Chapter 4.5, Problem 11PPS

To determine

The independent and dependent variable for the given set of data. Make a scatter plot to check is there any relationship between the given data and make aline of fit for the given data.

Answer to Problem 11PPS

The interval of eruptions is the dependent variable because it depends upon the duration of eruptions, which means that duration of eruptions is the independent variable.

Explanation of Solution

Given: The time to the next eruption of Old Faithful can be predicted by using the duration of current eruption as shown.

Duration of eruptions (minutes)	Interval of eruptions (minutes)
1.5	48
2	55
2.5	70
3	72
3.5	74
4	82
4.5	93
5	100

Graph: The graph for the scatter-plot and line of fit for the given bivariate datais shown as,

  

Conclusion: As the duration of eruptions increases, the interval of eruptions also increases. Hence, there is a positive correlation between the duration of eruptions and interval of eruptions.

To determine

If $\left(x\right)$ represents the duration of previous of interval and $\left(y\right)$ represents the time between the interval then write a equation for the line of fit in slope-intercept form. Also predict the interval after a 7.5 minutes eruption.

Answer to Problem 11PPS

The slope-intercept form for the line of fit is $y=15x+25$ . And, the interval after 7.5minutes eruptions will be 137.5minutes.

Explanation of Solution

Given: The time to the next eruption of Old Faithful can be predicted by using the duration of current eruption as shown.

Duration of eruptions (minutes)	Interval of eruptions (minutes)
1.5	48
2	55
2.5	70
3	72
3.5	74
4	82
4.5	93
5	100

Calculation: From the given data, consider any two points on line of fit say $\left(2,55\right)\text{ and }\left(5,100\right)$ . Therefore, the slope $\left(m\right)$ of line of fit is evaluated as,

  $\begin{array}{c}m=\frac{100-55}{5-2}\\ =15\end{array}$

Now, the equation for the line of fit having slope $\left(m=15\right)$ and passing through the point $\left(5,100\right)$ is evaluated as,

  $\begin{array}{c}y-100=15\left(x-5\right)\mathrm{......}\left(\text{Point-slope form}\right)\\ y-100=15x-75\\ y=15x+\mathrm{25......}\left(\text{Slope-intercept form}\right)\mathrm{....}\left(1\right)\end{array}$

The interval after 7.5minutes eruption can be evaluated by plugging $\left(x=7.5\right)$ in slope-intercept form (1).

  $\begin{array}{c}y=15\left(7.5\right)+25\\ =137.5\text{ minutes}\end{array}$

To determine

The critical judgement using the equation for line of fit to predict the duration of next eruption. Is this equation is a useful model for this task.

Answer to Problem 11PPS

No, the equation is not useful model for the prediction of the duration of next eruption.

Explanation of Solution

Given: The time to the next eruption of Old Faithful can be predicted by using the duration of current eruption as shown.

Duration of eruptions (minutes)	Interval of eruptions (minutes)
1.5	48
2	55
2.5	70
3	72
3.5	74
4	82
4.5	93
5	100

Calculation: As, the duration of eruption is not dependent on the interval of eruption because the eruption is a physical phenomenon which is not controllable. Therefore, it is not possible to predict the duration of next eruption. Hence, the model for the line of fit is not useful for this task.