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

Explanation:

Step1: Recall Power Regression Formula

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

Step2: Calculate Logarithmic Values

First, we calculate \( \ln(x) \) and \( \ln(y) \) for each data point:

• For \( x = 2 \), \( y = 5399 \): \( X_1=\ln(2)\approx0.6931 \), \( Y_1=\ln(5399)\approx8.5933 \)
• For \( x = 4 \), \( y = 6140 \): \( X_2=\ln(4)\approx1.3863 \), \( Y_2=\ln(6140)\approx8.7213 \)
• For \( x = 8 \), \( y = 6910 \): \( X_3=\ln(8)\approx2.0794 \), \( Y_3=\ln(6910)\approx8.8495 \)
• For \( x = 9 \), \( y = 7075 \): \( X_4=\ln(9)\approx2.1972 \), \( Y_4=\ln(7075)\approx8.8673 \)
• For \( x = 12 \), \( y = 7726 \): \( X_5=\ln(12)\approx2.4849 \), \( Y_5=\ln(7726)\approx8.9503 \)
• For \( x = 14 \), \( y = 8366 \): \( X_6=\ln(14)\approx2.6391 \), \( Y_6=\ln(8366)\approx9.0333 \)
• For \( x = 17 \), \( y = 8200 \): \( X_7=\ln(17)\approx2.8332 \), \( Y_7=\ln(8200)\approx9.0150 \)

Step3: Perform Linear Regression on Transformed Data

Using a linear regression calculator (or software) on the data points \( (X_i, Y_i) \), we find the slope \( b \) and the intercept \( A \).

After performing the linear regression, we get:

• Slope \( b\approx0.16 \)
• Intercept \( A\approx\ln(a)\approx8.522 \)

Step4: Find \( a \)

Since \( A = \ln(a) \), we solve for \( a \) by exponentiating both sides: \( a = e^A \). Substituting \( A\approx8.522 \), we get \( a = e^{8.522}\approx5034.73 \).

Step5: Write the Power Regression Equation

Substituting \( a\approx5034.73 \) and \( b\approx0.16 \) into the power regression formula \( y = ax^{b} \), we get \( y = 5034.73x^{0.16} \).

Answer:

\( y = 5034.73x^{0.16} \)