Find the exponential regression equation that best fits the
data (2, 7), (3, 10), (5, 50), and (8, 415).
y = 2. 89(1. 00)x
y = 1. 00(2. 89)x
y = 1. 47(2. 02)x
y = 2. 02(1. 47)x
b
The correct exponential regression equation that best fits the given data is y = 1.47(2.02)x.
To find the exponential regression equation that best fits the given data, you can follow these steps:
Step 1: Write down the given data pairs: (2, 7), (3, 10), (5, 50), and (8, 415).
Step 2: Use a graphing calculator or a spreadsheet software to plot the given data points.
Step 3: Determine if the data shows exponential growth or decay. Exponential growth is represented by a positive relationship between x and y values, while exponential decay is represented by a negative relationship.
Step 4: Calculate the exponential regression equation using the exponential model, which is of the form: y = ab^x, where a and b are constants.
Step 5: Apply logarithms to the given data points to linearize the exponential equation. Take the natural logarithm (ln) of both sides of the equation to obtain ln(y) = ln(a) + x * ln(b).
Step 6: Create a new set of data using the logarithmic transformations of the original data points.
Step 7: Use linear regression to fit a line to the new logarithmic data. This will give you the values of ln(a) and ln(b).
Step 8: Convert the linear equation obtained from the linear regression back to its exponential form by taking the inverse natural logarithm (e^x) of both sides.
By following these steps, you can calculate the exponential regression equation that best fits the given data. In this case, the correct answer is option:
y = 2.02(1.47)^x.
To find the exponential regression equation that best fits the given data, you can use the exponential regression formula:
y = ab^x
Let's compute the values of a and b using the given data points:
Using the point (2, 7):
7 = ab^2
Using the point (3, 10):
10 = ab^3
Using the point (5, 50):
50 = ab^5
Using the point (8, 415):
415 = ab^8
Now, let's solve this system of equations:
From the first equation, we can solve for a:
a = 7 / b^2
Substituting this value into the second equation, we get:
10 = (7 / b^2)b^3
10 = 7b
Solving for b:
b = 10/7
Now, substituting this value of b into the first equation, we can solve for a:
7 = a(10/7)^2
a = 7 / (10/7)^2
a = 7 / (100/49)
a = 7 * 49 / 100
a = 3.43
Therefore, the exponential regression equation that best fits the given data is:
y = 3.43(10/7)^x
Simplifying further:
y = 3.43(1.43)^x
So the correct answer is:
y = 1.47(2.02)^x