hw4
Adithya Parupudi
my homework 4
Author

Adithya Parupudi

Published

April 24, 2023

Code 
library(tidyverse)
library(ggplot2)
library(stats)
library(alr4)
library(smss)

knitr::opts_chunk$set(echo = TRUE)

Question 1

A

Code 
Pred_selling_price <-  -10536 + 53.8 * 1240 + 2.84 * 18000
Pred_selling_price
[1] 107296
Code 
Residual <- Pred_selling_price - 145000
Residual
[1] -37704

From the above result, we can say that the house was sold for 37704 dollars greater than predicted.

B

Using the prediction equation ŷ = -10536 + 53.8x1 + 2.84x2, where x2 equals lot size, the house selling price is expected to increase by 53.8 dollars per each square-foot increase in home size given the lot sized is fixed. This is because a fixed lot size would make 2.84x2 a set number in the prediction equation. Therefore, we would not need to factor in a change in the output based on any input. Then, we are left with the coefficient for the home size variable, which is 53.8. For x1 = 1, representing one square-foot of home size, the output would increase by 53.8 * 1 = 53.8.

C

For fixed home size, 53.8 * 1 = 2.84x2

Code 
result <- 53.8/2.84
result
[1] 18.94366

An increase in lot size of about 18.94 square-feet would have the same impact as an increase of 1 square-foot in home size on the predicted selling price.

Question 2

Code 
data("salary")
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  28516
7      PhD  Prof Female    0    32  24900
8  Masters  Prof   Male   16    18  31909
9      PhD  Prof   Male   13    30  31850
10     PhD  Prof   Male   13    31  32850
11 Masters  Prof   Male   12    22  27025
12 Masters Assoc   Male   15    19  24750
13 Masters  Prof   Male    9    17  28200
14     PhD Assoc   Male    9    27  23712
15 Masters  Prof   Male    9    24  25748
16 Masters  Prof   Male    7    15  29342
17 Masters  Prof   Male   13    20  31114
18     PhD Assoc   Male   11    14  24742
19     PhD Assoc   Male   10    15  22906
20     PhD  Prof   Male    6    21  24450
21     PhD  Asst   Male   16    23  19175
22     PhD Assoc   Male    8    31  20525
23 Masters  Prof   Male    7    13  27959
24 Masters  Prof Female    8    24  38045
25 Masters Assoc   Male    9    12  24832
26 Masters  Prof   Male    5    18  25400
27 Masters Assoc   Male   11    14  24800
28 Masters  Prof Female    5    16  25500
29     PhD Assoc   Male    3     7  26182
30     PhD Assoc   Male    3    17  23725
31     PhD  Asst Female   10    15  21600
32     PhD Assoc   Male   11    31  23300
33     PhD  Asst   Male    9    14  23713
34     PhD Assoc Female    4    33  20690
35     PhD Assoc Female    6    29  22450
36 Masters Assoc   Male    1     9  20850
37 Masters  Asst Female    8    14  18304
38 Masters  Asst   Male    4     4  17095
39 Masters  Asst   Male    4     5  16700
40 Masters  Asst   Male    4     4  17600
41 Masters  Asst   Male    3     4  18075
42     PhD  Asst   Male    3    11  18000
43 Masters Assoc   Male    0     7  20999
44 Masters  Asst Female    3     3  17250
45 Masters  Asst   Male    2     3  16500
46 Masters  Asst   Male    2     1  16094
47 Masters  Asst Female    2     6  16150
48 Masters  Asst Female    2     2  15350
49 Masters  Asst   Male    1     1  16244
50 Masters  Asst Female    1     1  16686
51 Masters  Asst Female    1     1  15000
52 Masters  Asst Female    0     2  20300

A

Code 
summary(lm(salary ~ sex, data = salary))

Call:
lm(formula = salary ~ sex, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-8602.8 -4296.6  -100.8  3513.1 16687.9 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    24697        938  26.330   <2e-16 ***
sexFemale      -3340       1808  -1.847   0.0706 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5782 on 50 degrees of freedom
Multiple R-squared:  0.0639,    Adjusted R-squared:  0.04518 
F-statistic: 3.413 on 1 and 50 DF,  p-value: 0.0706

The null hypothesis would be that mean salary for men and mean salary for women are equal, and the alternative hypothesis would be that the salaries are not equal. I ran a regression with sex as the explanatory variable and salary as the outcome variable. The female coefficient is -3340, which means that women do make less than men not considering any other variables. However, if we consider the other variables and also there is a significance level of 0.07, so we fail to reject the null hypothesis and therefore cannot conclude that there is a difference between mean salaries for men and women.

