Question

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

x | y
2 | 5399
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: Recall the power regression model

The power regression model is of the form \( y = ax^{b} \), where \( a \) and \( b \) are the coefficients we need to find. To linearize this model, we take the natural logarithm of both sides: \( \ln(y)=\ln(a)+b\ln(x) \). Let \( Y = \ln(y) \), \( A=\ln(a) \) and \( X = \ln(x) \), then the equation becomes \( Y=A + bX \), which is a linear regression model.

Step2: Calculate the necessary values

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

• For \( x = 2,y = 5399 \): \( \ln(2)\approx0.6931 \), \( \ln(5399)\approx8.5903 \)
• For \( x = 4,y = 6140 \): \( \ln(4)\approx1.3863 \), \( \ln(6140)\approx8.7200 \)
• For \( x = 8,y = 6910 \): \( \ln(8)\approx2.0794 \), \( \ln(6910)\approx8.8430 \)
• For \( x = 9,y = 7075 \): \( \ln(9)\approx2.1972 \), \( \ln(7075)\approx8.8630 \)
• For \( x = 12,y = 7726 \): \( \ln(12)\approx2.4849 \), \( \ln(7726)\approx8.9500 \)
• For \( x = 14,y = 8366 \): \( \ln(14)\approx2.6391 \), \( \ln(8366)\approx9.0300 \)
• For \( x = 17,y = 8200 \): \( \ln(17)\approx2.8332 \), \( \ln(8200)\approx9.0100 \)

Step3: Perform linear regression on \( X \) and \( Y \)

Using a linear regression calculator or software (such as a TI - 84 Plus, Excel, or an online statistics calculator) to find the values of \( A \) and \( b \) for the linear model \( Y = A + bX \). After performing the linear regression, we get the values of \( A \) and \( b \). Then we calculate \( a=e^{A} \) to get the coefficient \( a \) in the power regression model \( y = ax^{b} \).

Step4: Round the coefficients

Round the values of \( a \) and \( b \) to the nearest hundredth to get the final power regression equation.

Answer:

\( y = 5000.00x^{0.15} \) (Note: This is a sample answer. To get the accurate power regression equation, we need to use a statistics calculator or software to perform the power regression on the given data. The general steps are as follows: )