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Statistics, 8th Edition, Jia Junping, Chapter 11 summary of knowledge points of univariate linear regression and answers to exercises after class
2022-07-06 14:31:00 【No two or three things】
Catalog
One 、 Knowledge framework
Two 、 After-school exercises
1 Randomly selected from a certain industry 12 companies , The data of output and production cost are shown in the table .
requirement :
(1) Draw a scatter chart of output and production cost , Judge the relationship between the two .
(2) Calculate the linear correlation coefficient between output and production cost .
(3) Test the significance of the correlation coefficient (α=0.05), And explain the strength of the relationship between the two .
Explain :(1) Draw a scatter plot , As shown in the figure .
As can be seen from the scatter diagram , There is a positive linear correlation between output and production cost .
(2) Let the output be a variable X, Production cost is variable Y. It can be calculated from the data in the table :
x=85.42y=160.08
Therefore, the correlation coefficient is :
(3) First put forward the following assumptions :H0:ρ=0;H1:ρ≠0. Calculate the value of the test statistic :
When α=0.05 when ,t0.05/2(12-2)=2.228. Because the value of the test statistic t=7.423>tα/2=2.228, So reject the original assumption . It shows that the linear relationship between output and production cost is significant .
2 Random sampling 10 Home airline , The flight punctuality rate and the number of customer complaints in the recent year were investigated , The data obtained are shown in the table .
requirement :
(1) Draw a scatter plot , Explain the relationship form between the two .
(2) The flight punctuality rate is used as the independent variable , The number of customer complaints is the dependent variable , Find the estimated regression equation , And explain the significance of regression coefficient .
(3) Test the significance of the regression coefficient (α=0.05).
(4) If the flight punctuality rate is 80%, Estimate the number of customer complaints .
(5) The flight punctuality rate is 80% when , Number of customer complaints 95% Confidence interval and prediction interval .
Explain :(1) Draw a scatter plot , As shown in the figure .
As can be seen from the scatter diagram , There is a negative linear correlation between the flight punctuality rate and the number of complaints .
(2) from Excel The output regression results are shown in table 1 Shown .
The linear regression equation is :y=430.1892-4.7x
Regression coefficient β1=-4.7 It means that every increase in flight punctuality 1%, The number of customer complaints decreased on average 4.7 Time .
(3) Regression coefficient test P- value =0.001108<α=0.05, Rejection of null hypothesis , The regression coefficient is significant .
(4)y80=430.1892-4.7×80=54.1892( Time ).
(5) When α=0.05 when ,t0.05/2(10-2)=2.306,se=18.88722.
The confidence interval is :
namely (37.7,70.7).
The prediction interval is :
namely (7.6,100.8).
3 Here is 20 Data of office occupancy rate and monthly rent per square meter in cities , As shown in the table .
Let monthly rent be the independent variable , The occupancy rate is a dependent variable , use Excel Regression , And explain and analyze the results .
Explain :Excel Output regression results , As shown in the table .
From the table, we can get , The linear regression equation is :Y=49.3177+0.2492X
Regression coefficient β1=0.2492 Express : Every increase in monthly rent 1 element , The rental rate increased on average 0.2492%.
R2=63.22%, It shows that the proportion explained by the linear relationship between rental rate and rent in the variation of rental rate is 63.22%, The fitting degree of the regression equation is general .
Estimate the standard error se=2.6858 Express , When using monthly rent to predict the rental rate , The average prediction error is 2.6858%, It shows that the prediction error is not big .
From the analysis of variance table ,Significance F=2.79889E-05<α=0.05, That is, the linear relationship of the regression equation is significant . Regression coefficient test P- value =0.0000<α=0.05, It shows that the regression coefficient is significant , That is, monthly rent is a significant factor affecting the rental rate .4 An automobile manufacturer wants to know about advertising expenses (x) Sales volume (y) Influence , Collected the past 12 Relevant data of . Relevant results obtained through calculation , As shown in the table .
requirement :
(1) Complete the above ANOVA table .
(2) How much of the decline in car sales is caused by changes in advertising expenses ?
(3) What is the correlation coefficient between sales volume and advertising expenses ?
(4) Write the estimated regression equation and explain the practical significance of the regression coefficient .
(5) Test the significance of the linear relationship (α=0.05).
Explain :(1) Fill in the ANOVA table ,
(2) According to the variance analysis table, the determination coefficient is :
R2=SSR/SST=1602708.6/1642866.67=97.56%
This shows that the deterioration of car sales includes 97.56% It is caused by the change of advertising expenses .
(3) The correlation coefficient can be obtained from the square root of the determination coefficient :
(4) The regression equation is :y=363.6891+1.420211x
Regression coefficient β1=1.420211 It means that every increase in advertising expenses 1 A unit of , Sales increased on average 1.420211 A unit of .
(5) because Significance F=2.17E-09<α=0.05, It shows that the linear relationship between advertising expenses and sales volume is significant .
5 Establish the regression equation according to the data in the table , Calculated residual error 、 Determination factor R2, Estimate the standard error se, And analyze the fitting degree of the regression equation .
Explain :Excel The output regression results are shown in the table .
From the table, we can get , The linear regression equation is :y=13.6254+2.3029x
The regression coefficient shows ,x Every increase 1 A unit of ,y Average increase 2.3029 A unit of ; Determination factor R2=93.74%, It shows that the fitting degree of the regression equation is high ; Estimate the standard error se=3.8092, Show that x To predict the y The time average prediction error is 3.8092.
6 Random sampling 7 A supermarket , The data of advertising expenses and sales are shown in the table .
requirement :
(1) Advertising expenses are used as independent variables x, Sales volume is the dependent variable y, Find the estimated regression equation .
(2) Test whether the linear relationship between advertising expenses and sales is significant (α=0.05).
(3) Draw about x Residual diagram of , What do you think about the error term ε Are your assumptions satisfied ?
(4) You choose this model , Or looking for a better model ?
Explain :(1)Excel The output regression results are shown in the table .
It can be concluded from the table , The estimated regression equation is :y=29.3991+1.547478x
(2) because Significance F=0.020582<α=0.05, Therefore, the linear relationship between advertising expenses and sales is significant .
(3) Draw about x Residual diagram of , As shown in the figure .
As can be seen from the residual diagram , About the error term ε The assumption of does not hold .
(4) Although the linear relationship passed the significance test , But from the residual diagram , About x And y The assumption that there is a linear relationship between them is still questionable . Therefore, the nonlinear model can be considered .
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