Chapter 10, Problem 16E

(a)

To determine

To Explain: the reason for the linear model is not appropriate for using the length in the inch for the pendulum to expect the number of swings in 20 sec.

Explanation of Solution

Given:

Length (in)	Number of Swings
6.5	22
9	20
11.5	17
14.5	16
18	14
21	13
24	13
27	12
30	11
37.5	10

Graph:

  

Calculation:

The regression equation is

  $\widehat{y}$ = -0.3727x + 22.217

  

By seeing the shape of the residual plot, it can be estimated that the there curve in the figure, therefore the linear model is not suitable.

(b)

To determine

To conduct: the re-express the data list to be straighten for the scatterplot.

Explanation of Solution

Graph:

  

Residual plot

  

  $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=\widehat{y}=0.0018x+0.0367$

It may say from the above plot that this is a suitable model to fit, as the residuals are scattered. It may say from this that the reciprocal number of swings linearizes the plot.

(c)

To determine

To construct: an appropriate model.

Explanation of Solution

This is the appropriate model for the given data

  $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=\widehat{y}=0.0018x+0.0367$

(d)

To determine

To calculate: the number of swings for the pendulum of 4-inch string.

Answer to Problem 16E

22.8624

Explanation of Solution

Given:

  $x=4inch$

Formula used:

  $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=0.0018x+0.0367$

Calculation:

  $\begin{array}{c}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=0.0018x+0.0367\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=0.0018\left(4\right)+0.0367\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=0.04374\\ \text{Number}\text{of}\text{swing}s=\frac{1}{0.04374}\\ =22.8624\end{array}$

(e)

To determine

To calculate: the number of swings for a 48-inch string pendulum.

Answer to Problem 16E

8.2522

Explanation of Solution

Given:

  $x=48inch$

Formula used:

  $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=0.0018x+0.0367$

Calculation:

  $\begin{array}{c}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=0.0018x+0.0367\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=0.0018\left(48\right)+0.0367\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\text{Number}\text{of}\text{swings}$}\right.=0.12118\\ \text{Number}\text{of}\text{swing}s=\frac{1}{0.12118}\\ =8.2522\end{array}$

(f)

To determine

To explain: the confidence does place in this prediction.

Explanation of Solution

Because $R^2=0.981$ and s=0.002531, this is too much small, thus it could say that this is fairly confident.