knitr::opts_chunk$set(echo = TRUE, warning=FALSE, message=FALSE)
library(readxl)
library(tidyverse)
library(ggplot2)
library(dplyr)
library(stringr)
library(alr4)
library(smss)Homework 4
hw4
1a.
Prediction equation: ŷ = −10,536 + 53.8x1 + 2.84x2.
#plugging in size of home and lot size into prediction equation
-10536 + 53.8*1240 + 2.84*18000[1] 107296#Calculating residual, actual-predicted
145000-107296[1] 37704Predicted selling price $107,296 Residual: $37,704
1b.
According to the prediction equations, for a fixed lot size, the price of the house is predicted to increase $53.80 per square foot of house size.
1c.
53.8/2.84[1] 18.94366For a fixed home size, lot size would need to increase by 18.94366 feet to have the same impact as a one square foot increase in home size.
#Check:
-10536 + 53.8*1400 + 2.84*20000[1] 121584-10536 + 53.8*1400+ 2.84*20018.94366[1] 121637.8121637.8-121584[1] 53.82a.
H0: Salary(male)=Salary(female) Ha:Salary(male) not equal to Salary(female)
data(salary)
head(salary) degree rank sex year ysdeg salary
1 Masters Prof Male 25 35 36350
2 Masters Prof Male 13 22 35350
3 Masters Prof Male 10 23 28200
4 Masters Prof Female 7 27 26775
5 PhD Prof Male 19 30 33696
6 Masters Prof Male 16 21 28516t.test(salary ~ sex, data=salary)
Welch Two Sample t-test
data: salary by sex
t = 1.7744, df = 21.591, p-value = 0.09009
alternative hypothesis: true difference in means between group Male and group Female is not equal to 0
95 percent confidence interval:
-567.8539 7247.1471
sample estimates:
mean in group Male mean in group Female
24696.79 21357.14 According to this two sample t.test, there is not evidence for a difference in salary between male and female professors at the 5% significance level. At the 10% significance level, there is a difference.
2b.
#creating a model
summary(lm(salary ~ degree + rank + sex + year + ysdeg, data=salary))
Call:
lm(formula = salary ~ degree + rank + sex + year + ysdeg, data = salary)
Residuals:
Min 1Q Median 3Q Max
-4045.2 -1094.7 -361.5 813.2 9193.1
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 15746.05 800.18 19.678 < 2e-16 ***
degreePhD 1388.61 1018.75 1.363 0.180
rankAssoc 5292.36 1145.40 4.621 3.22e-05 ***
rankProf 11118.76 1351.77 8.225 1.62e-10 ***
sexFemale 1166.37 925.57 1.260 0.214
year 476.31 94.91 5.018 8.65e-06 ***
ysdeg -124.57 77.49 -1.608 0.115
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2398 on 45 degrees of freedom
Multiple R-squared: 0.855, Adjusted R-squared: 0.8357
F-statistic: 44.24 on 6 and 45 DF, p-value: < 2.2e-16#assigning model to an object
prof_fit_1<- lm(salary ~ degree + rank + sex + year + ysdeg, data=salary)
#Creating a confidence interval for coefficients in the model
confint(prof_fit_1) 2.5 % 97.5 %
(Intercept) 14134.4059 17357.68946
degreePhD -663.2482 3440.47485
rankAssoc 2985.4107 7599.31080
rankProf 8396.1546 13841.37340
sexFemale -697.8183 3030.56452
year 285.1433 667.47476
ysdeg -280.6397 31.49105The 95% confidence interval for the difference in salary between male and females is -697.82 and 3030.56.
2c.
Degree: The p-value for degree is not statistically significant. However, acccoring to this regression equation, for a faculty member with a PhD, their predicted salary is $1,388.61 higher than a faculty member with a masters degree (all other variables held constant).
Rank: The baseline category is Asst professor. For a faculty member of rank Associate, all other variables held constant, their predicted salary is $5,292.36 more than an Asst Professor.
For a faculty member of rank Professor, the predicted salary is $11,118.76 more than a faculty of rank Asst Professor (all other variables held constant).
These salary differences are statistically significant at the 0.0001 alpha level for both Asst and Professor rank.
