Question

A realtor for a residential real estate company in a large city has the business objective to develop a more accurate estimate of the monthly rental cost for apartments. The agent would like to use 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 (see Rent).

b. Run a linear regression using size (in square feet) to predict rent. Report your regression equation.

c. Is the regression model you ran statistically significant? How can you tell?

D. Interpret your slope and y-intercept within the context of the problem.

E. Using the regression equation you generated in (b) predict the rent of a 1900 square foot apartment.

F. What is the relationship (correlation) between square footage and rent?

G. How much variance in rent is explained by square footage?

Rent($)	Size(sq. ft)
700	1875
718	1950
726	1935
755	2200
850	1950
896	2150
956	2150
1000	1800
1040	2650
1085	2200
1100	2400
1136	2650
1150	2200
1175	2400
1200	2750
1225	2450
1232	2500
1245	2100
1259	2700
1285	2650
1361	2600
1369	2800
1450	2600
1485	2700
1985	3300

Answer 1

Notice that size of an apartment is being used to predict rent hence size of an apartment is independent variable and rent is dependent variable.

Independent variable(X) : Size(in square feet) of an apartment.

Dependent variable(Y): Rent(in dollars) of an apartment.

b. To run a linear regression we will use R software (you can do it manually also).

R software codes

>x=c(1875,1950,1935,2200,1950,2150,2150,1800,2650,2200,2400,2650,2200,2400,2750,2450,2500,2100,2700,2650,2600,2800,2600,2700,3300)

>y=c(700,718,726,755,850,896,956,1000,1040,1085,1100,1136,1150,1175,1200,1225,1232,1245,1259,1285,1361,1369,1450,1485,1985)

>model=lm(y~x)

>summary(model)

We created data vectors x and y. lm() function is used to fit linear regression model. summary() function is used to obtain the summary statistics of the regression model.

R software output

Call: lm( formula = y - x) Residuals: Min 1Q -274.15 -89.07 Max Median -18.94 30 80.77 303.98 Coefficients: Estimate Std. Error t value P r>t (Intercept) -483.63556 211.45030 -2.287 0.0317 * Х 0.67841 0.08765 7.740 7.52e-08 *** 0 I*** 0.001 '**' 0.01 I* Signif. codes: 0.05 '. 0.1 1 1 Residual standard error: 155.3 on 23 degree s of freedom Multiple R-squared: 0.7226, Adjusted R- squared: 0.7105 F-statistic: 59.91 on 1 and 23 DF, p-valu e: 7.518e-08

Observe the above output. Using above output regression equation is

Y= -483.63556 + 0.67841X

In words,

Rent = -483.63556 + 0.67841(Size)

c. Regression model in part b above is statistically significant. Observe the above output. Notice that the p-value is 7.518×10^-08 which is very small as compare to level of significance 0.01 or 0.05. We reject null hypothesis Ho if p-value is less than given level of significance. Here

Null hypothesis Ho: Fitted regression model is not significant.

Alternative hypothesis H₁: Fitted model is significant.

Since p-value is less than the level of significance we reject null hypothesis. Hence the Fitted regression model is statistically significant.

D.

Slope of the regression line is interpreted as the change in the dependent variable for every unit change in the independent variable.

In this context, slope is 0.67841 which can be interpreted as rent of apartment increases by 0.67841 dollars if the area of apartment is increased by 1 square foot.

y-intercept is interpreted as mean of dependent variable if independent variable assumes zero value.

Here, y-intercept is -483.63556 which can be interpreted as mean rent of apartment is -483.63556 when the size of apartment is zero foot.

Note that in the context of our problem, y-intercept is not meaningful.

E. To predict rent of an 1900 square feet apartment, just substitute X=1900 in the regression equation in part b.

So, rent= -483.63556+(0.67841×1900) = 805.34344.

So, rent of an 1900 square feet apartment is 805.34344 dollars.

F. To comment on the relationship between square footage and rent we will compute correlation coefficient between them using R software (you can do it manually also)

R software codes

>x=c(1875,1950,1935,2200,1950,2150,2150,1800,2650,2200,2400,2650,2200,2400,2750,2450,2500,2100,2700,2650,2600,2800,2600,2700,3300)

>y=c(700,718,726,755,850,896,956,1000,1040,1085,1100,1136,1150,1175,1200,1225,1232,1245,1259,1285,1361,1369,1450,1485,1985)

>cor(x,y)

Data vectors x and y are created. cor() function is used to compute correlation coefficient.

R software output

[1] 0.8500608

Observe the above output. Correlation coefficient between Rent and square footage is 0.8500608 which suggests that there is a strong positive linear relationship between rent and square footage.

That is, increase in square footage implies increase in the rent of an apartment.

G. Multiple R-squared gives the amount of variation in dependent variable explained by independent variable. Observe the above output, Multiple R-squared is 0.7226 which implies that 72.26 % of variation in rent is explained by square footage.

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