1)
First, the data is entered in Excel:
Sales | Ad_Dollars | Accounts | Competitors | Potential |
79.3 | 5.5 | 31 | 10 | 8 |
200.1 | 2.5 | 55 | 8 | 6 |
163.2 | 8 | 67 | 12 | 9 |
200.1 | 3 | 50 | 7 | 16 |
146 | 3 | 38 | 8 | 15 |
177.7 | 2.9 | 71 | 12 | 17 |
30.9 | 8 | 30 | 12 | 8 |
291.9 | 9 | 56 | 5 | 10 |
160 | 4 | 42 | 8 | 4 |
339.4 | 6.5 | 73 | 5 | 16 |
159.6 | 5.5 | 60 | 11 | 7 |
86.3 | 5 | 44 | 12 | 12 |
237.5 | 6 | 50 | 6 | 6 |
107.2 | 5 | 39 | 10 | 4 |
155 | 3.5 | 55 | 10 | 4 |
291.4 | 8 | 70 | 6 | 14 |
100.2 | 6 | 40 | 11 | 6 |
135.8 | 4 | 50 | 11 | 8 |
223.3 | 7.5 | 62 | 9 | 13 |
195 | 7 | 59 | 9 | 11 |
73.4 | 6.7 | 53 | 13 | 5 |
47.7 | 6.1 | 38 | 13 | 10 |
140.7 | 3.6 | 43 | 9 | 17 |
93.5 | 4.2 | 26 | 8 | 3 |
259 | 4.5 | 75 | 8 | 19 |
331.2 | 5.6 | 71 | 4 | 9 |
The data is first visualized through a matrix of scatterplots:
The above diagram shows that the Sales is weakly correlated with Ad Dollars and mildly with Potential, positively correlated with Accounts, and negatively correlated with Competitors. We first build a linear model as function of all variables:
> tt <- read.csv("clipboard",header=TRUE,sep="\t")
> pairs(tt)
> sales_lm <- lm(Sales~.,tt)
> summary(sales_lm)
Call:
lm(formula = Sales ~ ., data = tt)
Residuals:
Min
1Q Median
3Q Max
-19.0906 -5.9796 0.8968 6.5667 14.7985
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 178.3203 12.9603 13.759 5.62e-12
***
Ad_Dollars 1.8071
1.0810 1.672
0.109
Accounts
3.3178 0.1629 20.368 2.60e-15 ***
Competitors -21.1850 0.7879 -26.887 <
2e-16 ***
Potential 0.3245
0.4678 0.694
0.495
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 9.604 on 21 degrees of freedom
Multiple R-squared: 0.9892, Adjusted R-squared:
0.9871
F-statistic: 479.1 on 4 and 21 DF, p-value: < 2.2e-16
The model is significant, and the covariates Accounts and Potential are significant variables.
The overall elasticity is 40.68815.
>
predict(sales_lm,newdata=list(Ad_Dollars=8,Accounts=30,Competitors=12,Potential=8)
+ )
1
40.68815
2)
a)
family |
square feet |
income |
family size |
senior parent |
parent education |
1 |
2240 |
60.8 |
2 |
0 |
4 |
2 |
2380 |
68.4 |
2 |
1 |
6 |
3 |
3640 |
104.5 |
3 |
0 |
7 |
4 |
3360 |
89.3 |
1 |
1 |
0 |
5 |
3080 |
72.2 |
4 |
0 |
2 |
6 |
2940 |
114.3 |
1 |
1 |
10 |
7 |
4480 |
125.4 |
6 |
0 |
6 |
8 |
2520 |
83.6 |
3 |
0 |
8 |
9 |
4200 |
133.5 |
5 |
0 |
2 |
10 |
2800 |
95.3 |
3 |
0 |
6 |
Regression Analysis: square feet versus income family ....nt, Education
Stepwise selection of terms
α to enter = 0.15, α to remove = 0.15
Analysis of Variance
Source DF Adj SS Adj MS F-value P-value
Regression 2 4619699 2309849 30.90 0.000
income 1 380692 380692 5.09 0.059
family size 1 962595 962595 12.88 0.009
Error 7 523341 74763
Total 9 5143040
Model Summary
S R-sq R-sq(adj) R-sq(pred)
273.428 89.82% 86.92% 82.48%
Coefficients
Term Coef SE Coef T-value p-value VIF
Constant 713 363 1.96 0.091
income 12.15 5.38 2.26 0.0059 2.08
family size 372 104 3.59 0.009 2.08
Regression Equation
Square feet = 713 + 12.15 income + 372 family size
Fits and Diagnostics for Unusual Observations
Obs Square feet fit Resid Std Resid
3 3640 3098 542 2.25 R
R large residual
From the above output, the multiple linear regression equation is,
Y = 713 + 12.15 (income, X1) + 372 (Family size, X2)
b)
Family size and income are independent variables. That should be in the final model.
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