Mathematics — Applications And Interp.

Mathematics — Applications and Interp.

Problem

## Explorations 63. (a) Compute the power regression model for the following data.

$x$	4	6.5	8.5	10
$y$	2816	31,908	122,019	275,000

(b) Predict the

y

-value associated with

x=7.1

using the power regression model. (c) Re-express the data in terms of their natural logarithms and make a scatter plot of

(\ln x, \ln y)

. (d) Compute the linear regression model

(\ln y)=a(\ln x)+b

for

(\ln x, \ln y)

. (e) Confirm that

y=e^{b} \cdot x^{a}

is the power regression model found in (a).

Intuitive explanation

Power regression models relationships of the form $y = Cx^a$ , where the exponent $a$ controls how steeply $y$ grows with $x$ . The key insight is that taking natural logarithms of both sides transforms this into a linear relationship: $\ln y = a \ln x + \ln C$ . By fitting a line to the $(\ln x, \ln y)$ data, you recover the exponent and coefficient, then convert back to get your power model.

Intuitive explanation

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