Sex: For a faculty member who is female, their predicted salary is $1166.37 more than a faculty member who is male. However, his coefficient is not statistically significant at any alpha level.
Year: For each year in their current rank, the salary is expected to increase by $478.31. The coeffiticent is significant at the 0.0001 alpha level.
ysdegree: For eah year after completion of their highest degree, salary is expected to decrease by $124.57. However this coefficient is not significant at any alpha level.
2d.
salary$rank<- relevel(salary$rank, ref = 'Assoc')
summary(lm(salary ~ degree + rank + sex + year + ysdeg, data=salary))
Call:
lm(formula = salary ~ degree + rank + sex + year + ysdeg, data = salary)
Residuals:
Min 1Q Median 3Q Max
-4045.2 -1094.7 -361.5 813.2 9193.1
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 21038.41 1109.12 18.969 < 2e-16 ***
degreePhD 1388.61 1018.75 1.363 0.180
rankAsst -5292.36 1145.40 -4.621 3.22e-05 ***
rankProf 5826.40 1012.93 5.752 7.28e-07 ***
sexFemale 1166.37 925.57 1.260 0.214
year 476.31 94.91 5.018 8.65e-06 ***
ysdeg -124.57 77.49 -1.608 0.115
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2398 on 45 degrees of freedom
Multiple R-squared: 0.855, Adjusted R-squared: 0.8357
F-statistic: 44.24 on 6 and 45 DF, p-value: < 2.2e-16The baseline category is now Assoc. According to these coefficients, faculty of rank asst are expected to make $5292.36 less than Associate professors. Faculty of rank Professor are expected to make $5826.40 more than Associate professors.
2e.
summary(lm(salary ~ degree + sex + year + ysdeg, data=salary))
Call:
lm(formula = salary ~ degree + sex + year + ysdeg, data = salary)
Residuals:
Min 1Q Median 3Q Max
-8146.9 -2186.9 -491.5 2279.1 11186.6
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 17183.57 1147.94 14.969 < 2e-16 ***
degreePhD -3299.35 1302.52 -2.533 0.014704 *
sexFemale -1286.54 1313.09 -0.980 0.332209
year 351.97 142.48 2.470 0.017185 *
ysdeg 339.40 80.62 4.210 0.000114 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3744 on 47 degrees of freedom
Multiple R-squared: 0.6312, Adjusted R-squared: 0.5998
F-statistic: 20.11 on 4 and 47 DF, p-value: 1.048e-09Excluding the rank variable reveals a difference between male and female salaries with females making $1286.54 less than men. However, this difference is not signficant at any standard alpha levels.
2f.
#creating a dummy variable new and old dean
salary<-mutate(salary, dean= case_when(ysdeg < 15 ~"new",
ysdeg >=15 ~"old"))summary(lm(salary ~ dean + degree + sex + rank +year, data=salary))
Call:
lm(formula = salary ~ dean + degree + sex + rank + year, data = salary)
Residuals:
Min 1Q Median 3Q Max
-3588.0 -1532.2 -232.2 565.7 9132.5
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 20468.7 951.7 21.507 < 2e-16 ***
deanold -2421.6 1187.9 -2.038 0.0474 *
degreePhD 1073.5 843.3 1.273 0.2096
sexFemale 1046.7 858.0 1.220 0.2289
rankAsst -5012.5 1002.3 -5.001 9.16e-06 ***
rankProf 6213.3 1045.0 5.946 3.76e-07 ***
year 450.7 81.5 5.530 1.55e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2360 on 45 degrees of freedom
Multiple R-squared: 0.8597, Adjusted R-squared: 0.841
F-statistic: 45.95 on 6 and 45 DF, p-value: < 2.2e-16I excluded ysdegree after creating the new variable to prevent multicollinearity. (Because one variable is made from the other.)
According to this equation, faculty hired by the old dean make $2421.60 less than new faculty when we control for other factors. This is significant at the 0.05 alpha level.
