Question
In: Statistics and Probability
Correlation To determine how the number of housing starts is affected by mortgage rates an economist...
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 |
- What is the dependent and independent variable?
- Prepare a scatter plot. What is the direction of correlation?
- What is the equation of the regression line?
- What is the coefficient of determination (r2) and interpret the meaning of it?
- Calculate the correlation coefficient (r) and interpret the meaning of it.
- Predict the number of housing starts if the interest rate is 8.65.
- Would it be possible to use this regression model to predict the number of housing starts for an interest rate of 2.45? Why?
Solutions
Expert Solution
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
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1992 8.3
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1995 7.21
1996 7.06
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