Chapter 4, Problem 5CS

To determine

To calculate: The proportion of variation in unemployment explained by inflation.

Answer to Problem 5CS

The proportion of the variation in unemployment is explained by inflation is $0.027454$ .

Explanation of Solution

Given information:

The below table presents the inflation rate and unemployment rate, both in percent, for the year $1988-2015$ .

  $\boxed{\begin{array}{ccc}\begin{array}{l}\begin{array}{l}\text{Year} \hfill \end{array}\\ \begin{array}{r}\hfill 1988\\ \hfill 1989\\ \hfill 1990\\ \hfill 1991\\ \hfill 1992\\ \hfill 1993\\ \hfill 1994\\ \hfill 1995\\ \hfill 1996\\ \hfill 1997\\ \hfill 1998\\ \hfill 1999\\ \hfill 2000\\ \hfill 2001\\ \hfill 2002\\ \hfill 2003\\ \hfill 2004\\ \hfill 2005\\ \hfill 2006\\ \hfill 2007\\ \hfill 2008\\ \hfill 2009\\ \hfill 2010\\ \hfill 2011\\ \hfill 2012\\ \hfill 2013\\ \hfill 2014\\ \hfill 2015\end{array}\end{array}& \begin{array}{l}\text{Inflation }\left(x\right)\hfill \\ 4.4\hfill \\ 4.6\hfill \\ 6.1\hfill \\ 3.1\hfill \\ 2.9\hfill \\ 2.7\hfill \\ 2.7\hfill \\ 2.5\hfill \\ 3.3\hfill \\ 1.7\hfill \\ 1.6\hfill \\ 2.7\hfill \\ 3.4\hfill \\ 1.6\hfill \\ 2.4\hfill \\ 1.9\hfill \\ 3.3\hfill \\ 3.4\hfill \\ 2.5\hfill \\ 4.1\hfill \\ 0.1\hfill \\ 2.7\hfill \\ 1.5\hfill \\ 3\hfill \\ 1.7\hfill \\ 1.5\hfill \\ 0.8\hfill \\ 0.8\hfill \end{array}& \begin{array}{l}\text{Unemployment }\left(y\right)\hfill \\ 5.5\hfill \\ 5.3\hfill \\ 5.6\hfill \\ 6.8\hfill \\ 7.5\hfill \\ 6.9\hfill \\ 6.1\hfill \\ 5.6\hfill \\ 5.4\hfill \\ 4.9\hfill \\ 4.5\hfill \\ 4.2\hfill \\ 4\hfill \\ 4.7\hfill \\ 5.8\hfill \\ 6\hfill \\ 5.5\hfill \\ 5.1\hfill \\ 4.6\hfill \\ 4.6\hfill \\ 5.8\hfill \\ 9.3\hfill \\ 9.6\hfill \\ 8.9\hfill \\ 8.1\hfill \\ 7.4\hfill \\ 6.2\hfill \\ 5.3\hfill \end{array}\end{array}}$

Concept Used:

The correlation coefficient is given as,

  $r=\frac{1}{n-1}{\displaystyle \sum \left(\frac{x-\overline{x}}{s_x}\right)}\left(\frac{y-\overline{y}}{s_y}\right)$

The coefficient of variation is denoted by $r^2$ .

Where,

  $\begin{array}{l}n\text{is the number of observations}\\ \overline{x}\text{ is the}\text{average of }x\\ \overline{y}\text{is the}\text{average of }y\\ s_x\text{is the}\text{standard deviation of }x\\ s_y\text{is the}\text{standard deviation of }y\end{array}$

Calculation:

Here,

  $x=\mathrm{in}\text{flation and }y=\text{unemployment}$

  $\begin{array}{l}n=28\\ \overline{x}=2.60714\\ \overline{y}=6.04286\\ s_x=1.27452\\ s_y=1.51154\end{array}$