B

Code 
model <- lm(salary ~ ., data = salary)
summary(model)

Call:
lm(formula = salary ~ ., 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
Code 
confint(model)
                 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    31.49105

Assuming there is no interaction between sex and other predictors, we can be 95% confident that the difference in salary of women compared to men falls between -697.8183 dollars and 3030.56452 dollars.

C

For degree as the predictor, a PHD would be expected to increase salary by 1388.61 dollars in reference to a Masters degree salary. However, at a significance level of 0.18, we cannot conclude that degree level has a statistically significant impact on salary.

For the rank variable, an Associate can expect a 5292.36 dollar increase in salary compared to Assistant, while a Professor can expect a 11118.76 dollar salary increase compared to Assistant. Both ranks have significance levels well below 0.05 and we can determine that rank does have a statistically significant impact on salary. For the variable of sex, a Female can expect a salary increase of 1166.37 dollars in comparison to Male salary, but the significance level is 0.214, so this is not a statistically significant relationship.

For year, a faculty member can expect a salary increase of 476.31 dollars for an increase in 1 year of employment in his/her/their position. Additionally, the level of significance is less than 0.01 so the relationship between year and salary appears to be significant.

For the ysdeg variable, an increase in years since earning highest degree can expect a decrease in salary, with a coefficient of -124.57. However, with a 0.115 level of significance, this relationship cannot be found to be statistically significant.

D

Code 
salary$rank <- relevel(salary$rank, ref = "Prof")
summary(lm(salary ~ rank, salary))

Call:
lm(formula = salary ~ rank, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-5209.0 -1819.2  -417.8  1586.6  8386.0 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  29659.0      669.3  44.316  < 2e-16 ***
rankAsst    -11890.3      972.4 -12.228  < 2e-16 ***
rankAssoc    -6483.0     1043.0  -6.216 1.09e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2993 on 49 degrees of freedom
Multiple R-squared:  0.7542,    Adjusted R-squared:  0.7442 
F-statistic: 75.17 on 2 and 49 DF,  p-value: 1.174e-15

After changing the baseline category for the rank variable, an Associate can expect a 6483.0 dollar decrease in salary compared to Professor, while a Assistant can expect a 11890.3 dollar salary decrease compared to Professor. Both ranks have significance levels well below 0.05 and we can determine that rank does have a statistically significant impact on salary.

E

Code 
summary(lm(salary ~ degree + sex + year + ysdeg, 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-09

When removing the variable “rank”, the coefficient for sex is -1286.54 compared to the above regression that included rank with a coefficient for sex at 1166.37. The new coefficient predicts that a female salary would be 1286.54 less than a male salary, when excluding the variable of rank. However, the significance level is 0.332, which is very high and therefore the results cannot be found to be statistically significant. While the change of the coefficient to negative upon removal of rank is interesting, the significance level would likely prevent these results from holding up in court as an indication of discrimination on the basis of sex.

F

Code 
salary <- salary %>%
  mutate(hired = case_when(ysdeg <= 15 ~ "1", ysdeg > 15 ~ "0"))
summary(lm(salary ~ hired, data = salary))

Call:
lm(formula = salary ~ hired, data = salary)

Residuals:
   Min     1Q Median     3Q    Max 
 -8294  -3486  -1772   3829  10576 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  27469.4      913.4  30.073  < 2e-16 ***
hired1       -7343.5     1291.8  -5.685 6.73e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4658 on 50 degrees of freedom
Multiple R-squared:  0.3926,    Adjusted R-squared:  0.3804 
F-statistic: 32.32 on 1 and 50 DF,  p-value: 6.734e-07
Code 
summary(lm(salary ~ sex + rank + degree + hired, data = salary))

Call:
lm(formula = salary ~ sex + rank + degree + hired, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-6187.5 -1750.9  -438.9  1719.5  9362.9 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  29511.3      784.0  37.640  < 2e-16 ***
sexFemale     -829.2      997.6  -0.831    0.410    
rankAsst    -11925.7     1512.4  -7.885 4.37e-10 ***
rankAssoc    -7100.4     1297.0  -5.474 1.76e-06 ***
degreePhD     1126.2     1018.4   1.106    0.275    
hired1         319.0     1303.8   0.245    0.808    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3023 on 46 degrees of freedom
Multiple R-squared:  0.7645,    Adjusted R-squared:  0.7389 
F-statistic: 29.87 on 5 and 46 DF,  p-value: 2.192e-13

I created a dummy variable called “hired” which coded those employed for 15 years or less (thus hired by the new Dean) as 1 and those who have been employed for over 15 years as 0. Then, I fit a new regression model and decided to include the variables of sex, rank, degree, and hired. I omitted the year and ysdeg variables to prevent overlapping or multicollinearity. Multicollinearity can be a concern when variables are highly correlated or related in some way. The idea of regression is to observe how each variable partially effects the output while holding the other variables fixed. We cannot reasonably change the year or ysdeg or hired variables individually while holding the other two fixed since they tend to “grow” in similar manners. Since the variable hired is a product of the ysdeg variable, we could not include both.

