） Whether it's consistent with ？
（5） Calculation x1 And x2 The correlation coefficient between , What does the result mean ？
（6） Whether there is multicollinearity in the model ？ What suggestions do you have for the model ？

Explain ：（1） from Excel Of “CORREL” Coefficient of function calculation

The test statistics are ：

take α＝0.05,t0.05/2（15－2）＝2.160. Because the test statistics t1＝1.1712＜tα/2＝2.160,t2＝0.0044＜tα/2＝2.160. Therefore, there is no evidence that the sales price and the purchase price 、 There is a linear relationship between selling price and selling expense .
（2） Because there is no evidence that the sales price and the purchase price 、 There is a linear relationship between selling price and selling expense , Therefore, it is useless to predict the sales price by using the purchase price and sales expenses .

（3） from Excel Output regression results , As shown in the table .

Regression results

Analysis of variance

So the regression equation is ：y＝375.6018＋0.5378x1＋1.4572x2
because Significance F＝0.073722＞α＝0.05, The linear relationship is not significant .
（4）R2＝35.25%,Ra2＝24.45%, It shows that the sales price is in the total variation , The proportion explained by the estimated regression equation of sales is 24.45%, It shows that the linear relationship is not significant , Conclusions and problems （2） Agreement .
（5） from Excel Of “CORREL” Coefficient of function calculation

The two independent variables are highly negatively correlated .
（6） Because the two independent variables are highly negatively correlated , Therefore, there is multicollinearity in the model , It is suggested to eliminate an independent variable from the model .