Question

An agent for a real estate company wanted to predict the monthly rent for apartments based on the size of the apartment. The data for a sample of

25 apartments is available below. Perform a t test for the slope to determine if a significant linear relationship between the size and the rent exists.

a. At the 0.05 level of​ significance, is there evidence of a linear relationship between the size of the apartment and the monthly​ rent?

b. Construct a​ 95% confidence interval estimate of the population​ slope, betaβ1.

Size_(sq._ft)	Rent_($)
850	1950
1450	2575
1085	2225
1232	2500
718	1975
1495	2675
1126	2675
726	1935
700	1875
966	2150
1110	2400
1285	2650
1985	3275
1359	2775
1165	2425
1235	2475
1255	2075
1269	2725
1160	2200
896	2175
1351	2625
1040	2650
765	2175
1000	1825
1200	2775

Answer 1

Given:

Size_(sq._ft) x	Rent_($) y
850	1950
1450	2575
1085	2225
1232	2500
718	1975
1495	2675
1126	2675
726	1935
700	1875
966	2150
1110	2400
1285	2650
1985	3275
1359	2775
1165	2425
1235	2475
1255	2075
1269	2725
1160	2200
896	2175
1351	2625
1040	2650
765	2175
1000	1825
1200	2775

a)

Equation for simple linear regression is given as

where is the slope.

and of the above equation is estimated by solving the below simultaneous equation

n = number of samples = 25

The above required parameters are calculated below

x	y	x2	xy
850	1950	722500	1657500
1450	2575	2102500	3733750
1085	2225	1177225	2414125
1232	2500	1517824	3080000
718	1975	515524	1418050
1495	2675	2235025	3999125
1126	2675	1267876	3012050
726	1935	527076	1404810
700	1875	490000	1312500
966	2150	933156	2076900
1110	2400	1232100	2664000
1285	2650	1651225	3405250
1985	3275	3940225	6500875
1359	2775	1846881	3771225
1165	2425	1357225	2825125
1235	2475	1525225	3056625
1255	2075	1575025	2604125
1269	2725	1610361	3458025
1160	2200	1345600	2552000
896	2175	802816	1948800
1351	2625	1825201	3546375
1040	2650	1081600	2756000
765	2175	585225	1663875
1000	1825	1000000	1825000
1200	2775	1440000	3330000

Plugging in the above values into the equations we have

Multiplying the equation (1) by 28423 and equation (2) by 25 we have

Subtracting (3) from (4) we have

Substituting in equation 1 we have

Hence, the simple linear equation to predict the rent is given as

Rent = 1207.2481 + 1.0407 Size

Hence, for each sample for the observed size, the predicted equation is calculated as below

850	1950	2091.8431
1450	2575	2716.2631
1085	2225	2336.4076
1232	2500	2489.3905
718	1975	1954.4707
1495	2675	2763.0946
1126	2675	2379.0763
726	1935	1962.7963
700	1875	1935.7381
966	2150	2212.5643
1110	2400	2362.4251
1285	2650	2544.5476
1985	3275	3273.0376
1359	2775	2621.5594
1165	2425	2419.6636
1235	2475	2492.5126
1255	2075	2513.3266
1269	2725	2527.8964
1160	2200	2414.4601
896	2175	2139.7153
1351	2625	2613.2338
1040	2650	2289.5761
765	2175	2003.3836
1000	1825	2247.9481
1200	2775	2456.0881

Null hypothesis:

H0: There is no linear relationship between the size of the apartment and the monthly rent,

Alternative hypothesis:

H1: There is a linear relationship between the size of the apartment and the monthly rent,

Level of significance:

Test statistic:

where, is the mean. Therefore, the above deviations are calculated as below

850	1950	2091.8431	-286.92	82323.09	-141.843	20119.47
1450	2575	2716.2631	313.08	98019.09	-141.263	19955.26
1085	2225	2336.4076	-51.92	2695.686	-111.408	12411.65
1232	2500	2489.3905	95.08	9040.206	10.6095	112.5615
718	1975	1954.4707	-418.92	175494	20.5293	421.4522
1495	2675	2763.0946	358.08	128221.3	-88.0946	7760.659
1126	2675	2379.0763	-10.92	119.2464	295.9237	87570.84
726	1935	1962.7963	-410.92	168855.2	-27.7963	772.6343
700	1875	1935.7381	-436.92	190899.1	-60.7381	3689.117
966	2150	2212.5643	-170.92	29213.65	-62.5643	3914.292
1110	2400	2362.4251	-26.92	724.6864	37.5749	1411.873
1285	2650	2544.5476	148.08	21927.69	105.4524	11120.21
1985	3275	3273.0376	848.08	719239.7	1.9624	3.851014
1359	2775	2621.5594	222.08	49319.53	153.4406	23544.02
1165	2425	2419.6636	28.08	788.4864	5.3364	28.47716
1235	2475	2492.5126	98.08	9619.686	-17.5126	306.6912
1255	2075	2513.3266	118.08	13942.89	-438.327	192130.2
1269	2725	2527.8964	132.08	17445.13	197.1036	38849.83
1160	2200	2414.4601	23.08	532.6864	-214.46	45993.13
896	2175	2139.7153	-240.92	58042.45	35.2847	1245.01
1351	2625	2613.2338	214.08	45830.25	11.7662	138.4435
1040	2650	2289.5761	-96.92	9393.486	360.4239	129905.4
765	2175	2003.3836	-371.92	138324.5	171.6164	29452.19
1000	1825	2247.9481	-136.92	18747.09	-422.948	178885.1
1200	2775	2456.0881	63.08	3979.086	318.9119	101704.8

Applying the above estimated values in to the SE formula we have

Plugging in the SE into t-statistic we have

Critical Value:

The critical value of t-statistic is calculated from the students t-table for 5% level of significance with (n-2) degrees of freedom with two-tailed test. Therefore, for the above hypothesis we have n-2 = 25 -2 = 23 degrees of freedom. Hence, at 5% level of significance for two-tailed test with 23 degrees of freedom the t-value from the students t distribution table is given as 2.069. (It can also be calculated using excel with the function TINV. The inputs for the function are level of significance which is 0.05 and degrees of freedom which is 23. Therefore, the final function in excel is given as =TINV(0.05,23))

Inference:

The calculated t-value of 7.3809 is greater than the critical t-value of 2.069. Therefore, the null hypothesis is rejected and we conclude that there is an evidence of linear relationship between the size of the apartment and the monthly rent.

b)

95% Confidence interval for the slope is constructed as below

From section a we have

= 1.0407

t for 95% CI with 23 degrees of freedom = 2.069

= 0.141

Therefore,