Concept explainers
To calculate: The
Answer to Problem 12CS
0.3275
Explanation of Solution
Given information:
A model in which previous values of a variable are used to predict future values of the same variable is called an autoregressive model. The following table presents the data needed to fit this model.
Year | This Year’sUnemployment | Next Year’sUnemployment |
1985 | 7.2 | 7.0 |
1986 | 7.0 | 6.2 |
1987 | 6.2 | 5.5 |
1988 | 5.5 | 5.3 |
1989 | 5.3 | 5.6 |
1990 | 5.6 | 6.8 |
1991 | 6.8 | 7.5 |
1992 | 7.5 | 6.9 |
1993 | 6.9 | 6.1 |
1994 | 6.1 | 5.6 |
1995 | 5.6 | 5.4 |
1996 | 5.4 | 4.9 |
1997 | 4.9 | 4.5 |
1998 | 4.5 | 4.2 |
1999 | 4.2 | 4.0 |
2000 | 4.0 | 4.7 |
2001 | 4.7 | 5.8 |
2002 | 5.8 | 6.0 |
2003 | 6.0 | 5.5 |
2004 | 5.5 | 5.1 |
2005 | 5.1 | 4.6 |
2006 | 4.6 | 4.6 |
2007 | 4.6 | 5.8 |
2008 | 5.8 | 9.3 |
2009 | 9.3 | 9.6 |
2010 | 9.6 | 8.9 |
2011 | 8.9 | 8.1 |
Formula Used:
The
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 | ||||
7.2 | 7.0 | 1.17778 | 1.38716 | 0.94444 | 0.89198 |
7.0 | 6.2 | 4.12963 | 17.05384 | 0.14444 | 0.02086 |
6.2 | 5.5 | 3.32963 | 11.08643 | -0.55556 | 0.30864 |
5.5 | 5.3 | 2.62963 | 6.91495 | -0.75556 | 0.57086 |
5.3 | 5.6 | 2.42963 | 5.90310 | -0.45556 | 0.20753 |
5.6 | 6.8 | 2.72963 | 7.45088 | 0.74444 | 0.55420 |
6.8 | 7.5 | 3.92963 | 15.44199 | 1.44444 | 2.08642 |
7.5 | 6.9 | 4.62963 | 21.43347 | 0.84444 | 0.71309 |
6.9 | 6.1 | 4.02963 | 16.23791 | 0.04444 | 0.00198 |
6.1 | 5.6 | 3.22963 | 10.43051 | -0.45556 | 0.20753 |
5.6 | 5.4 | 2.72963 | 7.45088 | -0.65556 | 0.42975 |
5.4 | 4.9 | 2.52963 | 6.39903 | -1.15556 | 1.33531 |
4.9 | 4.5 | 2.02963 | 4.11940 | -1.55556 | 2.41975 |
4.5 | 4.2 | 1.62963 | 2.65569 | -1.85556 | 3.44309 |
4.2 | 4.0 | 1.32963 | 1.76791 | -2.05556 | 4.22531 |
4.0 | 4.7 | 1.12963 | 1.27606 | -1.35556 | 1.83753 |
4.7 | 5.8 | 1.82963 | 3.34754 | -0.25556 | 0.06531 |
5.8 | 6.0 | 2.92963 | 8.58273 | -0.05556 | 0.00309 |
6.0 | 5.5 | 3.12963 | 9.79458 | -0.55556 | 0.30864 |
5.5 | 5.1 | 2.62963 | 6.91495 | -0.95556 | 0.91309 |
5.1 | 4.6 | 2.22963 | 4.97125 | -1.45556 | 2.11864 |
4.6 | 4.6 | 1.72963 | 2.99162 | -1.45556 | 2.11864 |
4.6 | 5.8 | 1.72963 | 2.99162 | -0.25556 | 0.06531 |
5.8 | 9.3 | 2.92963 | 8.58273 | 3.24444 | 10.52642 |
9.3 | 9.6 | 6.42963 | 41.34014 | 3.54444 | 12.56309 |
9.6 | 8.9 | 6.72963 | 45.28791 | 2.84444 | 8.09086 |
8.9 | 8.1 | 6.02963 | 36.35643 | 2.04444 | 4.17975 |
Hence, the standard deviation is given by:
And,
Consider,
Hence, the table for calculating coefficient of correlation is given by:
x | y | |||
7.2 | 7.0 | 1.17778 | 0.94444 | 1.11235 |
7.0 | 6.2 | 4.12963 | 0.14444 | 0.59650 |
6.2 | 5.5 | 3.32963 | -0.55556 | -1.84979 |
5.5 | 5.3 | 2.62963 | -0.75556 | -1.98683 |
5.3 | 5.6 | 2.42963 | -0.45556 | -1.10683 |
5.6 | 6.8 | 2.72963 | 0.74444 | 2.03206 |
6.8 | 7.5 | 3.92963 | 1.44444 | 5.67613 |
7.5 | 6.9 | 4.62963 | 0.84444 | 3.90947 |
6.9 | 6.1 | 4.02963 | 0.04444 | 0.17909 |
6.1 | 5.6 | 3.22963 | -0.45556 | -1.47128 |
5.6 | 5.4 | 2.72963 | -0.65556 | -1.78942 |
5.4 | 4.9 | 2.52963 | -1.15556 | -2.92313 |
4.9 | 4.5 | 2.02963 | -1.55556 | -3.15720 |
4.5 | 4.2 | 1.62963 | -1.85556 | -3.02387 |
4.2 | 4.0 | 1.32963 | -2.05556 | -2.73313 |
4.0 | 4.7 | 1.12963 | -1.35556 | -1.53128 |
4.7 | 5.8 | 1.82963 | -0.25556 | -0.46757 |
5.8 | 6.0 | 2.92963 | -0.05556 | -0.16276 |
6.0 | 5.5 | 3.12963 | -0.55556 | -1.73868 |
5.5 | 5.1 | 2.62963 | -0.95556 | -2.51276 |
5.1 | 4.6 | 2.22963 | -1.45556 | -3.24535 |
4.6 | 4.6 | 1.72963 | -1.45556 | -2.51757 |
4.6 | 5.8 | 1.72963 | -0.25556 | -0.44202 |
5.8 | 9.3 | 2.92963 | 3.24444 | 9.50502 |
9.3 | 9.6 | 6.42963 | 3.54444 | 22.78947 |
9.6 | 8.9 | 6.72963 | 2.84444 | 19.14206 |
8.9 | 8.1 | 6.02963 | 2.04444 | 12.32724 |
Plugging the values in the formula,
Therefore, the correlation coefficient for the given data is 0.3275
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Chapter 4 Solutions
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