Question

An agent for a residential real estate company in a large city has the business objective of developing more accurate estimates of the monthly rental cost for apartments. Toward that goal, the agent would like to use the size of an apartment, as defined by square footage to predict the monthly rental cost. The agent selects a sample of 25 apartments in a particular residential neighborhood and collects the following data: Size (square feet) Rent ($) 850 1950 1450 2600 1085 2200 1232 2500 718 1950 1485 2700 1136 2650 726 1935 700 1875 956 2150 1100 2400 1285 2650 1985 3300 1369 2800 1175 2400 1225 2450 1245 2100 1259 2700 1150 2200 896 2150 1361 2600 1040 2650 755 2200 1000 1800 1200 2750 (a) Determine the coefficient of determination, r2, and interpret its meaning. (b) Determine the standard error of the estimate (Syx). (c) How useful do you think this regression model is for predicting the monthly rent? (d) Can you think of other variables that might explain the variation in monthly rent?

Answer 1

Solution

Let x = size of apartment (in sq.ft) and y = rent (in $)

Back-up Theory

The linear regression model Y = β₀ + β₁X + ε, ……………………………………....................................…..(1)

where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ².

Estimated Regression of Y on X is given by: Y = β_0cap + β_1capX, ……………...................................……….(2)

where β_1cap = Sxy/Sxx = r.√(Syy/Sxx) = r.(SDy/SDx) and β_0cap = Ybar – β_1cap.Xbar………...................…..(3)

Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i ………………………………………….………...................................….(4)

Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i ………………………………..................................………….……….….(5)

Sxx = Σ(i = 1 to n)(x_i – Xbar)² …………………………………………..................................……..…………....(6)

Syy = Σ(i = 1 to n)(y_i – Ybar)² ………………………………………...................................……..………………(7)

Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – Ybar)} ……………………………......................................………………….(8)

Correlation coefficient, r = Sxy/sqrt(Sxx. Syy) ……………………...........................................……………….. (10)

Estimate of σ² is given by s² = (Syy – β_1cap²Sxx)/(n - 2)…………...........................…………………………..(11)

Standard Error of yicap = s√[(1/n) + {(xi – Xbar)²/Sxx}] …………….................................……………………..(12)

Now to work out the solution,

Summary of Excel Calculations is given below. The answers are derived from these calculations.

n	25
Xbar	1135.32
ybar	2386.4
Sxx	1999747.44
Syy	3139726
Sxy	2130018.8
β1cap	1.06514391
β0cap	1177.12082
s^2	37867.3676
s	194.595395
r	0.8500608
r^2	0.72260336

Part (a)

Coefficient of determination = r² = 0.72226   Answer 1

Interpretation: r² represents the proportion of the variation in the response variable that is explained by the variation in the predictor variable. In the present scenario, this would imply that 72% of the variation in the rent is determined by the size of the apartment, which in turn can be taken to mean that size of the apartment may be used to get fairly good predictions of the rent. Answer 2   

Part (b)

Vide (11) and (12),

Standard Error of yicap at x = x*, say = 194.5954√[(1/25) + {(x* – 1135.32)²/1999747.44}] Answer 3

Part (c)

As already explained under interpretation of r², the present regression is quite useful in predicting the rent.

Answer 4

Part (d)

Other variables, which could impact the rent are:

1. Average income of the residents
2. Proximity (distance in miles) of the residential complex to shopping malls and other service providers
3. Proximity (distance in miles) of the residential complex to highway
4. Access to transport facilities. Answer 5

DONE