Chapter 1.6, Problem 7E

a.

To determine

To graph: the data on the scatter plot.

Answer to Problem 7E

  

Explanation of Solution

Given information:

The table of values given is as follows:

Year (x)	Personal income in dollar (y)
1995	23255
1996	24270
1997	25412
1998	26893
1999	27880
2000	29760
2001	30413
2002	30832
2010	?

Calculation:

The graph for the given data is shown below.

  

b.

To determine

To find: best fit line using two ordered pairs.

Answer to Problem 7E

  $y=1084x-2139336$

Explanation of Solution

Given information:

The table of values given is as follows:

Year (x)	Personal income in dollar (y)
1995	23255
1996	24270
1997	25412
1998	26893
1999	27880
2000	29760
2001	30413
2002	30832
2010	?

Calculation:

Let’s select any two ordered pair like $\left(1997,25412\right)$ and $\left(2002,30832\right)$ . Plug in the values in the slope equation and solve m as follows:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{30832-25412}{2002-1997}\\ =\frac{5420}{5}\\ =1084\end{array}$

Now, plug the value of m and any one point coordinate into the slope intercept equation and evaluate b as follows:

  $\begin{array}{c}y=mx+b\\ 25412=\left(1084\right)\left(1997\right)+b\\ 25412=2164748+b\\ b=25412-2164748\\ =-2139336\end{array}$

Now, plug the values of m and b in the slope intercept formula as follows:

  $\begin{array}{l}y=mx+b\\ y=1084x-2139336\end{array}$

Thus, the line equation is $y=1084x-2139336$ .

c.

To determine

To find: the equation for regression line and correlation value using the graphing calculator.

Answer to Problem 7E

  $\begin{array}{c} y=1164.1071428571x-2299128.75\\ r=0.9921999084378\end{array}$

Explanation of Solution

Given information:

The table of values given is as follows:

Year (x)	Personal income in dollar (y)
1995	23255
1996	24270
1997	25412
1998	26893
1999	27880
2000	29760
2001	30413
2002	30832
2010	?

Calculation:

The graph for the above equation is as follows:

  

Using the graphing calculator, the regression line equation is as follows:

  $y=1164.1071428571x-2299128.75$

Using the graphing calculator, the correlation coefficient comes out to be, $r=0.9921999084378$

d.

To determine

To find: the missing value if regression line equation has strong or moderate relationship, and explains whether the prediction is reliable.

Answer to Problem 7E

  $\approx 40726$

Prediction is reliable.

Explanation of Solution

Given information:

The table of values given is as follows:

Year (x)	Personal income in dollar (y)
1995	23255
1996	24270
1997	25412
1998	26893
1999	27880
2000	29760
2001	30413
2002	30832
2010	?

  $\begin{array}{c} y=1164.1071428571x-2299128.75\\ r=0.9921999084378\end{array}$

Calculation:

The equation of regression line shows strong relationship because $0.75\le r\le 1$ .

Plugging $x=2010$ into the regression formula and evaluate y as follows:

  $\begin{array}{c} y=1164.1071428571x-2299128.75\\ y=1164.1071428571\left(2010\right)-2299128.75\\ =\text{2339855}\text{.357142771}-2299128.75\\ =40726.60714277066\\ \approx 4076\end{array}$

The regression line represents that the rate of increase in personal income per year is slower than that of inflation. On average, money doubles every seven years. As minimum wage does not increase quickly as compared to inflation and only a small fraction of the population has the large amount of money. So, this data seems reliable and the predicted value is also reliable.