Question

The following table gives the average monthly exchange rate between the U.S. dollar and the euro for 2009. It shows that 1 euro was equivalent to 1.324 U.S. dollars in January 2009. Develop a trend line that could be used to predict the exchange rate for 2010. Use this model to predict the exchange rate for January 2010 and February 2010.

MONTH EXCHANGE RATE

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} x & y \\ \hline 1 & 1.289 \\ \hdashline 2 & 1.324\\ \hdashline 3 & 1.321 \\ \hdashline 4 & 1.317\\ \hdashline 5 & 1.280 \\ \hdashline 6 & 1.254\\ \hdashline 7 & 1.230 \\ \hdashline 8 & 1.240\\ \hdashline 9 & 1.287 \\ \hdashline 10 & 1.298\\ \hdashline 11& 1.283 \\ \hdashline 12 & 1.311\\ \end{array}$$

$$\bar{x}=\dfrac{\sum_ix_i}{n}=\dfrac{78}{12}=6.5$$

$$\bar{y}=\dfrac{\sum_iy_i}{n}=\dfrac{15.434}{12}=1.286167$$

$$SS_{xx}=\sum_ix_i^2-\dfrac{1}{n}(\sum_ix_1)^2=650-\dfrac{1}{12}(78)^2$$

$$=143$$

$$SS_{yy}=\sum_iy_i^2-\dfrac{1}{n}(\sum_iy_1)^2=19.861426-\dfrac{1}{12}(15.434)^2$$

$$=0.010730$$

$$SS_{xy}=\sum_ix_iy_i-\dfrac{1}{n}(\sum_ix_1)(\sum_iy_i)$$

$$=100.03-\dfrac{1}{12}(78)(15.434)=−0.291000$$

$$slope=m=\dfrac{SS_{xy}}{SS_{xx}}=\dfrac{−0.291000}{143}=-0.002$$

$$b=\bar{y}-m\cdot\bar{x}=1.286167-(-0.002)(6.5)$$

$$=1.2994$$

The equation of the trend line is

$$y=-0.002x+1.2994$$

January 2010

$$y=-0.002(13)+1.2994$$

$$y=1.273$$

February 2010

$$y=-0.002(14)+1.2994$$

$$y=1.271$$