Question

Estimating & Forecasting. Assume you believe that the biggest factor affecting the selling price of a house is its size. To test your hypothesis, you randomly sample fifteen houses on sale in your neighborhood and collect the following data:

Observation (i)	Selling Price (x $1,000) (Y)	Size (x 100 ft2) (X)
1	265.2	12.0
2	279.6	20.2
3	311.2	27.0
4	328.0	30.0
5	352.0	30.0
6	281.2	21.4
7	288.4	21.6
8	292.8	25.2
9	356.0	37.2
10	263.2	14.4
11	272.4	15.0
12	291.2	22.4
13	299.6	23.9
14	307.6	26.6
15	320.4	30.7

1. What are the components of a regression line? Hint: there are a total of 5 in this case.
2. How would you specify this linear model? In other words, what would the regression equation look like. Hint: at this moment do not calculate anything. I am not interested in the actual coefficients, but how you would write out the equation using names of variables.
3. What are the estimates for the intercept and independent variable? Interpret them.
4. Plot sample data and estimated regression line in one graph.

Answer 1

(a) The five components of a regression line are:

• independent variable (X)
• dependent variable (Y)
• intercept coefficient ()
• slope coefficient ()
• error/residual term (u)

(b) A regression line is expressed algebraically as , which is how two variable linear model is specified. This can equivalently be represented as .

(c) The intercept term is the expected value of dependent variable for the independent variable be zero. The slope term is the expected change in the value of dependent variable for a unit increase in the independent variable. The estimated regression model would be as . The mean values of X and Y would be as and .

We have or or . Since or or .

The intercept is , meaning that the expected selling price of a house with size 0 sq. feet is $206.278 thousand. Note that in this case, the intercept is not meaningful, since for a house of size 0 sq. feet would mean the house doesn't exist, and price of that hypothetical house would not exist.

The slope or independent variable's coefficient is , meaning that for a unit increase in X, the Y would increase by 3.9559 units, ie if the size of house increases by 100 sq. feet then the selling price of that house would increase in response by $3.9559 thousand.

(d) The graph is as below.

The graph below is zoomed for appropriate viewing.