Chapter 4.5, Problem 31E

a.

To determine

To calculate: The equation of line of best fit using graphing calculator, then to further identify and interpret the correlation coefficient.

Answer to Problem 31E

Equation of y $\text{line of best fit is y= 513}\text{.5x - 297}\text{.7}$ .

  $r$ is $0.99$ .

Explanation of Solution

Given Information: Data values of year, $x$ and text messages (in billions), $y$ are given.

Calculation:

Entering the data and using the linear regression feature in the graphing calculator,

  $\begin{array}{l}\text{LinReg}\\ \text{ y=ax+b}\\ \text{ a= 513}\text{.5}\\ \text{ b= -297}\text{.7}\\ {\text{ r}}^2\text{= 0}\text{.9860248400752258}\\ \text{ r= 0}\text{.9929878348072678}\end{array}$

As $y=ax+b$ ,

Thus, the equation of $\text{line of best fit is y= 513}\text{.5x - 297}\text{.7}$ .

Interpretation:

Graphing calculator results display that $\text{ r= 0}\text{.99}$ .

As $\text{r}$ is near about $\text{1}$ , this implies that there is strong positive correlation and points lie around a positive slope. In this context, as the time progressed and years passed by, number of messages increased, $\therefore$ such a strong positive correlation is seen.

Thus, $r$ is $0.99$ .

b.

To determine

The existence of a causal relationship.

Answer to Problem 31E

Yes, there is causal relationship.

Explanation of Solution

Given information: Data values of year, $x$ and text messages (in billions), $y$ are given.

Yes, this is a casual relationship. As the years passed by the use of mobile phones surged and it resulted in more and more messages being sent.

c.

To determine

To calculate: The residuals and then to make scatter plot of the residuals and interpret results.

Answer to Problem 31E

The equation $\text{y= 513}\text{.5x - 297}\text{.7}$ is a good fit.

Explanation of Solution

Given Information: Data values of year, $x$ and text messages, $y$ .Equation $\text{y= 513}\text{.5x - 297}\text{.7}$ .

Calculation:

Calculating the $y(from\_\mathrm{mod}el)$ and residuals and arranging the results in a table,

  $\begin{array}{llll}x\hfill & y\hfill & y(\mathrm{mod}el)\hfill & \mathrm{Re}sidual\hfill \\ 1\hfill & 241\hfill & 215.8\hfill & 25.2\hfill \\ 2\hfill & 601\hfill & 729.3\hfill & -128.3\hfill \\ 3\hfill & 1360\hfill & 1242.8\hfill & 117.2\hfill \\ 4\hfill & 1806\hfill & 1756.3\hfill & 49.7\hfill \\ 5\hfill & 2206\hfill & 2269.8\hfill & -63.8\hfill \end{array}$

Graph:

  

The points are evenly distributed about the horizontal axis, $\therefore$ the equation $\text{y= 513}\text{.5x - 297}\text{.7}$ is a good fit for the data.

Thus, the equation $\text{y= 513}\text{.5x - 297}\text{.7}$ is a good fit.

d.

To determine

To compare: Methods used in part $(a)$ and part $(c)$ .

Answer to Problem 31E

Method implemented in part $(a)$ is better.

Explanation of Solution

Given Information: Part $(a)$ and part $(c)$ are to be compared.

Method used in part $(a)$ is preferable because making use of a computerized graphing calculator leaves no scope for errors and correlation coefficient is a solid parameter to determine the relationship explicitly.