Question

write a power regression equation for the following data, rounding all coefficients to the nearest hundredth.

x	y
5	191
6	168
7	146
10	114
14	85
18	72

copy values for calculator
open statistics calculator

Explanation:

Step1: Recall Power Regression Formula

A power regression equation has the form \( y = ax^b \), where \( a \) and \( b \) are constants. To find \( a \) and \( b \), we can take the natural logarithm of both sides: \( \ln(y)=\ln(a)+b\ln(x) \). This transforms the power relationship into a linear relationship of the form \( Y = A + bX \), where \( Y=\ln(y) \), \( A = \ln(a) \), and \( X=\ln(x) \).

Step2: Calculate \( \ln(x) \) and \( \ln(y) \) for Each Data Point

For each pair \((x,y)\) in the data:

• When \( x = 1 \), \( y = 742 \): \( X=\ln(1)=0 \), \( Y=\ln(742)\approx6.60 \)
• When \( x = 5 \), \( y = 191 \): \( X=\ln(5)\approx1.61 \), \( Y=\ln(191)\approx5.25 \)
• When \( x = 6 \), \( y = 168 \): \( X=\ln(6)\approx1.79 \), \( Y=\ln(168)\approx5.12 \)
• When \( x = 7 \), \( y = 146 \): \( X=\ln(7)\approx1.95 \), \( Y=\ln(146)\approx5.09 \)
• When \( x = 10 \), \( y = 114 \): \( X=\ln(10)\approx2.30 \), \( Y=\ln(114)\approx4.74 \)
• When \( x = 14 \), \( y = 85 \): \( X=\ln(14)\approx2.64 \), \( Y=\ln(85)\approx4.44 \)
• When \( x = 18 \), \( y = 72 \): \( X=\ln(18)\approx2.89 \), \( Y=\ln(72)\approx4.28 \)

Step3: Perform Linear Regression on \( (X,Y) \)

Using a linear regression calculator (or software) to find the slope \( b \) and the intercept \( A \) of the line \( Y = A + bX \):

• The slope \( b \approx - 0.49 \)
• The intercept \( A \approx \ln(a)\approx6.60 \) (since when \( X = 0 \), \( Y=\ln(a) \))

Step4: Find \( a \) from \( A \)

Since \( A=\ln(a) \), we solve for \( a \) by exponentiating both sides: \( a = e^A \). Substituting \( A\approx6.60 \), we get \( a = e^{6.60}\approx738.79 \)

Step5: Write the Power Regression Equation

Substituting \( a\approx738.79 \) and \( b\approx - 0.49 \) into the power regression formula \( y = ax^b \), we get \( y = 738.79x^{-0.49} \)

Answer:

\( y = 738.79x^{-0.49} \)