The TimesObserver is a daily newspaper in Metro City Like ma

The Times-Observer is a daily newspaper in Metro City. Like many city newspapers, the Times-

Observer is suffering through difficult financial times. The circulation manager is studying other

papers in similar cities in the United States and Canada. She is particularly interested in what

variables relate to the number of subscriptions to the paper. She is able to obtain the following

sample information on 25 newspapers in similar cities.

a. What is the dependent variable?

b. What are the independent variables?

c. Create a scatter diagram for each independent variable acting on the dependent variable.

Comment.

d. What is the Null Hypothesis for the overall model?

e. What is the Alternative Hypothesis for the overall model?

f. What is the multiple regression equation?

g. What is the standard error? Describe what this means as related to the problem.

h. For your initial model equation, discuss whether R2 or Adjusted R2 should be used in interpreting

how good the linear equation is. Then, interpret the appropriate R2 value.

i. What does the overall statistical significance explain with respect to the model?

j. Evaluate the p-value for each independent variable. Should any of the independent variables be

dropped? Why or why not?

k. Develop a correlation matrix for the independent variables. Should any of the independent

variables be dropped? Why or why not?

l. Is this a good model?

Subscriptions	Population	Adv	Income
37.95	523.76	13.68	33.47
37.66	523.84	13.99	33.62
37.55	528.95	14.04	33.63
38.78	529.46	14.16	33.64
37.67	536.91	15.29	33.64
38.23	539.15	16.20	33.93
36.90	546.44	16.79	33.99
38.28	566.57	17.03	34.02
38.95	575.44	18.51	34.06
39.27	575.76	18.78	34.45
38.30	577.79	19.12	34.65
38.84	577.86	20.40	34.82
38.14	589.78	20.83	34.90
38.39	593.69	20.99	34.99
37.29	599.17	21.82	35.28
39.15	599.78	21.83	35.37
38.29	602.51	21.85	35.40
38.09	611.09	21.87	35.44
37.83	624.21	21.94	35.48
39.37	631.36	23.11	35.64
37.81	643.64	23.49	35.65
37.42	653.29	24.37	35.68
38.83	658.38	25.77	35.69
38.33	673.55	26.15	35.90
40.24	675.00	26.50	36.00
Independent variable	Population	Income
Dependent Variables	Subscriptions	Advertising

Solution

a. What is the dependent variable?

Here dependent variable (Y) is Subscriptions.

b. What are the independent variables?

And there are three independent variables as Population, Adv and Income.

Let X1 = Population

X2 = Adv

X3 = Income.

c. Create a scatter diagram for each independent variable acting on the dependent variable and comment.

Scatter diagram by using MINITAB

Steps :

START --> GRAPH --> Scatter plots --> simple --> ok --> Y-variable : Y --> X-variable -X1 -->Y-variable : Y --> X-variable -X2 --> Y-variable : Y --> X-variable -X3 --> ok

This will gives us three scatter diargram.

and from the scatter diagram we say that in all the three cases there is positive correalation between dependent and independent variables.

d. What is the Null Hypothesis for the overall model?

There are three independent variables X1,X2 and X3.

Null hypothesis is,

H0: B1 = B2 = B3 = 0

where B1, B2 and B3 are population slope.

e. What is the Alternative Hypothesis for the overall model?

Alternative hypothesis is,

H1 : At least one B is not zero

f. What is the multiple regression equation?

Now we can done this by using MINITAB.

Steps :

STAT --> Regression --> Regression --> Response : Y --> Predicors --> X1,X2 and X3 --> Results : select second option --> ok --> ok

Output is,

Regression Analysis: Y versus X1, X2, X3

The regression equation is
Y = 63.6 - 0.0034 X1 + 0.282 X2 - 0.829 X3

Predictor Coef SE Coef T P
Constant 63.56 27.42 2.32 0.031
X1 -0.00344 0.01690 -0.20 0.841
X2 0.2815 0.2798 1.01 0.326
X3 -0.8293 0.8602 -0.96 0.346

S = 0.735304 R-Sq = 16.4% R-Sq(adj) = 4.5%

Analysis of Variance

Source DF SS MS F P
Regression 3 2.2341 0.7447 1.38 0.277
Residual Error 21 11.3541 0.5407
Total 24 13.5883

The regression equation is,

Y = 63.6 - 0.0034 X1 + 0.282 X2 - 0.829 X3

g. What is the standard error? Describe what this means as related to the problem.

Standard error is  0.735304  and is the square root of the error variance. S represents the average distance that the observed values fall from the regression line. Conveniently, it tells you how wrong the regression model is on average using the units of the response variable. Smaller values are better because it indicates that the observations are closer to the fitted line.

h. For your initial model equation, discuss whether R2 or Adjusted R2 should be used in interpreting

R2 = 16.4% = 16.4/100 = 0.164

Adjusted R2 = 4.5% = 4.5/100 = 0.045

R2 expresses the proportion of the variation in Y which is explained by X1, X2 and X3.

i. What does the overall statistical significance explain with respect to the model?

In the ANOVA table,

P-value = 0.277

Assume alpha = 0.05

P-value > alpha

Accept H0 at 5% level of significance.

Conclusion : All the slopes are equal to 0.

j. Evaluate the p-value for each independent variable. Should any of the independent variables be

dropped? Why or why not?

P-value for X1 is 0.841.

P-value for X is 0.326

P-value for X3 is 0.346

All the P-values are greator than 0.05.

Accept H0.

neither X1 or X2 and X3 belonged in the model.

k. Develop a correlation matrix for the independent variables. Should any of the independent

variables be dropped? Why or why not?

correlation matrix we can form in EXCEL.

syntax :

=CORREL(X1,X1)

=CORREL(X1,X2)

=CORREL(X1,X3)

=CORREL(X2,X1)

=CORREL(X2,X2)

=CORREL(X2,X3)

=CORREL(X3,X1)

=CORREL(X3,X2)

=CORREL(X3,X3)

	X1	X2	X3
X1	1	0.982272	0.958614
X2	0.982272	1	0.978668
X3	0.958614	0.978668	1