Question

A real estate agent wants to study the relationship between the size of an apartment and its monthly rent price. The table below presents the size in square feet and the monthly rent in dollars, for a sample of apartments in a suburban neighborhood.

Rent ($)	720	595	915	760	1000	790	880	845	650	748	685	755	815	745	715	885
Size (Square Feet)	1000	900	1200	810	1210	860	1135	960	800	960	650	970	1000	1000	1000	1180

Calculate the correlation between these two variables.

If a linear regression model were fit, what is the value of the slope and the value of the y-intercept?

In a test for the slope of the regression line being equal to zero versus the two-sided alternate, what is the value of the test statistic and the p-value?

Answer 1

	Size X	Rent ($) Y	X * Y	X2	Y2
	1000	720	720000	1000000	518400
	900	595	535500	810000	354025
	1200	915	1098000	1440000	837225
	810	760	615600	656100	577600
	1210	1000	1210000	1464100	1000000
	860	790	679400	739600	624100
	1135	880	998800	1288225	774400
	960	845	811200	921600	714025
	800	650	520000	640000	422500
	960	748	718080	921600	559504
	650	685	445250	422500	469225
	970	755	732350	940900	570025
	1000	815	815000	1000000	664225
	1000	745	745000	1000000	555025
	1000	715	715000	1000000	511225
	1180	885	1044300	1392400	783225
Total	15635	12503	12403480	15637025	9934729

r = 0.7647

Equation of regression line is Ŷ = a + bX
b = ( n Σ(XY) - (ΣX* ΣY) ) / ( n Σ X² - (ΣX)² )
b = ( 16 * 12403480 - 15635 * 12503 ) / ( 16 * 15637025 - ( 15635 )²)
b = 0.5177

a =( ΣY - ( b * ΣX ) ) / n
a =( 12503 - ( 0.5177 * 15635 ) ) / 16
a = 275.5298
Equation of regression line becomes Ŷ = 275.5298 + 0.5177 X

To Test :-

Sxx =Σ (Xi - X̅ )	Syy = Σ( Yi - Y̅ )	Sxy = Σ (Xi - X̅ ) * (Yi - Y̅)
520.4102	3774.566	-1401.54
5957.91	34758.94	14390.64
49645.41	17838.94	29759.39
27951.66	459.5664	3584.082
54201.66	47769.57	50884.08
13732.91	73.31641	-1003.42
24904.79	9714.566	15554.39
295.4102	4040.191	-1092.48
31395.41	17275.82	23289.08
295.4102	1118.066	574.707
107051.7	9300.191	31553.14
51.66016	698.9414	190.0195
520.4102	1126.441	765.6445
520.4102	1327.691	-831.23
520.4102	4413.941	-1515.61
41132.91	10725.19	21003.77
358698.4	164415.9	185704.7

X̅ = Σ (Xi / n ) = 15635/16 = 977.1875
Y̅ = Σ (Yi / n ) = 12503/16 = 781.4375

H1 :-

H0 :-

Test Statistic :-

S² = ( 164415.9375 - 0.5177 * 185704.6875 ) / 16 - 2
S² = 4876.9015
S = 69.8348



t = 4.4399

P - value = P ( t > 4.4399 ) = 0.0006