  $\boxed{\begin{array}{ccc}\begin{array}{l}\begin{array}{l}\begin{array}{l}\text{Year}\\ \end{array}\hfill \end{array}\\ \begin{array}{r}\hfill 1988\\ \hfill 1989\\ \hfill 1990\\ \hfill 1991\\ \hfill 1992\\ \hfill 1993\\ \hfill 1994\\ \hfill 1995\\ \hfill 1996\\ \hfill 1997\\ \hfill 1998\\ \hfill 1999\\ \hfill 2000\\ \hfill 2001\\ \hfill 2002\\ \hfill 2003\\ \hfill 2004\\ \hfill 2005\\ \hfill 2006\\ \hfill 2007\\ \hfill 2008\\ \hfill 2009\\ \hfill 2010\\ \hfill 2011\\ \hfill 2012\\ \hfill 2013\\ \hfill 2014\\ \hfill 2015\end{array}\end{array}& \begin{array}{l}\begin{array}{l}\text{Inflation }\left(x\right)\\ \end{array}\hfill \\ 4.4\hfill \\ 4.6\hfill \\ 6.1\hfill \\ 3.1\hfill \\ 2.9\hfill \\ 2.7\hfill \\ 2.7\hfill \\ 2.5\hfill \\ 3.3\hfill \\ 1.7\hfill \\ 1.6\hfill \\ 2.7\hfill \\ 3.4\hfill \\ 1.6\hfill \\ 2.4\hfill \\ 1.9\hfill \\ 3.3\hfill \\ 3.4\hfill \\ 2.5\hfill \\ 4.1\hfill \\ 0.1\hfill \\ 2.7\hfill \\ 1.5\hfill \\ 3\hfill \\ 1.7\hfill \\ 1.5\hfill \\ 0.8\hfill \\ 0.8\hfill \end{array}& \begin{array}{l}\begin{array}{l}\text{Unemployment }\left(y\right)\\ \end{array}\hfill \\ 5.5\hfill \\ 5.3\hfill \\ 5.6\hfill \\ 6.8\hfill \\ 7.5\hfill \\ 6.9\hfill \\ 6.1\hfill \\ 5.6\hfill \\ 5.4\hfill \\ 4.9\hfill \\ 4.5\hfill \\ 4.2\hfill \\ 4\hfill \\ 4.7\hfill \\ 5.8\hfill \\ 6\hfill \\ 5.5\hfill \\ 5.1\hfill \\ 4.6\hfill \\ 4.6\hfill \\ 5.8\hfill \\ 9.3\hfill \\ 9.6\hfill \\ 8.9\hfill \\ 8.1\hfill \\ 7.4\hfill \\ 6.2\hfill \\ 5.3\hfill \end{array}\end{array}\begin{array}{ccc}\begin{array}{l}\frac{x-\overline{x}}{s_x}\hfill \\ 1.406694\hfill \\ 1.563616\hfill \\ 2.74053\hfill \\ 0.386702\hfill \\ 0.229781\hfill \\ 0.072859\hfill \\ 0.072859\hfill \\ -0.08406\hfill \\ 0.543624\hfill \\ -0.71175\hfill \\ -0.79021\hfill \\ 0.072859\hfill \\ 0.622085\hfill \\ -0.79021\hfill \\ -0.16252\hfill \\ -0.55483\hfill \\ 0.543624\hfill \\ 0.622085\hfill \\ -0.08406\hfill \\ 1.171312\hfill \\ -1.96712\hfill \\ 0.072859\hfill \\ -0.86867\hfill \\ 0.308242\hfill \\ -0.71175\hfill \\ -0.86867\hfill \\ -1.4179\hfill \\ -1.4179\hfill \end{array}& \begin{array}{l}\frac{y-\overline{y}}{s_y}\hfill \\ -0.35914\hfill \\ -0.49146\hfill \\ -0.29299\hfill \\ 0.500906\hfill \\ 0.96401\hfill \\ 0.567064\hfill \\ 0.037803\hfill \\ -0.29299\hfill \\ -0.4253\hfill \\ -0.75609\hfill \\ -1.02072\hfill \\ -1.21919\hfill \\ -1.35151\hfill \\ -0.88841\hfill \\ -0.16067\hfill \\ -0.02836\hfill \\ -0.35914\hfill \\ -0.62377\hfill \\ -0.95456\hfill \\ -0.95456\hfill \\ -0.16067\hfill \\ 2.154849\hfill \\ 2.353322\hfill \\ 1.890218\hfill \\ 1.360956\hfill \\ 0.897853\hfill \\ 0.10396\hfill \\ -0.49146\hfill \end{array}& \begin{array}{l}\left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)\hfill \\ -0.50521\hfill \\ -0.76845\hfill \\ -0.80294\hfill \\ 0.193702\hfill \\ 0.221511\hfill \\ 0.041316\hfill \\ 0.002754\hfill \\ 0.024629\hfill \\ -0.2312\hfill \\ 0.538147\hfill \\ 0.806585\hfill \\ -0.08883\hfill \\ -0.84075\hfill \\ 0.702028\hfill \\ 0.026113\hfill \\ 0.015732\hfill \\ -0.19524\hfill \\ -0.38804\hfill \\ 0.080243\hfill \\ -1.11809\hfill \\ 0.316059\hfill \\ 0.157\hfill \\ -2.04427\hfill \\ 0.582644\hfill \\ -0.96866\hfill \\ -0.77994\hfill \\ -0.14741\hfill \\ 0.696839\hfill \end{array}\end{array}}$

  $\begin{array}{l}r=\frac{1}{n-1}{\displaystyle \sum \left(\frac{x-\overline{x}}{s_x}\right)}\left(\frac{y-\overline{y}}{s_y}\right)\\ =\frac{-4.47372}{28-1}\\ =-0.16569\\ \\ r^2={(}^{-}\\ =0.027454\end{array}$

Calculation:

In regression, the coefficient of determination explains the proportion of variation in the dependent variable.

It was calculated that the coefficient of determination

  $r^2=0.027454$

Therefore the proportion of the variation in unemployment is explained by inflation is $0.027454$ .