Question

Correlation

To determine how the number of housing starts is affected by mortgage rates an economist recorded the average mortgage rate and the number of housing starts in a large country for the past 10 years. These data are listed here.

	Rate	Starts
year #1	8.5	115
year #2	7.8	111
year #3	7.6	185
year #4	7.5	201
year #5	8	206
year #6	8.4	167
year #7	8.8	155
year #8	8.9	117
year #9	8.5	133
year #10	8	150

         

1. What is the dependent and independent variable?

2. Prepare a scatter plot. What is the direction of correlation?

3. What is the equation of the regression line?

4. What is the coefficient of determination (r²) and interpret the meaning of it?

5. Calculate the correlation coefficient (r) and interpret the meaning of it.

6. Predict the number of housing starts if the interest rate is 8.65.

7. Would it be possible to use this regression model to predict the number of housing starts for an interest rate of 2.45? Why?

Answer 1

a. Here we need to determine how the number of housing starts is affected by mortgage rates.

Hence mortgage rates is independent variable and number of housing starts is dependent variable

b.

As there is decreasing trend so there is negative correlation

c.

Sum of X = 82
Sum of Y = 1540
Mean X = 8.2
Mean Y = 154
Sum of squares (SSX) = 2.16
Sum of products (SP) = -84.6

Regression Equation = ŷ = bX + a

b = SP/SSX = -84.6/2.16 = -39.1667

a = MY - bMX = 154 - (-39.17*8.2) = 475.1667

ŷ = -39.1667X + 475.1667

d.

X Values
∑ = 82
Mean = 8.2
∑(X - Mx)2 = SSx = 2.16

Y Values
∑ = 1540
Mean = 154
∑(Y - My)2 = SSy = 11240

X and Y Combined
N = 10
∑(X - Mx)(Y - My) = -84.6

R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))

r = -84.6 / √((2.16)(11240)) = -0.543

So r^2=-0.543^2=0.295

Hence 29.5% of variation in starts is explained by rates

e. X Values
∑ = 82
Mean = 8.2
∑(X - Mx)2 = SSx = 2.16

Y Values
∑ = 1540
Mean = 154
∑(Y - My)2 = SSy = 11240

X and Y Combined
N = 10
∑(X - Mx)(Y - My) = -84.6

R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))

r = -84.6 / √((2.16)(11240)) = -0.543

Hence there is moderate negative correlation between starts and rates

f. For x=8.65, ŷ = (-39.1667*8.65) + 475.1667=136.3747