Question

If an exponential model was used to fit the data set below, which of the following would be the best prediction for the output of the model if the input was x=20?

Answer 1

Answer:

The equation is found to be: $y = 50.6e^{0.16x}$

y(20) = 1241.34

Step-by-step explanation:

The given data is:

x:   3          7         11        14          17

y: 83      142      301     450      722

Now, we find sum summation values, relevant to the formula of exponential regression model, using calculator:

∑ ln y = 27.77305, ∑x ln y = 308.1494, ∑x = 52, ∑ x² = 664

and, n = no. of data points = 5

Now, we use formulae of exponential regression model to find out values of constant:

b = (n∑x lny - ∑x ∑ln y)/[n∑x² - (∑x)²]

b = [(5)(308.1494) - (52)(27.77305)]/[(5)(664) - (52)²]

b = 0.16

Now, for a;

a = (∑ln y - b∑x)/n

Therefore,

a = [(27.77305) - (0.16)(52)]/5

a = 3.9

For, α:

α = e^a = e^3.9

α = 50.6

So, the final equation of exponential regression model is given as:

$y = \alpha e^{bx}\\ y = 50.6e^{0.16x}$

Now, we find value of y for x = 20:

y(20) = (50.6) e^(0.16*20)

y(20) = 1241.34