Question

An agent for a residential real estate company has the business objective of developing more accurate estimates of the monthly rental cost of apartments. Toward that goal, the agent would like to use the size of the apartment as defined by square footage to predict monthly rental cost. The agent selects a sample of 100 one-bedroom apartments and collects the data.

1. Construct a scatter plot of the data. Comment on the relationship between square footage (x) and monthly cost (y).

2. Fit the simple linear regression model ? = ?" + ?#? + ?, where ? denotes cost, ? denotes square footage and ?~?(0, ?!). Using the data and the method of ordinary least squares, determine the estimates for ?" and ?#, say ?" and ?#.

3. Compute the coefficient of determination and interpret this measure.

4. Construct a 95% confidence interval on the slope of the linear regression model. Is there evidence that the

   model slope is not equal to zero? Explain.

5. Plot the standardized residuals versus ?, as well as a normal probability plot of the standardized residuals.

   What do these plots suggest about the model assumptions?

6. Predict the mean monthly rent for an apartment that has 850 square feet.

7. Construct a 95% confidence interval on the mean monthly rent for an apartment that has 850 square feet.

8. Construct a 95% prediction interval on monthly rent for an apartment that has 850 square feet.

Observation	Cost (y)	Sq Ft (x)
1	1375.45	932.57
2	1101.84	748.51
3	1117.36	802.89
4	1264.47	863.70
5	1153.78	820.81
6	1284.78	880.18
7	1347.72	889.99
8	1153.94	757.00
9	1227.91	832.32
10	940.92	636.79
11	1451.17	964.54
12	1227.46	787.09
13	1223.59	729.62
14	1299.92	824.61
15	1033.04	707.14
16	1332.67	847.91
17	1167.35	793.93
18	1424.59	1067.78
19	1212.05	963.85
20	730.11	600.31
21	1318.49	894.13
22	1004.32	710.19
23	1165.73	824.49
24	1304.73	866.44
25	1322.26	924.77
26	1171.99	822.70
27	1573.41	1007.63
28	1282.09	801.91
29	1568.79	882.75
30	1405.95	916.47
31	1294.61	858.52
32	1350.73	938.10
33	1304.96	882.32
34	1153.32	771.59
35	829.62	669.46
36	1529.05	1035.86
37	1260.27	789.55
38	1234.43	860.34
39	1409.90	906.32
40	1392.18	861.36
41	1248.01	759.53
42	1283.53	803.23
43	1031.28	837.51
44	1441.41	997.90
45	1237.03	763.92
46	1433.17	928.47
47	1322.78	880.86
48	1284.61	826.61
49	1230.46	744.30
50	1248.76	821.59
51	1187.13	841.33
52	1037.57	703.06
53	1284.05	869.22
54	1242.33	767.77
55	1303.09	840.58
56	1381.18	883.62
57	1155.09	759.53
58	1196.53	821.17
59	1334.62	885.01
60	1022.29	666.41
61	1291.58	953.60
62	1657.56	1092.45
63	1512.67	945.94
64	1217.62	818.42
65	1497.53	892.86
66	1176.65	746.40
67	1515.65	1037.79
68	1401.71	944.07
69	1366.58	928.73
70	1168.73	762.41
71	1399.46	881.99
72	1163.06	794.17
73	1275.13	818.86
74	1366.03	793.00
75	1215.29	747.43
76	1222.21	759.13
77	1163.03	829.01
78	967.75	680.11
79	1391.40	910.76
80	1248.27	838.22
81	1218.37	919.92
82	1165.00	876.96
83	1452.76	899.43
84	977.72	701.69
85	1330.79	747.97
86	986.75	805.30
87	1337.33	860.97
88	1445.37	962.87
89	1156.88	821.00
90	1505.67	976.16
91	1267.93	897.54
92	1538.10	967.41
93	1301.73	862.69
94	1013.54	784.32
95	811.32	701.86
96	1270.94	865.55
97	1513.36	931.86
98	1250.23	820.74
99	1200.31	795.92
100	1298.48	819.14

Answer 1

from the results in R studio,

From the scattered plot we conclude that there is linear relationship between rent and the falt size.

From fitted model regression etsimates are,

Bo=-0.2926

B1=1.4968

For slope B1 the p value is less than 0.05( level of significance. Therefore slope is significantly different from 0.

R square value is 0.70.

We can say that 70% of variation in y explained by the x.

Predicted rent for 850 sqft flat size is 1272.01

Confidence interval for slope and prediction are given in r output.