3a.
data(house.selling.price)
house.selling.price case Taxes Beds Baths New Price Size
1 1 3104 4 2 0 279900 2048
2 2 1173 2 1 0 146500 912
3 3 3076 4 2 0 237700 1654
4 4 1608 3 2 0 200000 2068
5 5 1454 3 3 0 159900 1477
6 6 2997 3 2 1 499900 3153
7 7 4054 3 2 0 265500 1355
8 8 3002 3 2 1 289900 2075
9 9 6627 5 4 0 587000 3990
10 10 320 3 2 0 70000 1160
11 11 630 3 2 0 64500 1220
12 12 1780 3 2 0 167000 1690
13 13 1630 3 2 0 114600 1380
14 14 1530 3 2 0 103000 1590
15 15 930 3 1 0 101000 1050
16 16 590 2 1 0 70000 770
17 17 1050 3 2 0 85000 1410
18 18 20 3 1 0 22500 1060
19 19 870 2 2 0 90000 1300
20 20 1320 3 2 0 133000 1500
21 21 1350 2 1 0 90500 820
22 22 5616 4 3 1 577500 3949
23 23 680 2 1 0 142500 1170
24 24 1840 3 2 0 160000 1500
25 25 3680 4 2 0 240000 2790
26 26 1660 3 1 0 87000 1030
27 27 1620 3 2 0 118600 1250
28 28 3100 3 2 0 140000 1760
29 29 2070 2 3 0 148000 1550
30 30 830 3 2 0 69000 1120
31 31 2260 4 2 0 176000 2000
32 32 1760 3 1 0 86500 1350
33 33 2750 3 2 1 180000 1840
34 34 2020 4 2 0 179000 2510
35 35 4900 3 3 1 338000 3110
36 36 1180 4 2 0 130000 1760
37 37 2150 3 2 0 163000 1710
38 38 1600 2 1 0 125000 1110
39 39 1970 3 2 0 100000 1360
40 40 2060 3 1 0 100000 1250
41 41 1980 3 1 0 100000 1250
42 42 1510 3 2 0 146500 1480
43 43 1710 3 2 0 144900 1520
44 44 1590 3 2 0 183000 2020
45 45 1230 3 2 0 69900 1010
46 46 1510 2 2 0 60000 1640
47 47 1450 2 2 0 127000 940
48 48 970 3 2 0 86000 1580
49 49 150 2 2 0 50000 860
50 50 1470 3 2 0 137000 1420
51 51 1850 3 2 0 121300 1270
52 52 820 2 1 0 81000 980
53 53 2050 4 2 0 188000 2300
54 54 710 3 2 0 85000 1430
55 55 1280 3 2 0 137000 1380
56 56 1360 3 2 0 145000 1240
57 57 830 3 2 0 69000 1120
58 58 800 3 2 0 109300 1120
59 59 1220 3 2 0 131500 1900
60 60 3360 4 3 0 200000 2430
61 61 210 3 2 0 81900 1080
62 62 380 2 1 0 91200 1350
63 63 1920 4 3 0 124500 1720
64 64 4350 3 3 0 225000 4050
65 65 1510 3 2 0 136500 1500
66 66 4154 3 3 0 381000 2581
67 67 1976 3 2 1 250000 2120
68 68 3605 3 3 1 354900 2745
69 69 1400 3 2 0 140000 1520
70 70 790 2 2 0 89900 1280
71 71 1210 3 2 0 137000 1620
72 72 1550 3 2 0 103000 1520
73 73 2800 3 2 0 183000 2030
74 74 2560 3 2 0 140000 1390
75 75 1390 4 2 0 160000 1880
76 76 5443 3 2 0 434000 2891
77 77 2850 2 1 0 130000 1340
78 78 2230 2 2 0 123000 940
79 79 20 2 1 0 21000 580
80 80 1510 4 2 0 85000 1410
81 81 710 3 2 0 69900 1150
82 82 1540 3 2 0 125000 1380
83 83 1780 3 2 1 162600 1470
84 84 2920 2 2 1 156900 1590
85 85 1710 3 2 1 105900 1200
86 86 1880 3 2 0 167500 1920
87 87 1680 3 2 0 151800 2150
88 88 3690 5 3 0 118300 2200
89 89 900 2 2 0 94300 860
90 90 560 3 1 0 93900 1230
91 91 2040 4 2 0 165000 1140
92 92 4390 4 3 1 285000 2650
93 93 690 3 1 0 45000 1060
94 94 2100 3 2 0 124900 1770
95 95 2880 4 2 0 147000 1860
96 96 990 2 2 0 176000 1060
97 97 3030 3 2 0 196500 1730
98 98 1580 3 2 0 132200 1370
99 99 1770 3 2 0 88400 1560
100 100 1430 3 2 0 127200 1340summary(lm(Price ~ Size + New, data= house.selling.price))
Call:
lm(formula = Price ~ Size + New, data = house.selling.price)
Residuals:
Min 1Q Median 3Q Max
-205102 -34374 -5778 18929 163866
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -40230.867 14696.140 -2.738 0.00737 **
Size 116.132 8.795 13.204 < 2e-16 ***
