Question

interest rates the table showing the u.s. federal reserves prime interest rate on january 1 of various years.

year	2006	2007	2008	2009	2010
prime rate (%)	7.25	8.25	7.25	3.25	3.25

source: federal reserve board
a. write the equation in slope - intercept form for the best - fit line for the data. let x represent the number of years since 2006. round to the nearest hu
y =
x+

b. find the correlation coefficient. round to the nearest thousandth, if necessary.
r =

Explanation:

Step1: Organize data points

Let \(x = 0\) for year 2006, \(x = 1\) for year 2007, \(x = 2\) for year 2008, \(x=3\) for year 2009, \(x = 4\) for year 2010. The data - points are \((0,7.25)\), \((1,8.25)\), \((2,7.25)\), \((3,3.25)\), \((4,3.25)\).

Step2: Calculate means of \(x\) and \(y\)

\(\bar{x}=\frac{0 + 1+2+3+4}{5}=2\)
\(\bar{y}=\frac{7.25 + 8.25+7.25+3.25+3.25}{5}=\frac{29.25}{5}=5.85\)

Step3: Calculate slope \(m\)

\[

$$\begin{align*} m&=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\\ \sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(0 - 2)(7.25-5.85)+(1 - 2)(8.25 - 5.85)+(2 - 2)(7.25 - 5.85)+(3 - 2)(3.25 - 5.85)+(4 - 2)(3.25 - 5.85)\\ &=(- 2)\times1.4+(-1)\times2.4+0\times1.4+1\times(-2.6)+2\times(-2.6)\\ &=-2.8-2.4 + 0-2.6-5.2\\ &=-13\\ \sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}&=(0 - 2)^{2}+(1 - 2)^{2}+(2 - 2)^{2}+(3 - 2)^{2}+(4 - 2)^{2}\\ &=4 + 1+0+1+4\\ &=10 \end{align*}$$

\]
\(m=\frac{-13}{10}=-1.3\)

Step4: Calculate \(y\) - intercept \(b\)

Using the point - slope form \(y - y_{1}=m(x - x_{1})\) with the point \((\bar{x},\bar{y})=(2,5.85)\) and \(m=-1.3\), we have \(y-5.85=-1.3(x - 2)\)
\(y=-1.3x+5.85 + 2.6=-1.3x+8.45\)

Step5: Calculate correlation coefficient \(r\)

\[

$$\begin{align*} r&=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\\ \sum_{i = 1}^{5}(y_{i}-\bar{y})^{2}&=(7.25-5.85)^{2}+(8.25 - 5.85)^{2}+(7.25 - 5.85)^{2}+(3.25 - 5.85)^{2}+(3.25 - 5.85)^{2}\\ &=1.4^{2}+2.4^{2}+1.4^{2}+(-2.6)^{2}+(-2.6)^{2}\\ &=1.96+5.76+1.96 + 6.76+6.76\\ &=23.2 \end{align*}$$

\]
\(r=\frac{-13}{\sqrt{10\times23.2}}=\frac{-13}{\sqrt{232}}\approx\frac{-13}{15.232}\approx - 0.854\)

Answer:

a. \(y=-1.3x + 8.45\)
b. \(r\approx - 0.854\)