Statistics, 8th Edition, Jia Junping, Chapter 11 summary of knowledge points of univariate linear regression and answers to exercises after class

Catalog

One 、 Knowledge framework

Two 、 After-school exercises

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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 .