Please help me..!! Question 1 – Real Estate The dataset RealState includes two r
ID: 3172848 • Letter: P
Question
Please help me..!!
Question 1 – Real Estate
The dataset RealState includes two random samples of 100 listings each that have been extracted from
a much larger original dataset. One sample has houses with fireplaces and the other has houses without
fireplaces. The spreadsheet also has: Price ($), Living Area (sq. ft.), and Age (years) for each listing.
a) Describe, in one short phrase each, the populations being examined.
b) Carry out a hypothesis test to see whether there is a real difference in the mean price of these two
populations. Assume the two population variances are unequal.
Note: The level of significance is not specified. Draw your conclusions by referring to the strength of
evidence indicated by the p-value. Remember the three alpha values most commonly used: 0.05 =
statistically significant, 0.01 = highly statistically significant, and 0.001 = very highly statistically
significant.
c) Find the corresponding 95% confidence interval for the difference in mean house prices.
d) Describe in one sentence how the hypothesis test in part b) and the confidence interval in part c)
reflect the relationship between confidence intervals and hypothesis tests as introduced in class.
Spreadsheet below..
Price ($)
Living Area (sq ft)
Age (years)
Fireplace
1
171171
1822
12
0
2
136499
1152
36
0
3
87852
1270
110
0
4
92436
1300
10
0
5
216018
1216
75
0
6
165213
1664
10
0
7
104371
1463
125
0
8
104984
1184
21
0
9
150909
1374
44
0
10
110738
1800
83
0
11
120476
1248
14
0
12
168739
1620
0
0
13
147625
1668
16
0
14
149078
1664
14
0
15
115566
1184
17
0
16
123064
1209
73
0
17
90446
1200
10
0
18
57678
924
104
0
19
102003
1184
20
0
20
87852
976
23
0
21
84291
924
15
0
22
155787
1540
16
0
23
147634
864
39
0
24
60238
1664
76
0
25
111546
1560
31
0
26
185323
1662
0
0
27
262461
2809
2
0
28
107306
1276
29
0
29
172795
1910
21
0
30
98652
1158
14
0
31
107743
1184
18
0
32
161923
1158
7
0
33
100839
1480
17
0
34
95708
912
20
0
35
111809
1711
112
0
36
91386
1120
14
0
37
92519
1368
34
0
38
146626
1561
18
0
39
111221
1056
27
0
40
153850
2400
26
0
41
131165
1564
13
0
42
110963
1568
18
0
43
119784
1560
25
0
44
130782
1274
14
0
45
122257
1400
17
0
46
283851
2346
1
0
47
263408
1738
5
0
48
87588
1392
17
0
49
102806
1168
21
0
50
97476
1219
19
0
51
135167
1610
17
0
52
54210
792
55
0
53
101227
1754
110
0
54
137681
1728
12
0
55
112153
1834
65
0
56
224888
1200
65
0
57
236215
1392
4
0
58
119751
1480
17
0
59
130574
1600
57
0
60
103633
1314
21
0
61
93846
1385
17
0
62
120115
967
55
0
63
116321
1486
16
0
64
111534
1032
4
0
65
113444
1656
94
0
66
137540
1551
97
0
67
111264
1142
18
0
68
133067
1326
7
0
69
146611
1580
26
0
70
115649
1422
20
0
71
111436
1380
2
0
72
94064
912
37
0
73
103672
1476
6
0
74
64552
1298
106
0
75
166222
1498
10
0
76
109578
1176
5
0
77
139104
1760
135
0
78
93174
1022
55
0
79
98632
1406
19
0
80
168497
1888
6
0
81
108785
1480
22
0
82
106524
1148
7
0
83
88092
929
22
0
84
136551
1880
7
0
85
152822
1043
104
0
86
157411
1666
2
0
87
167145
1852
16
0
88
122082
2256
124
0
89
115401
1480
18
0
90
124939
1099
9
0
91
55817
1688
119
0
92
74422
2708
21
0
93
200793
1940
9
0
94
155488
1747
1
0
95
94405
1375
22
0
96
98389
1184
18
