Use the exponential regression equation that best fits the data (10, 4) (12, 20) (13, 35) (16, 300), and to estimate the value of y when x = 14.
a. 48.4
b. 73.5
c. 132.3
d. 223.7
The exponential regression equation can be written in the form y = ab^x, where a and b are constants to be determined.
To find the values of a and b, we can use the given data points. We will set up a system of equations based on these points.
Using the first data point (10, 4), we get:
4 = ab^10
Using the second data point (12, 20), we get:
20 = ab^12
Using the third data point (13, 35), we get:
35 = ab^13
Using the fourth data point (16, 300), we get:
300 = ab^16
To solve this system of equations, we can divide each equation by the equation before it:
(20)/(4) = (ab^12)/(ab^10)
5 = b^2
(35)/(20) = (ab^13)/(ab^12)
1.75 = b
(300)/(35) = (ab^16)/(ab^13)
8.57 = b^3
Now we can solve for b by taking the square root of 5 (from b^2 = 5), which gives us b ≈ 2.236. Taking the cube root of 8.57 (from b^3 = 8.57), we get approximately b ≈ 2.116.
Now that we have the value of b, we can substitute it into any of the original equations to solve for a. Let's use the first equation (4 = ab^10):
4 = a(2.236)^10
4 = 102.595a
a ≈ 0.0389
So the exponential regression equation that best fits the data is y ≈ 0.0389(2.236)^x.
To estimate the value of y when x = 14, we substitute this value into the equation:
y ≈ 0.0389(2.236)^14
y ≈ 0.0389(227.57)
y ≈ 8.8497
Therefore, the estimated value of y when x = 14 is approximately 8.8497.
None of the given answer choices (48.4, 73.5, 132.3, 223.7) match this estimation, so none of the provided options (a, b, c, d) are correct.