Chapter ISG, Problem 4.6.2.3P

a.

To determine

To graph: The scatter plot for the given data and the line of best fit.

Answer to Problem 4.6.2.3P

The equation of the line of best fit is $y=\frac{1}{2}x+16$ and the correlation coefficient is $0.992$ (approx.)

Explanation of Solution

Given: The given data is-

Number of minutes past 7:00a.m ( x )	10	18	12	0	4
Drive time in minutes ( y )	21	25	21	16	18

Graph: The graph for the scatter plot and line of best fit are as follows-

  

Interpretation: The line of best fit passes through points $(10,21)$ and $(18,25)$.

Thus, slope of the line is given by-

  $m=\frac{25-21}{18-10}$

  $=\frac{1}{2}$

Hence, the equation of a line passing through $(0,16)$ and having slope $\frac{1}{2}$ is given by-

  $y-16=\frac{1}{2}(x-0)$

  $\Rightarrow y=\frac{1}{2}x+16$

The correlation coefficient can be found using the formula-

  $r=\frac{n\left({\displaystyle \sum xy}\right)-\left({\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{\sqrt{\left[n{\displaystyle \sum x^2-{\left({\displaystyle \sum x}\right)}^2}\right]\left[n{\displaystyle \sum y^2-{\left({\displaystyle \sum y}\right)}^2}\right]}}$ , where n is the sample size.

The values of the different variables in the formula can be procured from the following table-

	x	y	xy	$x^2$	$y^2$
	10	21	210	100	441
	18	25	450	324	625
	12	21	252	144	441
	0	16	0	0	256
	4	18	72	16	324
${\displaystyle \sum }$	44	101	984	584	2087

Hence, the correlation coefficient, taking n = 5, is given by-

  $r=\frac{5\times 984-44\times 101}{\sqrt{\left[5\times 584-{(44)}^2\right]\left[5\times 2087-{(101)}^2\right]}}$

  $=\frac{4920-4444}{\sqrt{\left[2920-1936\right]\left[10435-10201\right]}}$

  $=\frac{476}{\sqrt{984\times 234}}$

  $=0.992$ (approx.)

b.

To determine

The reason behind the longer duration of drive time.

Explanation of Solution

Given: The given data is-

Number of minutes past 7:00a.m ( x )	10	18	12	0	4
Drive time in minutes ( y )	21	25	21	16	18

The table shows that it takes a longer duration to drive to work if left later than usual.

Since during the weekdays everyone needs to hurry to work, the delay in leaving for work increases the chances of traffic on the roads. More traffic results in a longer drive time to work.