Question

which of the following is the most appropriate model for the given data?
$hat{y}=1.25(2)^x$
$hat{y}=1.4x + 0.5$
$hat{y}=2(1.25)^x$
$hat{y}=0.5x + 1.4$

Explanation:

Step1: Observe data trend

The data points seem to follow a linear - like pattern rather than an exponential one. Exponential functions like $y = a(b)^x$ would show a more rapid increase or decrease.

Step2: Test linear models

For the linear model $\hat{y}=mx + c$, we can check the slope and intercept. The general form of a linear equation is $\hat{y}=mx + c$, where $m$ is the slope and $c$ is the y - intercept.
We can estimate the slope by taking two points $(x_1,y_1)$ and $(x_2,y_2)$ and using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. By observing the graph, as $x$ increases by 1, $y$ seems to increase by approximately 1.4 on average. And when $x = 0$, $y$ is around 0.5.
The model $\hat{y}=1.4x + 0.5$ best fits this observation.

Answer:

$\hat{y}=1.4x + 0.5$