The intercept our model the base price the sale price

From Table 1, it can be seen that the average house sale price in thousands of dollars is $804.88. The median house sale price is $798.95 thousand dollar. The spread of the sale prices can be assessed from the standard deviation which is $137.13 thousand dollar. Therefore, the spread of the data is high. The skewness of the data is very low which is 0.09. Therefore, it can be commented that the data is symmetrical and follows normal distribution. It can also be seen from the difference between mean and media, which are nearly the same considering the size of data. Therefore, the spread of the data is high whereas skewness of the house prices is nearly symmetrical.

Figure 1 Scatter plot between price and size

As it can be seen from the figure that there is a positive relationship between house sale prices and size of the house. It can be interpreted that as the size of house in square meters increases the sale price of house also increases.

Q.4 The linear regression of below stated model is performed using excel,

price_i= β₀+ β₁size _i + β₂age _i+ β₃proximity_i + u_i


SUMMARY OUTPUT

Regression Statistics
Multiple R	0.93093204
R Square	0.866634463
Adjusted R Square	0.86623276
Standard Error	50.15287793
Observations	1000

ANOVA
	df	SS	MS	F
Regression	3	16279587.45	5426529	2157.399
Residual	996	2505249.92	2515.311
Total	999	18784837.37

	Coefficients	Standard Error	t Stat	P-value
Intercept	25.59282925	14.00464211	1.827453	0.067931
size	3.890134007	0.078081854	49.82123	1E-272
age	-0.601560515	0.168471364	-3.5707	0.000373
proximity	195.8438204	3.176667127	61.65072	0

$${\widehat{price}}_{i} = \ 25.5983 + \ 3.8901{(size\ }_{i}) - 0.6016({age\ }_{i}) + \ 195.8438{(proximity}_{i})$$

The adjusted R² of the model is 0.866, which is a measure of goodness of fit. The intercept of our model is the base price of the sale price. It is the average price level which is not dependent on any of the explanatory variables. The estimates of slope parameters of size, age and proximity are 3.8901, -0.6016 and 195.8438 respectively. The estimate of slope coefficient of size can be interpreted as the sale price is increased by $3.89 thousand if size increases by 1 square meters, keeping other variables constant. Similarly, if the age of the house is increased by 1 year the sale prices are decreased by $-0.6015 thousand while keeping the effect of size and proximity constant. If the house is near major business district then the proximity variable will take the value of one, therefore, the prices of house which are near business districts are on 195.8438 thousand more as compared to the houses which are not near the business district keeping other factors constant. The results are expected as, the houses with more size and which are near business districts will costs more. The age of house is also very important factor, the more old the age of the house, the prices of house will decrease. It is also important to note that at 5% only size, age and proximity estimates are significant.

**Table 4 Regression output**
SUMMARY OUTPUT

Regression Statistics
Multiple R	0.928131596
R Square	0.86142826
Adjusted R Square	0.860731219
Standard Error	0.064934719
Observations	1000

ANOVA
	df	SS	MS	F	Significance F
Regression	5	26.05462137	5.210924	1235.836	0
Residual	994	4.191218689	0.004217
Total	999	30.24584006

	Coefficients	Standard Error	t Stat	P-value
Intercept	2.175009803	0.090650516	23.99335	1E-100
log(size)	0.847233288	0.017580409	48.1919	2.5E-262
age	-0.000873122	0.000218377	-3.99823	6.85E-05
proximity	0.247529504	0.004113783	60.17077	0
pool	0.018570706	0.00510377	3.638625	0.000288
fireplace	0.007021951	0.004136736	1.697462	0.089922

Q.8 Null Hypothesis: all the variables in the model are not significant.

Alternate hypothesis: at least one coefficient is different from zero.