0
97
102086
1302
16
0
98
180927
2050
33
0
99
59003
3285
233
0
100
79893
2634
83
0
101
152404
1508
16
1
102
135011
1327
35
1
103
82556
1480
14
1
104
233876
2434
20
1
105
247839
1536
66
1
106
156530
1734
29
1
107
180032
2066
32
1
108
132311
1512
13
1
109
196846
2144
19
1
110
254944
2579
0
1
111
127740
1480
16
1
112
263226
3355
1
1
113
278936
3121
0
1
114
152491
1875
31
1
115
164419
1785
1
1
116
171885
1536
7
1
117
189654
2320
16
1
118
260460
3726
3
1
119
219262
2327
10
1
120
259999
2577
1
1
121
206688
2284
17
1
122
215824
2056
2
1
123
154314
1466
62
1
124
176846
2249
1
1
125
253433
3021
21
1
126
138297
1800
49
1
127
254327
2541
5
1
128
160007
1665
27
1
129
297024
3020
2
1
130
128273
1343
15
1
131
120639
1498
0
1
132
237367
2936
6
1
133
207347
2119
14
1
134
112825
1314
20
1
135
139418
1296
51
1
136
229055
2500
0
1
137
202751
2202
6
1
138
234954
2294
0
1
139
329484
2786
0
1
140
261846
2772
0
1
141
189836
1726
26
1
142
69814
840
57
1
143
246976
1652
4
1
144
103408
1034
14
1
145
145147
1832
47
1
146
102806
1248
46
1
147
267731
3001
0
1
148
136192
1623
36
1
149
387652
3504
0
1
150
149933
1664
25
1
151
135820
1475
20
1
152
257015
2093
1
1
153
162612
1348
11
1
154
142302
1242
2
1
155
148585
1734
34
1
156
189736
2588
31
1
157
240872
2348
7
1
158
144335
1274
20
1
159
236737
3239
1
1
160
106524
1144
16
1
161
251699
3081
1
1
162
354739
3982
6
1
163
163524
2516
40
1
164
170600
1728
0
1
165
214749
2286
11
1
166
93895
1480
17
1
167
194880
2278
0
1
168
85560
2068
104
1
169
218475
2748
0
1
170
174208
2372
2
1
171
98790
1480
16
1
172
251259
1932
0
1
173
261647
1850
16
1
174
224928
2185
14
1
175
126311
1370
27
1
176
261011
3015
1
1
177
277235
2950
1
1
178
144561
1292
4
1
179
170534
1820
18
1
180
157081
2310
33
1
181
185902
2756
33
1
182
113740
1920
34
1
183
130239
1052
15
1
184
207018
2147
9
1
185
94064
1480
16
1
186
230708
2548
3
1
187
103861
1700
25
1
188
247520
2445
18
1
189
207751
2628
20
1
190
120476
2238
114
1
191
198622
2024
25
1
192
233714
2338
16
1
193
122796
1612
34
1
194
102806
1480
15
1
195
217314
2498
14
1
196
379678
3720
0
1
197
140602
1693
26
1
198
184108
2702
31
1
199
141029
1874
31
1
200
314912
3423
0
1
Price ($)
Living Area (sq ft)
Age (years)
Fireplace
1
171171
1822
12
0
2
136499
1152
36
0
3
87852
1270
110
0
4
92436
1300
10
0
5
216018
1216
75
0
6
165213
1664
10
0
7
104371
1463
125
0
8
104984
1184
21
0
9
150909
1374
44
0
10
110738
1800
83
0
11
120476
1248
14
0
12
168739
1620
0
0
13
147625
1668
16
0
14
149078
1664
14
0
15
115566
1184
17
0
16
123064
1209
73
0
17
90446
1200
10
0
18
57678
924
104
0
19
102003
1184
20
0
20
87852
976
23
0
21
84291
924
15
0
22
155787
1540
16
0
23
147634
864
39
0
24
60238
1664
76
0
25
111546
1560
31
0
26
185323
1662
0
0
27
262461
2809
2
0
28
107306
1276
29
0
29
172795
1910
21
0
30
98652
1158
14
0
31
107743
1184
18
0
32
161923
1158
7
0
33
100839
1480
17
0
34
95708
912
20
0
35
111809
1711
112
0
36
91386
1120
14
0
37
92519
1368
34
0
38
146626
1561
18
0
39
111221
1056
27
0
40
153850
2400
26
0
41
131165
1564
13
0
42
110963
1568
18
0
43
119784
1560
25
0
44
130782
1274
14
0
45
122257
1400
17
0
46
283851
2346
1
0
47
263408
1738
5
0
48
87588
1392
17
0
49
102806
1168
21
0
50
97476
1219
19
0
51
135167
1610
17
0
52
54210
792
55
0
53
101227
1754
110
0
54
137681
1728
12
0
55
112153
1834
