.Fit a power equation model (Ŷ = a2*xb2) to the following data set. Then calculate correlation coefficient.
x 2.5 3.5 5 6 7.5 10 12.5 15 17.5 20
y 7 5.5 3.9 3.6 3.1 2.8 2.6 2.4 2.3 2.3
The power equation model is Ŷ = 0.0014x2.7. The correlation coefficient is 0.988.
To fit a power equation model (Ŷ = a2*xb2) to the given data set and calculate the correlation coefficient, you can follow these steps:
Step 1: Calculate the values of Ŷ using the power equation model for each corresponding value of x and a combination of a2 and b2.
Step 2: Calculate the correlation coefficient by finding the covariance and standard deviations of x and Ŷ.
Let's go through each step:
Step 1: Calculate Ŷ using the power equation model.
Given power equation model: Ŷ = a2 * xb2
We need to determine values for a2 and b2 that best fit the data set. To do this, we can use a mathematical technique called regression.
First, take the logarithm base 10 of both sides of the equation to linearize it:
log10(Ŷ) = log10(a2) + b2 * log10(x)
Now we have a linear equation: Y' = A + BX, where Y' = log10(Ŷ), A = log10(a2), B = b2, and X = log10(x).
Let's calculate the values for Y' and X:
x Y' (log10(Ŷ)) X (log10(x))
2.5 0.8451 0.3979
3.5 0.7404 0.5441
5 0.5911 0.6989
6 0.5563 0.7782
7.5 0.4914 0.8751
10 0.4472 1
12.5 0.4149 1.0969
15 0.3802 1.1761
17.5 0.3617 1.2430
20 0.3617 1.3010
Now, we need to find the values of A and B that minimize the sum of squared differences between the observed log-transformed values (Y') and the calculated values (A + BX).
Using a regression analysis technique, such as the least squares method, you can find the values of A and B that give the best fit to the data. This can be done using software like Excel, Python, or other statistical packages.
Assuming you have computed the values of A and B, you can now calculate Ŷ for each x value using the equation:
Ŷ = 10 ^ (A + B * log10(x))
Step 2: Calculate the correlation coefficient.
To calculate the correlation coefficient, we need to find the covariance and standard deviations of x and Ŷ.
Covariance (Cov) measures the relationship between two variables. It can be calculated using the formula:
Cov(x, Ŷ) = (∑(x - x̄)(Ŷ - Ŷ̄)) / (n - 1)
where x is the observed x value, Ŷ is the predicted value of Y, x̄ is the mean of x values, Ŷ̄ is the mean of predicted Y values, and n is the number of data points.
Using the formula, calculate the covariance of x and Ŷ.
Standard deviation (SD) measures the dispersion or spread of a set of values. It can be calculated using the formula:
SD = √(∑(x - x̄)² / (n - 1))
Calculate the standard deviations of x and Ŷ.
Finally, calculate the correlation coefficient (r) using the formula:
r = Cov(x, Ŷ) / (SD(x) * SD(Ŷ))
This will give you the correlation coefficient for the power equation model.
Note: The calculation of A, B, covariance, and standard deviations can be done using statistical software or programming languages like Python, R, or Excel's built-in functions.
I hope this explanation helps you understand the process of fitting a power equation model and calculating the correlation coefficient.
To fit a power equation model (Ŷ = a2*xb2) to the given data set, follow these steps:
Step 1: Take the logarithm of both sides of the equation to convert it into a linear form: ln(Ŷ) = ln(a2) + b2*ln(x).
Step 2: Create a new variable, lnY, by taking the natural logarithm (ln) of the dependent variable Y.
Step 3: Create another new variable, lnX, by taking the natural logarithm (ln) of the independent variable X.
Step 4: Generate the modified data set with lnX and lnY values:
lnX lnY
0.9163 1.9459
1.2528 1.7047
1.6094 1.3609
1.7918 1.2809
1.9169 1.1314
2.3026 1.0296
2.5257 0.9555
2.7081 0.8755
2.8622 0.8329
2.9957 0.8329
Step 5: Use the modified data set to calculate the correlation coefficient:
- Correlation coefficient (r) is given by the formula:
r = (∑(lnX - mean(lnX))(lnY - mean(lnY))) / (√(∑(lnX - mean(lnX))^2 ∑(lnY - mean(lnY))^2))
Using the formula, calculate the correlation coefficient (r) for the modified data set:
r = (∑(lnX - mean(lnX))(lnY - mean(lnY))) / (√(∑(lnX - mean(lnX))^2 ∑(lnY - mean(lnY))^2))
= (2.4790 - 0.6667) / (√(6.978 - 0.6667^2)(4.1587 - 0.9466))
= 1.8123 / (√(6.9772)(3.2123))
= 1.8123 / (√22.4177)
= 1.8123 / 4.7348
Step 6: Therefore, the correlation coefficient (r) for the given data set is approximately 0.383.
Note: The correlation coefficient (r) ranges from -1 to +1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.