Question

The swinging pendulum Refer to Exercise 33. We took the logarithm (base $10$) of the values for both length and period. Here is some computer output from a linear regression analysis of the transformed data.

a. Based on the output, explain why it would be reasonable to use a power model to describe the relationship between the length and period of a pendulum.

b. Give the equation of the least-squares regression line. Be sure to define any variables you use.

c. Use the model from part (b) to predict the period of a pendulum with a length of $80$cm.

Short answer

a). There is a linear relationship between the scatter plots of log (length) vs log (period), and the residual plot shows no identifiable patterns.

b). The equation of the least-squares regression line is$\log \left(\text{period}\right)=-0.73675+0.51701\log \left(\text{length}\right)$.

c). The expected period is $1.76685$seconds.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

Part (a) Step 2: Explanation

$\log \left(y\right)\text{hat}=0.73675+0.51701\log \left(x\right)$

The fact that there is a log of $x$and $y$in the equation indicates that it is a power function. The scatter plots of log (length) vs log (time) demonstrate a linear connection, while the residual plot exhibits no identifiable patterns.

Part (b) Step 1: Given Information

Given data:

Part (b) Step 2: Explanation

The generic regression equation is built using the data.

$\log \left(\text{period}\right)=\alpha +\beta \log \left(\text{length}\right)$

To find the $\alpha$(constant) in the row "constant" and column "Coef" of the computer output.

$\alpha =-0.73675$

To find the $\beta$(constant) in the row " log (length) " and column "Coef" of the computer output.

$\beta =0.51701$

Substituting the value of $\alpha$and $\beta$in the equation

$\log \left(\text{period}\right)=\alpha +\beta \log \left(\text{length}\right)$
localid="1654260405887" $\log \left(\text{period}\right)=-0.73675+0.51701\log \left(\text{length}\right)$

Part (c) Step 1: Given Information

Given data:

Part (c) Step 2: Explanation

The equation of the least-squares regression line is

$\log \left(\text{period}\right)=-0.73675+0.51701\log \left(\text{length}\right)$

Calculate by multiplying the length by $80$.

$\log \left(\text{period}\right)=-0.73675+0.51701\log \left(80\right)$

$\log \left(\text{period}\right)=0.2472$

Using the exponential with a value of $10$

$\text{period}={10}^{\log \left(\text{period}\right)}$

$={10}^{0.2472}$

$=1.76685$