Chapter 4.2, Problem 25P

Expand Your Knowledge: Logarithmic Transformations, Power Law Model Let x = boiler steam pressure in $\text{100 Lb/in}{\text{.}}^{\text{2}}$, and let y = critical sheer strength of boiler plate steel joints in ${\text{tons/in}}^{\text{2}}$. We have the following data for a series of factory boilers:

x	4	5	6	8	10
y	3.4	4.2	6.3	10.9	13.3

(a) Make the logarithmic transformations $x\text{'}\text{ = log }x\text{ and }y\text{'}\text{= log }y$. Then make a scatter plot of the $(x\text{'},y\text{'})$ values. Does a linear equation seem to be a good fit to this plot?

(b) Use the $(x\text{'},y\text{'})$ data points and a calculator with regression keys to find the least-squares equation $y\text{'}=a+bx\text{'}.$ What is the sample correlation coefficient?

(c) Use the results of part (b) to find estimates for $\alpha \text{ and }\beta$ in the power law $y=\alpha x^{\beta }$. Write the power equation for the relationship between steam pressure and sheer strength of boiler plate steel. Hint: See Problem 24.