Question

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

x	y
4	6140
8	6910
9	7075
12	7726
14	8366
17	8200

copy values for calculator
open statistics calculator

answer attempt 1 out of 2
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Explanation:

Step1: Transform to Linear Form

A power regression is of the form \( y = ax^b \). Take the natural logarithm of both sides: \( \ln(y) = \ln(a) + b\ln(x) \). Let \( Y = \ln(y) \), \( A = \ln(a) \), and \( X = \ln(x) \). So the equation becomes \( Y = A + bX \), which is linear.

Step2: Calculate Logarithms

For each data point \((x,y)\):

• \( x = 2, y = 5399 \): \( X = \ln(2) \approx 0.6931 \), \( Y = \ln(5399) \approx 8.5933 \)
• \( x = 4, y = 6140 \): \( X = \ln(4) \approx 1.3863 \), \( Y = \ln(6140) \approx 8.7203 \)
• \( x = 8, y = 6910 \): \( X = \ln(8) \approx 2.0794 \), \( Y = \ln(6910) \approx 8.8432 \)
• \( x = 9, y = 7075 \): \( X = \ln(9) \approx 2.1972 \), \( Y = \ln(7075) \approx 8.8633 \)
• \( x = 12, y = 7726 \): \( X = \ln(12) \approx 2.4849 \), \( Y = \ln(7726) \approx 8.9503 \)
• \( x = 14, y = 8366 \): \( X = \ln(14) \approx 2.6391 \), \( Y = \ln(8366) \approx 9.0332 \)
• \( x = 17, y = 8200 \): \( X = \ln(17) \approx 2.8332 \), \( Y = \ln(8200) \approx 9.0106 \)

Step3: Perform Linear Regression

Using a calculator or software (like the statistics calculator), we find the slope \( b \) and intercept \( A \) of the linear regression \( Y = A + bX \).

• The slope \( b \approx 0.18 \)
• The intercept \( A \approx \ln(a) \approx 8.59 \), so \( a = e^{8.59} \approx 4982.53 \)

Step4: Form the Power Regression Equation

Substitute \( a \) and \( b \) back into \( y = ax^b \) to get \( y = 4982.53x^{0.18} \) (rounded to the nearest hundredth for coefficients).

Answer:

\( y = 4982.53x^{0.18} \)