Chapter 4.5, Problem 31E

(a)

To determine

To find: The equation of the line of best fit and correlation coefficient

Answer to Problem 31E

The equation of the line of best fit is $y=482.3x-265.3$

The correlation coefficient is $0.98530$ , so it has a strong positive correlation

Explanation of Solution

Given:

Year, x	$1$	$2$	$3$	$4$	$5$
Text messages (billions), y	$241$	$601$	$1360$	$1806$	$2206$

Calculation:

Consider the table shows the numbers $y$ (in billions) of text messages sent each year in a five - year period, where $x=1$ represents the first year in the five - year period

To use the linear regression feature of the $TI-83$ calculator, let us first press $STAT$ the key, then $ENTER$ to go to the statistics menu

  

Now press the $1$ key and go to the edit menu and then write the data containing values of $x$ in $L1$ and data containing the value of $y$ in $L2$

  

Now to calculate the linear regression, that is $ax+b$ , first turn the diagnostics on, from the home screen , press $2nd$ key and then press $CATALOG$ . In catalog select “ Diagnostic on” and then press $ENTER$ twice. Finally press the $STAT$ key, go to right arrow in the $CALC$ menu, select $Lin\mathrm{Re}g(ax+b)$ and then press $ENTER$

In order to input the list to use while finding the linear regression, press $2nd$ key, then press $1$ to select $L1$ , then use comma $(,)$ o separate and finally press $2nd$ key and the $L2$ (by selecting) , press comma $(,)$

  $VARS$ ,right arrow, then $ENTER$ twice

  

Therefore, the equation of the line of best fit is $y=482.3x-265.3$

To make the graph using graphing calculator, press the $ZOOM$ key, select $"Zoomstat"$ and then press $ENTER$

The graph obtained is shown below

  

Therefore, the equation of the line of best fit is $y=482.3x-265.3$

The correlation coefficient is $0.98530$ , so it has a strong positive correlation

Conclusion:

The equation of the line of best fit is $y=482.3x-265.3$

The correlation coefficient is $0.98530$ , so it has a strong positive correlation

(b)

To determine

To find: The casual relationship

Answer to Problem 31E

There is no causal relationship between year and the number of text messages

Explanation of Solution

Given:

Year, x	$1$	$2$	$3$	$4$	$5$
Text messages (billions), y	$241$	$601$	$1360$	$1806$	$2206$

Calculation:

There may be no causal relationship between year and the number of text messages

Conclusion:

There may be no causal relationship between year and the number of text messages

(c)

To determine

To find: The residuals of the data set and to plot the graph

Answer to Problem 31E

There is no correlation

Explanation of Solution

Given:

Year, x	$1$	$2$	$3$	$4$	$5$
Text messages (billions), y	$241$	$601$	$1360$	$1806$	$2206$

Calculation:

Consider the equation and the data set

Find the residuals for the data set

Year, x	Text messages (billions), y	Y-value from model	residuals
$1$	$241$	$217$	$241-217=24$
$2$	$601$	$699.3$	$601-699.3=-98.3$
$3$	$1360$	$1181.6$	$1360-1181.6=178.4$
$4$	$1806$	$1663.9$	$1806-1663.9=142.1$
$5$	$2206$	$2146.2$	$2200-2146.2=53.8$

Now plot the graph for the data set with residuals

  

From the graph, there is no correlation

Conclusion:

There is no correlation

(d)

To determine

To find: whether the model is a good fit

Answer to Problem 31E

The graphical calculation using linear regression feature is most preferable as the accuracy is more and good model to use.

Explanation of Solution

Given:

Year, x	$1$	$2$	$3$	$4$	$5$
Text messages (billions), y	$241$	$601$	$1360$	$1806$	$2206$

Calculation:

Compare the two - graphical calculation and the residual methods

As the result is differed to each other, the graphical calculation using linear regression feature is most preferable as the accuracy is more and good model to use.

Conclusion:

The graphical calculation using linear regression feature is most preferable as the accuracy is more and good model to use.