Question

Perfect Properties have collected sales data from property sales in the northern suburbs of Cape Town for the past month. In the table below you are supplied with the selling price (SP) of the house in Rand, the size of the plot in m² (P) as well as the size of the house, also in m² (H). They are interested in understanding which of these two factors influence the selling price.

House	Selling price (SP)	Plot size in m2 (P)	House area in m2 (H)
1	R3 264 000	1012	118
2	R4 054 000	1922	268
3	R3 448 000	1214	179
4	R3 718 000	2023	189
5	R3 634 000	1619	294
6	R3 914 000	1821	170
7	R3 564 000	506	188
8	R3 972 000	1113	181
9	R4 288 000	2023	242
10	R3 824 000	1720	190
11	R3 218 000	708	189
12	R3 556 000	1012	233
13	R3 674 000	708	213
14	R3 416 000	1012	151
15	R3 292 000	607	262
16	R3 198 000	1821	123
17	R3 684 000	1214	255
18	R3 436 000	911	277
19	R3 696 000	1113	272
20	R3 904 000	708	276

Use the data in the sheet named “Perfect” and answer the following questions:

1. Decide which linear regression model (with size of plot (P) or size of house (H) or both as independent variable) is the better model in explaining the behaviour of selling prices (SP) and motivate your selection.

The remaining answers must be based on the model that you have selected.

2. What is the total variation in “Selling Price” that is explained by the independent variable(s) that you have selected?
3. Is the overall regression model statistically significant? Test at the 5% level of significance using the model that you have selected in a. For this test formulate the appropriate null and alternative hypothesis, determine the region of acceptance, use the appropriate test statistic and draw the statistical and management conclusions.
4. Write down the linear regression equation in algebraic format using SP for Selling price, P for plot size and H for the size of the house.
5. Compute the expected Selling price for a house that stands on a plot of 1500 m² and has a house that covers 245 m².

Compute the 95% confidence interval of the mean expense for a house that stands on a plot of 1500 m² and has a house that covers 245 m².

Answer 1

a.

When we regress with Plot size, we get the following results,

Multiple R	0.478655
R Square	0.22911
Adjusted R Square	0.186283

	Coefficients	Standard Error	t Stat	P-value
Intercept	3288784	162273.9	20.26687	7.66E-14
Plot size in m2 (P)	281.5316	121.7208	2.312929	0.032759

When we regress with House area, we get the following results,

Multiple R	0.36864
R Square	0.135895
Adjusted R Square	0.08789

	Coefficients	Standard Error	t Stat	P-value
Intercept	3199378	268100.2	11.93352	5.52E-10
House area in m2 (H)	2053.03	1220.225	1.682501	0.109737

When we regress with both the variables , we get the following results,

Multiple R	0.630593
R Square	0.397648
Adjusted R Square	0.326783

	Coefficients	Standard Error	t Stat	P-value
Intercept	2773533	278567.6	9.956408	1.65E-08
Plot size in m2 (P)	301.9988	111.1118	2.717972	0.014616
House area in m2 (H)	2294.545	1052.079	2.180962	0.043522

Here, we can see that, when we use both the independent variables, the behavior of selling price is explained better. The R² , t stat and Adjusted R²are higher as compared to regression with individual variables. P value is also lower as compared to regression with individual variables .

b. The R²in this case is 39.76%, which means that 39.76% of the total variation of the Selling Price is explained by the variation in the independent variables

c. in this case,

Null Hypothesis, H₀ : Selling Price is not affected by Plot Size & House area.

Alternate Hypothesis, H_a :  Selling Price is affected by Plot Size & House area.

Now, here level of significance is 5%, which means we will accept the Hypothesis if its probability of alternate hypothesis happening is 95% or probability of not happening is 5%( which is p value)

Here, we are running regression to check the alternate hypothesis. If the p value of the independent variables is less than 5% or 0.05 , we can conclude that the variable is significantly affecting the dependent variable.

Upon regression with both the independent variables, we get the following results

	Coefficients	Standard Error	t Stat	P-value
Intercept	2773532.52	278567.57	9.96	0.00
Plot size in m2 (P)	302.00	111.11	2.72	0.01
House area in m2 (H)	2294.54	1052.08	2.18	0.04

Here , we can see that all the variables have less than 0.05 P value .

Therefore, we can conclude that these variables are significantly affecting the selling price. Thus, we can accept the alternate hypothesis that Selling Price is affected by Plot Size & House area and reject the null hypothesis.

The other observation is Plot size and House area , both drive sales positively, which is the real case.

d. The regression equation is given as,

SP = 2773532.52 +  302*P + 2294*H,

where, SP = Selling Price

P = Plot size

H = House area.

Hope I clarified your query