Question

OrderNo.	DeliveryTime	NumberOfPizzas	Distance	Location
1	16.68	7	5.6	Downtown
2	11.5	3	2.2	Not Downtown
3	12.03	3	3.4	Downtown
4	14.88	8	0.8	Not Downtown
5	13.75	6	1.5	Not Downtown
6	18.11	7	3.3	Downtown
7	8	2	1.1	Downtown
8	17.83	7	2.1	Downtown
9	79.24	30	14.6	Not Downtown
10	21.5	5	6.05	Not Downtown
11	40.33	16	6.88	Downtown
12	21	10	2.15	Not Downtown
13	13.5	4	2.55	Not Downtown
14	19.75	6	4.62	Not Downtown
15	24	9	4.48	Not Downtown
16	29	10	7.76	Not Downtown
17	15.35	6	2	Not Downtown
18	19	7	1.32	Downtown
19	9.5	3	0.36	Not Downtown
20	35.1	17	7.7	Downtown
21	17.9	10	1.4	Downtown
22	52.32	26	8.1	Not Downtown
23	18.75	9	4.5	Downtown
24	19.83	8	6.35	Not Downtown
25	10.75	4	1.5	Downtown
26	26	9	7.3	Not Downtown
27	14.21	5	2.4	Not Downtown
28	21	8	1.4	Downtown
29	10	4	0.9	Not Downtown
30	36	18	8	Downtown
31	18.1	9	1.5	Downtown

1. What is the correlation between downtown variable (1= downtown, 0=not downtown) and each of the other three variables? What does each of the relationship suggest?

2. Model A. Run a linear regression that predicts delivery time using number of pizzas, distance, and downtown as independent variables. Summarize and show results in a table. Explain each relationship.

3. What is R2? What does it mean?

4. What are the business implications of the regression results?

5. Model B. Run a linear regression that predicts delivery time using number of pizzas and distance. Summarize and show results in a table. Explain any differences from the previous model.

Answer 1

> Book1 <- read_excel("Book1.xlsx")
> d=Book1
> r1=cor(d$Location,d$DeliveryTime);r2=cor(d$NumberOfPizzas,d$Location);r3=cor(d$Distance,d$Location)
> r1;r2;r3
[1] -0.09029982
[1] -0.006621049
[1] -0.1282853
> ##here correlation of location variable (downtown=1, not downtwn=0) with remaining three variables (delievery time, number of pizzas and distance) are negative so if we are changing our location from not downtown to downtown then remaining three variable will decreases (i.e delievry time, no. pizzas, distance will decreases). Decrease in number of pizza will be very very small and decrease in delivery time will be small and dicrease in distance will be significant comparatively
>
> #
> m=lm(d$DeliveryTime~d$NumberOfPizzas+d$Distance+d$Location)
> m

Call:
lm(formula = d$DeliveryTime ~ d$NumberOfPizzas + d$Distance +
d$Location)

Coefficients:
(Intercept) d$NumberOfPizzas d$Distance d$Location
2.922 1.619 1.344 -1.346

> # fitted linear regression is DeliveryTime=2.922 + 1.619*NumberOfPizzas + 1.344*Distance - 1.346*Location it means that for increase in number of pizza by one delivery time will increase by 1.619 units while increasing distance by one unit delivery time will increase by 1.344 units and if we change location from not downtown to downtown then delievery time will reduces by 1.346 units
> summary(m)

Call:
lm(formula = d$DeliveryTime ~ d$NumberOfPizzas + d$Distance +
d$Location)

Residuals:
Min 1Q Median 3Q Max
-5.4641 -1.5229 0.0274 1.1334 8.1359

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.9223 1.1123 2.627 0.014016 *
d$NumberOfPizzas 1.6189 0.1470 11.016 1.71e-11 ***
d$Distance 1.3435 0.2978 4.511 0.000113 ***
d$Location -1.3461 1.1381 -1.183 0.247219
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.088 on 27 degrees of freedom
Multiple R-squared: 0.9586,   Adjusted R-squared: 0.954
F-statistic: 208.5 on 3 and 27 DF, p-value: < 2.2e-16

> # from summary we can say that all independent variables except location are significant for explaining delievery time
> summary(m)$r.squared ##R^2 = 0.9586 indicates that 95.86% of the total error is explained by above regression line
[1] 0.9586213
>
> #
> m1=lm(d$DeliveryTime~d$NumberOfPizzas+d$Distance)
> m1

Call:
lm(formula = d$DeliveryTime ~ d$NumberOfPizzas + d$Distance)

Coefficients:
(Intercept) d$NumberOfPizzas d$Distance
2.275 1.591 1.415

> summary(m1)

Call:
lm(formula = d$DeliveryTime ~ d$NumberOfPizzas + d$Distance)

Residuals:
Min 1Q Median 3Q Max
-6.2376 -1.0912 0.0869 1.3644 8.5679

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.2746 0.9750 2.333 0.0271 *
d$NumberOfPizzas 1.5912 0.1461 10.890 1.42e-11 ***
d$Distance 1.4151 0.2937 4.818 4.56e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.11 on 28 degrees of freedom
Multiple R-squared: 0.9565,   Adjusted R-squared: 0.9534
F-statistic: 307.7 on 2 and 28 DF, p-value: < 2.2e-16

> summary(m1)$r.squared #R^2=0.9565 indicates that 95.65% of total variation is explianed by above regression line
[1] 0.9564774
> #Difference between modelA (model m) and modelB(model m1) is that coefficients of number of pizza and distance are changed. also R^2 for model m is greater(but not much significantly greater) than that for model m1 this is because nuber of independent variable in model m is more than that in model m1.
>