Question

use methods of descriptive statistics to summarize the data and comment on your findings -

Income ($1000s)	Household Size	Amount Charged ($)
54	3	4,016
30	2	3,159
32	4	5,100
50	5	4,742
31	2	1,864
55	2	4,070
37	1	2,731
40	2	3,348
66	4	4,764
51	3	4,110
25	3	4,208
48	4	4,219
27	1	2,477
33	2	2,514
65	3	4,214
63	4	4,965
42	6	4,412
21	2	2,448
44	1	2,995
37	5	4,171
62	6	5,678
21	3	3,623
55	7	5,301
42	2	3,020
41	7	4,828
54	6	5,573
30	1	2,583
48	2	3,866
34	5	3,586
67	4	5,037
50	2	3,605
67	5	5,345
55	6	5,370
52	2	3,890
62	3	4,705
64	2	4,157
22	3	3,579
29	4	3,890
39	2	2,972
35	1	3,121
39	4	4,183
54	3	3,730
23	6	4,127
27	2	2,921
26	7	4,603
61	2	4,273
30	2	3,067
22	4	3,074
46	5	4,820
66	4	5,149

Answer 1

> Income=scan()
1: 54   30   32   50   31   55   37   40   66   51   25   48   27   33   65   63   42   21   44   37   62   21   55   42   41   54   30   48   34   67   50   67   55   52   62   64   22   29   39   35   39   54   23   27   26   61   30   22   46   66
51:
Read 50 items
> Household=scan()
1: 3   2   4   5   2   2   1   2   4   3   3   4   1   2   3   4   6   2   1   5   6   3   7   2   7   6   1   2   5   4   2   5   6   2   3   2   3   4   2   1   4   3   6   2   7   2   2   4   5   4
51:
Read 50 items
> Amount=scan()
1: 4016   3159   5100   4742   1864   4070   2731   3348   4764   4110   4208   4219   2477   2514   4214   4965   4412   2448   2995   4171   5678   3623   5301   3020   4828   5573   2583   3866   3586   5037   3605   5345   5370   3890   4705   4157   3579   3890   2972   3121   4183   3730   4127   2921   4603   4273   3067   3074   4820   5149
51:
Read 50 items
> d=cbind(Income,Household,Amount)
> colMeans(d)
Income Household Amount
43.48 3.42 3964.06
> c(median(d[,1]),median(d[,2]),median(d[,3]))#medians
[1] 42 3 4090
> hist(d[,1])

> hist(d[,2])

> ##household is positively skewed i.e most of people have small houses. Number of people having large house size are less in number
> hist(d[,3])

> ##Amount is slightly negatively skewed
> cor(d)
Income Household Amount
Income 1.0000000 0.1725335 0.6309742
Household 0.1725335 1.0000000 0.7528432
Amount 0.6309742 0.7528432 1.0000000
> #household size and amount are highly correlated. Income and amount are more correlated as compare to that income and household size
> sqrt(round(c(var(d[,1]),var(d[,2]),var(d[,3])),5))#standard deviation
[1] 14.550742 1.738988 933.494082
> sqrt(round(c(var(d[,1]),var(d[,2]),var(d[,3])),5))#standard deviation
[1] 14.550742 1.738988 933.494082
> sqrt(round(c(var(d[,1]),var(d[,2]),var(d[,3])),5))/colMeans(d)#coefficient of variation
Income Household Amount
0.3346537 0.5084761 0.2354894
> #regression on amount charged using income and household as predictors
> m=lm(Amount~Income+Household)
> m

Call:
lm(formula = Amount ~ Income + Household)

Coefficients:
(Intercept) Income Household
1304.90 33.13 356.30

> summary(m)

Call:
lm(formula = Amount ~ Income + Household)

Residuals:
Min 1Q Median 3Q Max
-1180.62 -155.31 7.05 194.56 1309.66

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1304.905 197.655 6.602 3.29e-08 ***
Income 33.133 3.968 8.350 7.68e-11 ***
Household 356.296 33.201 10.732 3.12e-14 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 398.1 on 47 degrees of freedom
Multiple R-squared: 0.8256,   Adjusted R-squared: 0.8181
F-statistic: 111.2 on 2 and 47 DF, p-value: < 2.2e-16

>
>
> #model is Amount = 1304.905 + 33.133*Income + 356.296*Household
> #for unit increase in household size amount charged increases by 356.296 and for unit increase in Income amount increases by 33.133