65
0
56
224888
1200
65
0
57
236215
1392
4
0
58
119751
1480
17
0
59
130574
1600
57
0
60
103633
1314
21
0
61
93846
1385
17
0
62
120115
967
55
0
63
116321
1486
16
0
64
111534
1032
4
0
65
113444
1656
94
0
66
137540
1551
97
0
67
111264
1142
18
0
68
133067
1326
7
0
69
146611
1580
26
0
70
115649
1422
20
0
71
111436
1380
2
0
72
94064
912
37
0
73
103672
1476
6
0
74
64552
1298
106
0
75
166222
1498
10
0
76
109578
1176
5
0
77
139104
1760
135
0
78
93174
1022
55
0
79
98632
1406
19
0
80
168497
1888
6
0
81
108785
1480
22
0
82
106524
1148
7
0
83
88092
929
22
0
84
136551
1880
7
0
85
152822
1043
104
0
86
157411
1666
2
0
87
167145
1852
16
0
88
122082
2256
124
0
89
115401
1480
18
0
90
124939
1099
9
0
91
55817
1688
119
0
92
74422
2708
21
0
93
200793
1940
9
0
94
155488
1747
1
0
95
94405
1375
22
0
96
98389
1184
18
0
97
102086
1302
16
0
98
180927
2050
33
0
99
59003
3285
233
0
100
79893
2634
83
0
101
152404
1508
16
1
102
135011
1327
35
1
103
82556
1480
14
1
104
233876
2434
20
1
105
247839
1536
66
1
106
156530
1734
29
1
107
180032
2066
32
1
108
132311
1512
13
1
109
196846
2144
19
1
110
254944
2579
0
1
111
127740
1480
16
1
112
263226
3355
1
1
113
278936
3121
0
1
114
152491
1875
31
1
115
164419
1785
1
1
116
171885
1536
7
1
117
189654
2320
16
1
118
260460
3726
3
1
119
219262
2327
10
1
120
259999
2577
1
1
121
206688
2284
17
1
122
215824
2056
2
1
123
154314
1466
62
1
124
176846
2249
1
1
125
253433
3021
21
1
126
138297
1800
49
1
127
254327
2541
5
1
128
160007
1665
27
1
129
297024
3020
2
1
130
128273
1343
15
1
131
120639
1498
0
1
132
237367
2936
6
1
133
207347
2119
14
1
134
112825
1314
20
1
135
139418
1296
51
1
136
229055
2500
0
1
137
202751
2202
6
1
138
234954
2294
0
1
139
329484
2786
0
1
140
261846
2772
0
1
141
189836
1726
26
1
142
69814
840
57
1
143
246976
1652
4
1
144
103408
1034
14
1
145
145147
1832
47
1
146
102806
1248
46
1
147
267731
3001
0
1
148
136192
1623
36
1
149
387652
3504
0
1
150
149933
1664
25
1
151
135820
1475
20
1
152
257015
2093
1
1
153
162612
1348
11
1
154
142302
1242
2
1
155
148585
1734
34
1
156
189736
2588
31
1
157
240872
2348
7
1
158
144335
1274
20
1
159
236737
3239
1
1
160
106524
1144
16
1
161
251699
3081
1
1
162
354739
3982
6
1
163
163524
2516
40
1
164
170600
1728
0
1
165
214749
2286
11
1
166
93895
1480
17
1
167
194880
2278
0
1
168
85560
2068
104
1
169
218475
2748
0
1
170
174208
2372
2
1
171
98790
1480
16
1
172
251259
1932
0
1
173
261647
1850
16
1
174
224928
2185
14
1
175
126311
1370
27
1
176
261011
3015
1
1
177
277235
2950
1
1
178
144561
1292
4
1
179
170534
1820
18
1
180
157081
2310
33
1
181
185902
2756
33
1
182
113740
1920
34
1
183
130239
1052
15
1
184
207018
2147
9
1
185
94064
1480
16
1
186
230708
2548
3
1
187
103861
1700
25
1
188
247520
2445
18
1
189
207751
2628
20
1
190
120476
2238
114
1
191
198622
2024
25
1
192
233714
2338
16
1
193
122796
1612
34
1
194
102806
1480
15
1
195
217314
2498
14
1
196
379678
3720
0
1
197
140602
1693
26
1
198
184108
2702
31
1
199
141029
1874
31
1
200
314912
3423
0
1
Explanation / Answer
We shall analyse this using the open source statistical package R , the complete R snippet for the analysis is
####
# read the data into R dataframe
data.df<- read.csv("C:\Users\586645\Downloads\Chegg\property.csv",header=TRUE)
str(data.df)
data.df$Price....<- as.numeric(data.df$Price....)
