Q10. - Algebra 1

Question

The table shows the relationship between years of experience and teacher salary.

Years Experience	1	5	10	15	20
Salary (thousands of dollars)	28	31	42	49	64

a. Write an equation for the best-fit line.

b. Find the correlation coefficient and explain what it tells us about the relationship between experience and salary.

Step-by-Step Solution

Answer

a. The equation for the best fit line is $y=1.88x+23.624$.

b. The value of correlation coefficient is $0.9844$.

The correlation coefficient tells about the relationship between experience and salary that the type of correlation coefficient is positive correlation and the fit line obtained between the experience and salary is almost accurate. There is positive correlation between the experience and salary that implies as the experience increases, the salary increases.

Part a. Step 1. Write the formula for the equation of the best-fit line.

Let $x$ be the independent variable and $y$ be the dependent variable.

The equation of the best-fit line is:

$y=ax+b$

Where $a=\frac{n{\displaystyle \sum xy-\sum x\sum y}}{n{\displaystyle \sum x^2-{\left(\sum x\right)}^2}}$, $b=\frac{{\displaystyle \sum y-a\sum x}}{n}$ and $n$ is the number of observations.

Part a. Step 2. Write an equation for the best-fit line.

Let $x$ represent the years experience and $y$ represent the salary.

The number of observations is 5. Therefore, the value of $n$ is 5.

Therefore, it can be noticed that:

$x$	$y$	$xy$	$x^2$
1	28	28	1
5	31	155	25
10	42	420	100
15	49	735	225
20	64	1280	400
$\sum x=51$	$\sum y=214$	$\sum xy=2618$	$\sum x^2=751$

Therefore, 

$$\begin{gathered} a=\frac{n{\displaystyle \sum xy-\sum x\sum y}}{n{\displaystyle \sum x^2-{\left(\sum x\right)}^2}} \\ =\frac{5\left(2618\right)-51\left(214\right)}{5\left(751\right)-{\left(51\right)}^2} \\ =\frac{13090-10914}{3755-2601} \\ =\frac{2176}{1154} \\ =1.88 \end{gathered}$$

 

$$\begin{gathered} b=\frac{{\displaystyle \sum y-a\sum x}}{n} \\ =\frac{214-1.88\left(51\right)}{5} \\ =\frac{118.12}{5} \\ =23.624 \end{gathered}$$

 

Substitute the values of and in the equation $y=ax+b$ to find the equation of the best-fit line.

$$\begin{gathered} y=ax+b \\ =\left(1.88\right)x+23.624 \\ =1.88x+23.624 \end{gathered}$$

 

Therefore, the equation for the best fit line is $y=1.88x+23.624$.

Part b. Step 1. Write the formula for the correlation coefficient.

The correlation coefficient ($r$) is given by:

$r=\frac{{\displaystyle \sum \left(\left(x-\overline{x}\right)\left(y-\overline{y}\right)\right)}}{\sqrt{{\displaystyle \sum {\left(x-\overline{x}\right)}^2\sum {\left(y-\overline{y}\right)}^2}}}$

Where $\overline{x}$ and $\overline{y}$ are the means of $x$ and $y$ respectively.

Part b. Step 2. Find the correlation coefficient.

The mean of $x$ ($\overline{x}$) is given by:

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x}}{n}\\ =\frac{51}{5}\\ =10.2\end{array}$

The mean of $y$ ($\overline{y}$) is given by:

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum y}}{n}\\ =\frac{214}{5}\\ =42.8\end{array}$

 

It can be noticed that:

$x$	$y$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$	$y-\overline{y}$	${\left(y-\overline{y}\right)}^2$	$\left(x-\overline{x}\right)\left(y-\overline{y}\right)$
1	28	$-9.2$	$84.64$	$-14.8$	$219.04$	$136.16$
5	31	$-5.2$	$27.04$	$-11.8$	$139.24$	$61.36$
10	42	$-0.2$	$0.04$	$-0.8$	$0.64$	$0.16$
15	49	$4.8$	$23.04$	$6.2$	$38.44$	$29.76$
20	64	$9.8$	$96.04$	$21.2$	$449.44$	$207.76$
$\sum x=51$	$\sum y=214$		$\sum {\left(x-\overline{x}\right)}^2=230.8$		$\sum {\left(y-\overline{y}\right)}^2=846.8$	$\sum \left(\left(x-\overline{x}\right)\left(y-\overline{y}\right)\right)=435.2$

Therefore,

$$\begin{gathered} r=\frac{{\displaystyle \sum \left(\left(x-\overline{x}\right)\left(y-\overline{y}\right)\right)}}{\sqrt{{\displaystyle \sum {\left(x-\overline{x}\right)}^2\sum {\left(y-\overline{y}\right)}^2}}} \\ =\frac{435.2}{\sqrt{230.8\times 846.8}} \\ =\frac{435.2}{\sqrt{195441.44}} \\ =\frac{435.2}{442.087593} \\ =0.9844 \end{gathered}$$

Therefore, the value of correlation coefficient is $0.9844$.

Part b. Step 3. Explain what the correlation coefficient about the relationship between experience and salary.

The value of correlation coefficient is approximately equal to 1. Therefore, the fit line obtained is almost accurate. As the value of correlation coefficient is positive, therefore the type of correlation obtained is positive correlation.

Therefore, the correlation coefficient tells about the relationship between experience and salary that the type of correlation coefficient is positive correlation and the fit line obtained between the experience and salary is almost accurate. There is positive correlation between the experience and salary that implies as the experience increases, the salary increases.