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

The best prediction of the output of the model when the input is 20 is 770

How to find the prediction

To find the prediction, we need to find the equation of the model in the table.

Exponential function are functions of the form

y = a(b)ˣ

plugging in the values, using data from the table

83 = a(b)³

142 = a(b)⁷

dividing both equations as follows

142 / 83 = a(b)⁷ / a(b)³

142/83 = b⁴

b = 1.14

solving for a, by substituting b

83 = a(b)³

83 = a(1.14)³

83 = a(1.14)³

83 = 1.48a

a = 56.08

a = 56

We then solve for x = 20

y = a(b)ˣ

y = 56 * (1.14)²⁰

y = 769.64

y = 770

Answer 2

The equation is found to be:
`y = 50.6e^(0.16x)`

y(20) = 1241.34

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