Concept explainers
The following table, Reproduce from chapter introduction. Presents the inflation rate and unemployment rate, both in percent, for the years 1985-2012
We will investigate some methods for predicting unemployment first. We will try to predict the unemployment rate from the inflation rate.
Compute the least-squares line for predicting unemployment from inflation.
To calculate: To compute the least squares regression line for the given data set.
Answer to Problem 2CS
Explanation of Solution
Given information:
The following table presents the inflation rate and unemployment rate, both in percent, for the years 1985-2012.
Year | Inflation | Unemployment |
1985 | 3.8 | 7.2 |
1986 | 1.1 | 7.0 |
1987 | 4.4 | 6.2 |
1988 | 4.4 | 5.5 |
1989 | 4.6 | 5.3 |
1990 | 6.1 | 5.6 |
1991 | 3.1 | 6.8 |
1992 | 2.9 | 7.5 |
1993 | 2.7 | 6.9 |
1994 | 2.7 | 6.1 |
1995 | 2.5 | 5.6 |
1996 | 3.3 | 5.4 |
1997 | 1.7 | 4.9 |
1998 | 1.6 | 4.5 |
1999 | 2.7 | 4.2 |
2000 | 3.4 | 4.0 |
2001 | 1.6 | 4.7 |
2002 | 2.4 | 5.8 |
2003 | 1.9 | 6.0 |
2004 | 3.3 | 5.5 |
2005 | 3.4 | 5.1 |
2006 | 2.5 | 4.6 |
2007 | 4.1 | 4.6 |
2008 | 0.1 | 5.8 |
2009 | 2.7 | 9.3 |
2010 | 1.5 | 9.6 |
2011 | 3.0 | 8.9 |
2012 | 1.7 | 8.1 |
Formula Used:
The equation for least-square regression line:
Where
The correlation coefficient of a data is given by:
Where,
The standard deviations are given by:
The mean of x is given by:
The mean of y is given by:
Calculation:
The mean of x is given by:
The mean of y is given by:
The data can be represented in tabular form as:
x | y | ||||
3.8 | 7.2 | 0.97143 | 0.94367 | 1.10357 | 1.21787 |
1.1 | 7.0 | -1.72857 | 2.98796 | 0.90357 | 0.81644 |
4.4 | 6.2 | 1.57143 | 2.46939 | 0.10357 | 0.01073 |
4.4 | 5.5 | 1.57143 | 2.46939 | -0.59643 | 0.35573 |
4.6 | 5.3 | 1.77143 | 3.13796 | -0.79643 | 0.63430 |
6.1 | 5.6 | 3.27143 | 10.70224 | -0.49643 | 0.24644 |
3.1 | 6.8 | 0.27143 | 0.07367 | 0.70357 | 0.49501 |
2.9 | 7.5 | 0.07143 | 0.00510 | 1.40357 | 1.97001 |
2.7 | 6.9 | -0.12857 | 0.01653 | 0.80357 | 0.64573 |
2.7 | 6.1 | -0.12857 | 0.01653 | 0.00357 | 0.00001 |
2.5 | 5.6 | -0.32857 | 0.10796 | -0.49643 | 0.24644 |
3.3 | 5.4 | 0.47143 | 0.22224 | -0.69643 | 0.48501 |
1.7 | 4.9 | -1.12857 | 1.27367 | -1.19643 | 1.43144 |
1.6 | 4.5 | -1.22857 | 1.50939 | -1.59643 | 2.54858 |
2.7 | 4.2 | -0.12857 | 0.01653 | -1.89643 | 3.59644 |
3.4 | 4.0 | 0.57143 | 0.32653 | -2.09643 | 4.39501 |
1.6 | 4.7 | -1.22857 | 1.50939 | -1.39643 | 1.95001 |
2.4 | 5.8 | -0.42857 | 0.18367 | -0.29643 | 0.08787 |
1.9 | 6.0 | -0.92857 | 0.86224 | -0.09643 | 0.00930 |
3.3 | 5.5 | 0.47143 | 0.22224 | -0.59643 | 0.35573 |
3.4 | 5.1 | 0.57143 | 0.32653 | -0.99643 | 0.99287 |
2.5 | 4.6 | -0.32857 | 0.10796 | -1.49643 | 2.23930 |
4.1 | 4.6 | 1.27143 | 1.61653 | -1.49643 | 2.23930 |
0.1 | 5.8 | -2.72857 | 7.44510 | -0.29643 | 0.08787 |
2.7 | 9.3 | -0.12857 | 0.01653 | 3.20357 | 10.26287 |
1.5 | 9.6 | -1.32857 | 1.76510 | 3.50357 | 12.27501 |
3.0 | 8.9 | 0.17143 | 0.02939 | 2.80357 | 7.86001 |
1.7 | 8.1 | -1.12857 | 1.27367 | 2.00357 | 4.01430 |
Hence, the standard deviation is given by:
And,
Consider,
Hence, the table for calculating coefficient of correlation is given by:
x | y | |||
3.8 | 7.2 | 0.97143 | 1.10357 | 1.07204 |
1.1 | 7.0 | -1.72857 | 0.90357 | -1.56189 |
4.4 | 6.2 | 1.57143 | 0.10357 | 0.16276 |
4.4 | 5.5 | 1.57143 | -0.59643 | -0.93724 |
4.6 | 5.3 | 1.77143 | -0.79643 | -1.41082 |
6.1 | 5.6 | 3.27143 | -0.49643 | -1.62403 |
3.1 | 6.8 | 0.27143 | 0.70357 | 0.19097 |
2.9 | 7.5 | 0.07143 | 1.40357 | 0.10026 |
2.7 | 6.9 | -0.12857 | 0.80357 | -0.10332 |
2.7 | 6.1 | -0.12857 | 0.00357 | -0.00046 |
2.5 | 5.6 | -0.32857 | -0.49643 | 0.16311 |
3.3 | 5.4 | 0.47143 | -0.69643 | -0.32832 |
1.7 | 4.9 | -1.12857 | -1.19643 | 1.35026 |
1.6 | 4.5 | -1.22857 | -1.59643 | 1.96133 |
2.7 | 4.2 | -0.12857 | -1.89643 | 0.24383 |
3.4 | 4.0 | 0.57143 | -2.09643 | -1.19796 |
1.6 | 4.7 | -1.22857 | -1.39643 | 1.71561 |
2.4 | 5.8 | -0.42857 | -0.29643 | 0.12704 |
1.9 | 6.0 | -0.92857 | -0.09643 | 0.08954 |
3.3 | 5.5 | 0.47143 | -0.59643 | -0.28117 |
3.4 | 5.1 | 0.57143 | -0.99643 | -0.56939 |
2.5 | 4.6 | -0.32857 | -1.49643 | 0.49168 |
4.1 | 4.6 | 1.27143 | -1.49643 | -1.90260 |
0.1 | 5.8 | -2.72857 | -0.29643 | 0.80883 |
2.7 | 9.3 | -0.12857 | 3.20357 | -0.41189 |
1.5 | 9.6 | -1.32857 | 3.50357 | -4.65474 |
3.0 | 8.9 | 0.17143 | 2.80357 | 0.48061 |
1.7 | 8.1 | -1.12857 | 2.00357 | -2.26117 |
Plugging the values in the formula,
Plugging the values to obtain b1 ,
Plugging the values to obtain b0 ,
Hence, the least-square regression line is given by:
Therefore, the least squares regression line for the given data set is
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Chapter 4 Solutions
Elementary Statistics 2nd Edition
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