Based on the regression model, those hired by the current Dean are predicted to make 319 dollars more than those not hired by the Dean. When it comes to salary, this is a rather insignificant number. Furthermore, the level of significance for the hired variable is .81, which is astronomical and indicates that the relationship between hired and salary is not statistically significant. Based on these factors, I would state that findings do not indicate any favorable treatment by the Dean toward faculty that the Dean specifically hired.

Question 3

Code 
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 1340

A

Code 
summary(lm(Price ~ Size + New, 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-16

Both Size and New significantly positively predict selling price. As each predictor goes up by 1 unit, selling price rises by 116.132 dollars and 57736.283 dollars respectively.

B

Code 
new <- house.selling.price %>% 
  filter(New == 1)
summary(lm(Price ~ Size, data = new))

Call:
lm(formula = Price ~ Size, data = new)

Residuals:
   Min     1Q Median     3Q    Max 
-78606 -16092   -987  20068  76140 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -100755.31   42513.73  -2.370   0.0419 *  
Size            166.35      17.09   9.735 4.47e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 45500 on 9 degrees of freedom
Multiple R-squared:  0.9133,    Adjusted R-squared:  0.9036 
F-statistic: 94.76 on 1 and 9 DF,  p-value: 4.474e-06
Code 
old <- house.selling.price %>% 
  filter(New == 0)
summary(lm(Price ~ Size, data = old))

Call:
lm(formula = Price ~ Size, data = old)

Residuals:
    Min      1Q  Median      3Q     Max 
-175748  -29155   -7297   14159  192519 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -22227.808  15708.186  -1.415    0.161    
Size           104.438      9.538  10.950   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 52620 on 87 degrees of freedom
Multiple R-squared:  0.5795,    Adjusted R-squared:  0.5747 
F-statistic: 119.9 on 1 and 87 DF,  p-value: < 2.2e-16

Size significantly positively predicts price for both new and old houses, but by a greater magnitude for new houses. Adjusted R-squared for the model is also much higher (0.91 vs. 0.58).

New_Price = 166 * Size - 100755.31

Old_Price = 104 * Size - 22227.808

C

Code 
Size <- 3000
New_Price = 166 * Size - 100755.31
Old_Price = 104 * Size - 22227.808
New_Price
[1] 397244.7
Code 
Old_Price
[1] 289772.2

D

Code 
summary(lm(Price ~ Size*New, data = house.selling.price))

Call:
lm(formula = Price ~ 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-16

E

The predicted selling price, based on the new regression that includes interaction between Size and Newness, would look like:

New_Price = -22227.81 + 104.44 * Size - 78527.50 * 1 + 61.92 * Size * 1

Old_Price = -22227.81 + 104.44 * Size

F

Code 
Size <- 3000
New_Price = -22227.81 + 104.44 * Size - 78527.50 * 1 + 61.92 * Size * 1
Old_Price = -22227.81 + 104.44 * Size
New_Price
[1] 398324.7
Code 
Old_Price
[1] 291092.2

G

Code 
Size <- 1500
New_Price = -22227.81 + 104.44 * Size - 78527.50 * 1 + 61.92 * Size * 1
Old_Price = -22227.81 + 104.44 * Size
New_Price
[1] 148784.7
Code 
Old_Price
[1] 134432.2

As size of home goes up, the difference in predicted selling prices between old and new homes becomes larger.

H

The prediction model with interaction has a significantly large negative coefficient for the New variable. The adjusted r-squared for the model with interaction is 0.7363 and the adjusted r-squared for the first model without interaction is 0.7169. The increase in the adjusted r-squared with the interaction model could be due to an additional variable or could indicate a slightly better fit for the prediction of the data. Since the models do have similar adjusted r-squared values, I would prefer the model with interaction because the regression indicates that the interaction term is statistically significant to selling price prediction, so I feel it is necessary to utilize an equation that factors for this.