# seperate the data into 2 parts
fireplace <- data.df[which(data.df$Fireplace==1),]
nofireplace <- data.df[which(data.df$Fireplace==0),]
# question 1
summary(fireplace)
summary(nofireplace)
# perform a t test on the price
t.test(fireplace$Price....,nofireplace$Price....)
## 95% confidence interval of mean , we use z = 1.95 for alpha = 0.05
mean(fireplace$Price....) + 1.96*sd(fireplace$Price....) # upper
mean(fireplace$Price....) - 1.96*sd(fireplace$Price....) # lower
mean(nofireplace$Price....) + 1.96*sd(nofireplace$Price....) # upper
mean(nofireplace$Price....) - 1.96*sd(nofireplace$Price....) # lower
####
The results are
> data.df<- read.csv("C:\Users\586645\Downloads\Chegg\property.csv",header=TRUE)
> str(data.df)
'data.frame': 200 obs. of 4 variables:
$ Price.... : Factor w/ 195 levels "100839","101227",..: 99 55 177 183 123 92 10 11 76 17 ...
$ Living.Area..sq.ft.: int 1822 1152 1270 1300 1216 1664 1463 1184 1374 1800 ...
$ Age..years. : int 12 36 110 10 75 10 125 21 44 83 ...
$ Fireplace : int 0 0 0 0 0 0 0 0 0 0 ...
>
> data.df$Price....<- as.numeric(data.df$Price....)
>
> # seperate the data into 2 parts
> fireplace <- data.df[which(data.df$Fireplace==1),]
> nofireplace <- data.df[which(data.df$Fireplace==0),]
>
> # question 1
> summary(fireplace)
Price.... Living.Area..sq.ft. Age..years. Fireplace
Min. : 5.00 Min. : 840 Min. : 0.00 Min. :1
1st Qu.: 67.75 1st Qu.:1530 1st Qu.: 2.00 1st Qu.:1
Median :113.50 Median :2067 Median : 15.00 Median :1
Mean :106.09 Mean :2112 Mean : 18.16 Mean :1
3rd Qu.:142.25 3rd Qu.:2543 3rd Qu.: 26.25 3rd Qu.:1
Max. :195.00 Max. :3982 Max. :114.00 Max. :1
> summary(nofireplace)
Price.... Living.Area..sq.ft. Age..years. Fireplace
Min. : 1.00 Min. : 792 Min. : 0.00 Min. :0
1st Qu.: 29.75 1st Qu.:1184 1st Qu.: 12.75 1st Qu.:0
Median : 73.00 Median :1414 Median : 18.50 Median :0
Mean : 87.47 Mean :1485 Mean : 35.47 Mean :0
3rd Qu.:165.25 3rd Qu.:1664 3rd Qu.: 40.25 3rd Qu.:0
Max. :194.00 Max. :3285 Max. :233.00 Max. :0
The mean price for the house with no fireplace is 87.47 while with fireplace is 106.7 . This also provides the 5 point summary statitics for the houses of the 2 populations characterised by fireplaces
>
> # perform a t test on the price
>
> t.test(fireplace$Price....,nofireplace$Price....)
Welch Two Sample t-test
data: fireplace$Price.... and nofireplace$Price....
t = 2.3215, df = 179.09, p-value = 0.02139 , as th ep value of the test is less than 0.05 , hence the test is only STATISTICALLY SIGNIFICANT and there is indeed a difference in the average prices of the properties for the 2 groups , characterised by presence and absence of fireplaces
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
2.792616 34.447384
sample estimates:
mean of x mean of y
106.09 87.47
>
> ## the upper and lower limit of the confidence interval is calculated using the equation
mean +- z* SD , here the Z is 1.96 , assuming alpha =0.05 from the z table
> ## 95% confidence interval of mean , we use z = 1.95 for alpha = 0.05
>
> mean(fireplace$Price....) + 1.96*sd(fireplace$Price....) # upper
[1] 197.4247
> mean(fireplace$Price....) - 1.96*sd(fireplace$Price....) # lower
[1] 14.75533
>
>
> mean(nofireplace$Price....) + 1.96*sd(nofireplace$Price....) # upper
[1] 215.4234
> mean(nofireplace$Price....) - 1.96*sd(nofireplace$Price....) # lower
[1] -40.48343
d)
while confidence intervals gives the range of the values that would contain the TRUE MEAN of the prices for the 2 groups , the t test help us compare whether the difference between the 2 property types is just by chance or by a effect which we need to explore
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