Boyle’s law Refers to Exercise 34. We took the logarithm (base $10$) of the values for both volume and pressure. 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 pressure and volume.

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 pressure in the syringe when the volume is $17$cubic centimeters.

Given data:

According to the equation, the scatter plot of log (volume) and log (pressure) is linear, and the residual figure shows no discernible patterns.

Given data:

The general regression equation is derived from the data.

$\log \left(\text{pressure}\right)=\alpha +\beta \log \left(\text{volume}\right)$

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

$\alpha =1.11116$

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

$\beta =-0.81344$

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

$\log \left(\text{pressure}\right)=\alpha +\beta \log \left(\text{volume}\right)$$\log \left(\text{pressure}\right)=1.11116-0.81344\log \left(\text{volume}\right)$

Given data:

The equation of the least-squares regression line is

$\log \left(pressure\right)=1.11116-0.81344\log \left(volume\right)$

Calculate by multiplying the volume amount by $17$:

$\log \left(\text{pressure}\right)=1.11116-0.81344\log \left(17\right)$

$\log \left(\text{pressure}\right)=0.1103$

Using the exponential with a value of $10$:

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

$={10}^{0.1103}$

$=1.28914$