Ecom90009 Quantitative Methods For Business Assessment Answers

Tasks

Write down the estimated model and interpret the coefficient on the interaction term. Find the expression for the effect of an additional bathroom on house price and then compute this effect for a house of average lot size. Interpret this result.

Re-estimate the relationship for house prices from part (v) adding the indicator variable on basement as a regressor (i.e. in addition to your quantitative variables, interaction variable and intercept in part (v)). Report the estimates for your model in a table (Excel output table is fine) and write down the estimated model.

Provide a full interpretation of the coefficient for the newly included nominal variable. Comment on the fit of the model you have just estimated and compare it to the fit of the model from part (v). Using the Excel output, what can you say about the validity of the model?

Q1

Here are the descriptive statistics for sales price and lot size.

Price: the average price is 67.7689 or $67768 as we are given data in $1000 units. The standard deviation is very low at 1.128. the data is positively skewed, but only moderately at 1.107 value for skewness. The minimum price is $25000 and maximum is $190000, there is little difference between median and mode, though mean exceeds both of these values.

Lot size: the average lot size is 5104.69 square feet. This lies between mode of 6000 and median of 4505 square feet. The data is positively skewed with a value of 1.38. The range is very large at 14550 square feet.

 Q2

We show plots for association between

Price and lot size: the first scatterplot shows a positive association between the variables. This is shown by the upward sloping trend line. The degree of association is low as R2 value is only 0.293. Only 29% of variation in lot size explains the variation in prices. We need other factors that impact on prices.

Price and no of bedrooms: we have the option of 1. 2 or 3 bedrooms. The highest price is for the 3 bedroom house.  

Price and no of bathrooms: the least number of homes have 3 bathrooms, with 1 bathroom being most common in the sample as we show in table below. The dots for 1 bathroom was maximum in the scatterplot as well. .

Price and no of garages.300 out of 535 samples have zero garages.

None of the relations is linear as the data is discrete in 3 variables- bedrooms, bathrooms and garages. For lot sizes linearity is weak as the value of R2 tells us.

Q3.

The regression results are below:

 The overall fit is moderately good as R2 is .71

The high F value of 138.7 implies that the overall model is significant.

The coefficient of lot size is 0.0047 which means that for every 1 square feet increase in lot size the price rises by .0047*1000 =4.7$. The coefficient is significant as p value is almost zero.

The coefficient of bedrooms is 5.5454 which means that for every additional bedroom the price rises by 5.5454*1000 =$ 5545.4 The coefficient is significant as p value is almost zero.

The coefficient of bathrooms is 17.8066 which means that for every additional bathroom the price rises by 17.8066*1000 =$ 17806.6 The coefficient is significant as p value is almost zero.

The coefficient of garage is 6.0854 which means that for every additional garage the price rises by 6.0854*1000 =$ 6085.4 The coefficient is significant as p value is almost zero.

The marginal effect of bathroom is maximum among all variables.

Q4

The t test for bathroom variable is as follows:

Step 1:

Ho: coefficient for bathroom is = 0

H1: coefficient for bathroom is ≠ 0

Step 2:

Set significance level = 0.05

Step 3:

The t value in the result is 10.05 with p value of almost zero.

Step 4:

Compare p value with 0.05.  p value < 0.05

Step 5:

 As p value is lower the coefficient is significant.

Step 6: the bathroom variable must be part of the regression.

The coefficient value is 17.8066, which means that every additional bathroom will add 17.8066*1000 = 17806 to the expected price of the house. As this is higher than cost of construction at 16000, an additional bathroom must be constructed.

Q5

The regression results are :

The estimated model is price (P)

P = 10.49 +.002977*lot size +5.38*bedroom +10.10*bathroom +6.25*garage +.001358*(bath*lot size)

An additional bathroom, with average lot size of 5104.69 means that price will change by

 10.10 +.001358*5104.69 = 17.032 or 17.032*1000 = $17032. Price rises by $17032 with an additional bathroom for an average lot size.

The coefficient of interaction term is 0.001358. This is significant at 5% level as p value is 0.03, which is less than 5%. At 10% this coefficient will  be insignificant as 0.1 >0.03. as the coefficient is positive it tells us that the effect of lot size and bathrooms are not independent of each other. A unit change in lot size will lead to price change of .0029+.00135=  0.0043 or 4.3$. in the same way an additional bathroom now increases price by 10.15+.0013 = 10.151 or $10151.

Q6.

The assumptions/conditions are:

Model : Y = a +b₁*X₁+ b₂*X₂ + b₃*X₃+……… b_n*X_n+ error

• The model is linear in the relationship between explanatory and dependent variables.
• The error terms are normally distributed
• There is no correlation between the error terms
• The variance of errors is equal for all observations( 1 to n).

Q7

The correlation matrix is given below. The values of correlation do not exceed .5 in any set of explanatory variables, which shows lack of multicollinearity. A value of .54 exists between the dependent variable (price) and lot size, which is not concerned with multicollinearity.

Q8

The regression equation is now

P= 12.25 +.00252*lot size +5.048*bedroom +7.87*bathroom +6.45*garage+6.667*basement +.001687*(bath*lot size)

The coefficient of basement is positive which means that having a basement adds to price. The extent of addition is 6.667*1000 = or $6667.

To compare the model with and without basement variable we must look at the significance of the added variable as well the effect on adjusted R^2 value. Note that the p value of the coefficient of basement is almost zero which makes it significant.

Also a simple comparison of R^2 values show an improvement from 0.71 to 0.72- a marginal increase. The value of adjusted R^2 is also marginally better with the expanded model.( 052 against 0.51 in smaller model)

Q9.

A point prediction will use all values of coefficients along with given values of the explanatory variables:

Predicted value of price = 12.25 +.00252*lot size +5.048*bedroom +7.87*bathroom +6.45*garage+6.667*basement +.001687*(bath*lot size)

=12.25 +.00252*5000 +5.048*3 +7.87*1+6.45*1+6.667*0 +.001687*(3*5000) =79.619 or $79619.

 Now we include a basement which gives the model:

Predicted value of price = = 12.25 +.00252*lot size +5.048*bedroom +7.87*bathroom +6.45*garage+6.667*basement +.001687*(bath*lot size)+6.667*basement

= 0 .25 +.00252*5000 +5.048*3 +7.87*1+6.45*1+6.667*0 +.001687*(3*5000) +6.6667*1 =82.2865

 or $ 82286.5.

the additional price is equal to the coefficient of[ basement *1000] as the units of price are in 1000 dollars.

Q10.

From a marketing perspective it maybe sensible to ask which feature in a home will fetch the highest price. These features include bathroom, garage, bedroom or basement. Depending on the model we choose the highest coefficient of these variables will tell us the feature that fetches highest price. The marketing team can then sell this feature more aggressively as buyers will be more willing to pay for it based on the datta given to us.  

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