New 57736.283 18653.041 3.095 0.00257 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 53880 on 97 degrees of freedom
Multiple R-squared: 0.7226, Adjusted R-squared: 0.7169
F-statistic: 126.3 on 2 and 97 DF, p-value: < 2.2e-16According to the coefficient for size, the price of a house is expected to increase by $116.132 for each square foot increase in size. The coefficient is significant at the 0.0001 alpha level, meaning there is a strong correlation between size and price when the age status (new/old) is held fixed.
According to the coefficient for new, a new house is expected to cost $57,736.283 more than an old house. This variable is significant at the 0.001 level, meaning that whether a house is old or new has a strong positive impact on price of the house.
3b.
Y = -40230.867 + 116.132(X1) + 57736.283 (X2) where X1 represents size and X2 represents new/old.
For a new house: Y = -40230.867 + 116.132(size) + 57736.283
For an old house: Y = -40230.867 + 116.132(size)
3c.
#new:
-40230.867 + 116.132*3000 + 57736.283[1] 365901.4#not new:
-40230.867 + 116.132*3000 + 57736.283*0[1] 308165.1New: $365,901.40 Not new: $308165.10
3d.
summary(lm(Price ~ Size + New + Size*New, data= house.selling.price))
Call:
lm(formula = Price ~ Size + New + Size * New, data = house.selling.price)
Residuals:
Min 1Q Median 3Q Max
-175748 -28979 -6260 14693 192519
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -22227.808 15521.110 -1.432 0.15536
Size 104.438 9.424 11.082 < 2e-16 ***
New -78527.502 51007.642 -1.540 0.12697
Size:New 61.916 21.686 2.855 0.00527 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 52000 on 96 degrees of freedom
Multiple R-squared: 0.7443, Adjusted R-squared: 0.7363
F-statistic: 93.15 on 3 and 96 DF, p-value: < 2.2e-16Y = -2227.808 + 104.438(size) + 61.916(size:new) -78527.502(new) Both size and the interaction term between size and new are statistically significant. The new/coefficient is no longer statistically significant.
3e.
For a new house: Y = -2227.808 + 166.354(size) - 78527.502 Old: Y = -2227.808 + 104.438(size)
3f.
#new:
-2227.808 + 166.354*3000 - 78527.502[1] 418306.7#not new:
-2227.808 + 104.438*3000[1] 311086.2New: $418,306.70 Not new: $311,086.20
3g.
#new:
-2227.808 + 166.354*1500 - 78527.502[1] 168775.7#not new:
-2227.808 + 104.438*1500[1] 154429.2New: $168,775.70 Not new: $154429.20
According to this equation, houses that are larger are much greater in price, especially when comparing new large houses to small new houses. For larger houses, the difference in cost between new and not new is much larger, compared to smaller houses, where new/not new makes less of a difference in price.
3h.
I prefer the second model with the interaction term which provides a clearer picture of how increased square footage makes a larger difference in bigger sized houses. The model with the interaction term also has a larger adjusted R squared.
However, I would be skeptical using this model with small homes: for a home that is 1000 square feet, the predicted price for a new house is greater than for an old house. In other words, I do not feel this model would be good at predicting tiny home prices.
#for a 1000 sq foot home:
#New:
-2227.808 + 166.354*1000 - 78527.502[1] 85598.69#Not new:
-2227.808 + 104.438*1000[1] 102210.2