Homework 4

hw4

DACSS 603

Author

Alexa Potter

Published

April 24, 2023

Code

library(tidyverse)
library(car)
library(smss)
data("house.selling.price", package = "smss")
df <- house.selling.price
library(alr4)
data("salary", package = "alr4")
df <- salary

Question 1

For recent data in Jacksonville, Florida, on y = selling price of home (in dollars), x1 = size of home (in square feet), and x2 = lot size (in square feet), the prediction equation is ŷ = −10,536 + 53.8x1 + 2.84x2.

A

A particular home of 1240 square feet on a lot of 18,000 square feet sold for $145,000. Find the predicted selling price and the residual, and interpret.

Code

a_predicted = (-10536 + (53.8*(1240)) + (2.84*(18000)))
a_predicted

[1] 107296

Code

a_actual = 145000

a_residual = a_actual - a_predicted

a_residual

[1] 37704

The predicted selling price is $107,296. The residual is $37,704. This means the buyer bought the house $37,704 over recent data predicitons.

B

For fixed lot size, how much is the house selling price predicted to increase for each square foot increase in home size? Why?

Code

(-10536 + (53.8*(1000)) + (2.84*(1000)))

[1] 46104

Code

(-10536 + (53.8*(1001)) + (2.84*(1000)))

[1] 46157.8

The house selling price is predicted to increase by $53.80. This is the coefficient to the home size variable, each x1 unit increases by 53.8.

C

According to this prediction equation, for fixed home size, how much would lot size need to increase to have the same impact as a one-square-foot increase in home size?

Code

53.8/2.84

[1] 18.94366

Lot size would need to increase by 18.94 to have the same impact as a 1 square foot increase in home size.

Question 2

(Data file: salary in alr4 R package). The data file concerns salary and other characteristics of all faculty in a small Midwestern college collected in the early 1980s for presentation in legal proceedings for which discrimination against women in salary was at issue. All persons in the data hold tenured or tenure track positions; temporary faculty are not included. The variables include degree, a factor with levels PhD and MS; rank, a factor with levels Asst, Assoc, and Prof; sex, a factor with levels Male and Female; Year, years in current rank; ysdeg, years since highest degree, and salary, academic year salary in dollars.

Code

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

Test the hypothesis that the mean salary for men and women is the same, without regard to any other variable but sex. Explain your findings.

Code

t.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

We can use the independent t-test here because it is comparing the means of continuous and categorical variables.
The p value here is not significant which means we fail to reject the null hypothesis that all population means are equal.

B

Run a multiple linear regression with salary as the outcome variable and everything else as predictors, including sex. Assuming no interactions between sex and the other predictors, obtain a 95% confidence interval for the difference in salary between males and females.

Linear regression:

Code

lm_salary <- lm(salary ~ degree + rank + sex + year + ysdeg, data =  salary)
summary(lm_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

95% Confidence Interval for sexFemale is -697.8183 to 3030.56452

Code

confint(lm_salary)

                 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

C

Interpret your finding for each predictor variable; discuss (a) statistical significance, (b) interpretation of the coefficient / slope in relation to the outcome variable and other variables

degree

Degree is not statistically significant. Individuals with degree “PhD” make 1388.61 more than “Master’s”.

rank

Rank is statistically significant. Individuals with rank “Prof” make 11118.76 more than “Asst” (constant) and individuals with rank “Assoc” make 5292.36 more than “Asst”

sex
Sex is not statistically significant. sexFemale reflects women make 1166.37 more than men.

year
Year is statistically significant. With each increase in year an individual makes 476.31 more.

ysdeg
Years since degree is not statistically significant. It also have a negative relationship with salary. This means the more years away from their degree, an individual makes 124.57 less.

D

Change the baseline category for the rank variable. Interpret the coefficients related to rank again.

