Question

As a real estate broker you are interested in a quantitative, data-driven approach to setting rental prices so that you can effectively price and lease apartment units. To aid you in this endeavor you have sample data on recently signed residential leases in your area, presented below. Your objective is to learn if there is a significant relationship/pattern between the size of an apartment and the rent charged. If there is a relationship/pattern, we want to quantify that relationship and use it to predict the rent of a newly available apartment using its size.

Size Rent Questions: (place all answers on this workbook, well-organized and easy to locate) 850 950 1 Which is your dependent/response variable? 1450 1600 2 Which is your independent/explanatory variable? 1085 1200 3 Construct a scatterplot of the data putting the appropriate variable on the appropriate axis? 1232 1500 4 Calculate r, the correlation coefficient, using Google Sheets function =correl(). 718 950 5 Describe the direction, strength, and shape of the relationship between Rent and Size you learned from the scatterplot and r. 1485 1700 6 Report the values for b0 and for b1 from the XL-Miner Output and then write the least-squares equation in Ŷ = b0 + b1X form. Don’t use y and x use “Size” and “Rent” in the equation. 1136 1650 7 Test Ho: β1=0 versus H1: β1≠0 at 0.01 level of significance. From this result determine whether an apartment’s rent has a significant linear relationship with an apartment’s size. 726 935 8 Report R^2, the Coefficient of Determination, and interpret it. 700 875 9 Use the regression model you learned from question 6 to predict the Rent for a (i) 950 ft2 and (ii) 1500 ft2 apartment. 956 1150 10 Plot the least squares regression line on the original scatterplot you created in question 3. 1100 1400 1285 1650 1985 2300 1369 1800

Answer 1

Que.1

Responce variable is price of newly available appartment.

Que.2

Independent variable is size of aparment.

Que.3

Scatter plot:

Que.4

Correlation coefficient = r = 0.96259

Que.5

There is positive correlation between rent and size of apartment. The relationship is strong because correlation coefficient r is close to 1.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.96256
R Square	0.926521
Adjusted R Square	0.920398
Standard Error	116.9612
Observations	14
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2069934	2069934	151.3118	3.67E-08
Residual	12	164159.1	13679.93
Total	13	2234093
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	128.0033	108.3619	1.181257	0.260383	-108.097	364.1036
X Variable 1	1.111399	0.090351	12.30088	3.67E-08	0.91454	1.308257

Que.6

Least-squares equation is

Ŷ = 128.0033 + 1.111399 X

Que.7

Test statistic for testing hypothesis Ho: β1=0 versus H1: β1≠0 is t = 12.3009

and p-value is 3.67E-08 which is less than 0.01, hence we reject null hypothesis at 1% level of significance and conclude that there is significanct relationship between rent and size.

Que.8

R2 = 0.9265

It mans variable size will explain 92.65% variation in the variable rent.

Que.9

(i)For 950 ft2

Ŷ = 128.0033 + 1.111399 X

Ŷ = 128.0033 + 1.111399 (950)

Ŷ = 1183.83235

(ii) 1500 ft2 apartment.

Ŷ = 128.0033 + 1.111399 X

Ŷ = 128.0033 + 1.111399 (1500)

Ŷ = 17951018

Que.10