Code

salary$rank <- relevel(salary$rank, ref = "Prof")
lm_salary2 <- lm(salary ~ degree + rank + sex + year + ysdeg, data =  salary)
summary(lm_salary2)


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)  26864.81    1375.29  19.534  < 2e-16 ***
degreePhD     1388.61    1018.75   1.363    0.180    
rankAsst    -11118.76    1351.77  -8.225 1.62e-10 ***
rankAssoc    -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-16

Changing the baseline of rank to “Prof” does not change the statisitically significant variables. It does however, change the relationship with “Asst” and “Assoc” level of rank, as well as degree level of “Masters”. These variables now have a negative relationship.

E

Finkelstein (1980), in a discussion of the use of regression in discrimination cases, wrote, “a variable may reflect a position or status bestowed by the employer, in which case if there is discrimination in the award of the position or status, the variable may be ‘tainted.’” Thus, for example, if discrimination is at work in promotion of faculty to higher ranks, using rank to adjust salaries before comparing the sexes may not be acceptable to the courts.
Exclude the variable rank, refit, and summarize how your findings changed, if they did.

Code

lm_salary3 <- lm(salary ~ degree + sex + year + ysdeg, data =  salary)
summary(lm_salary3)


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

Without the variable of rank, the statistically significant variables are now deegree, year, and ysdeg. Without rank females also make 1286.54 less than males so. Years since degree now has a postive relationship.

Question 3

(Data file: house.selling.price in smss R package)

A

Using the house.selling.price data, run and report regression results modeling y = selling price (in dollars) in terms of size of home (in square feet) and whether the home is new (1 = yes; 0 = no). In particular, for each variable; discuss statistical significance and interpret the meaning of the coefficient.

Code

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

Code

house_lm <- lm(Price ~ Size + New, data = house.selling.price)
summary(house_lm)


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

This model shows both size and new variables are statistically significant to price. A new house has a price of57736.283 more than a not new home and for each unit increase of size, price increases 116.132.

B

Report and interpret the prediction equation, and form separate equations relating selling price to size for new and for not new homes.

y = -40230.867+116.132X1(size) + 57736.283X2(new)

If the house is not new, X2 is cancelled out by 0

C

Find the predicted selling price for a home of 3000 square feet that is (i) new, (ii) not new.
y = -40230.867+116.132X1(size) + 57736.283X2(new)

Code

#i

-40230.867+(116.132*3000) + (57736.283*1)

[1] 365901.4

Code

#ii  
-40230.867+(116.132*3000) + (57736.283*0)

[1] 308165.1

D

Fit another model, this time with an interaction term allowing interaction between size and new, and report the regression results

Code

house_lm2 <- lm(Price ~ Size*New, data = house.selling.price)
summary(house_lm2)


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

These results now show “new” is not statistically significant but “size” and “size:new” is.

E

Report the lines relating the predicted selling price to the size for homes that are (i) new, (ii) not new.

y = -22227.808 + 104.438X1(size) - 78527.502X2(new) + 61.916X1(size)X2(new)
y = -22227.808 + 104.438*X1(size)

F

Find the predicted selling price for a home of 3000 square feet that is (i) new, (ii) not new.

Code

#i

-22227.808 + (104.438*3000) - 78527.502 + (61.916*3000)

[1] 398306.7

Code

#ii
-22227.808 + (104.438*3000)

[1] 291086.2

G

Find the predicted selling price for a home of 1500 square feet that is (i) new, (ii) not new. Comparing to (F), explain how the difference in predicted selling prices changes as the size of home increases.

Code

#i

-22227.808 + (104.438*1500) - 78527.502 + (61.916*1500)

[1] 148775.7

Code

#ii
-22227.808 + (104.438*1500)

[1] 134429.2

With the interaction of size*new, predicted selling prices changes by an increase of over 200% as the size of home increases in these instances. As the size increases, the percentage also increases in predicted selling price.

H

Do you think the model with interaction or the one without it represents the relationship of size and new to the outcome price? What makes you prefer one model over another?

R squared in the interaction model is higher as well as the standard error number is lower. BAsed on these two factors I would determine the interaction model is preferred.