Question

data were gathered and displayed on a scatter plot. which of the following is the most appropriate equation to model the data? ( hat{y}=0.85(10.89)^{x} ) ( hat{y}=10.89(0.85)^{x} ) ( hat{y}=-0.85x + 10.89 ) ( hat{y}=-10.89x + 0.85 )

Explanation:

Step1: Analyze the scatter - plot trend

The scatter - plot shows a decreasing trend as \(x\) increases. This indicates a negative relationship between \(x\) and \(y\). So, we can rule out the exponential growth models \(y = 0.85(10.89)^{x}\) and \(y = 10.89(0.85)^{x}\) since \(y = a\cdot b^{x}\) with \(b>1\) is exponential growth and \(0 < b<1\) is exponential decay but still positive - valued for positive \(a\).

Step2: Consider the form of linear model

The general form of a linear model is \(y=mx + c\), where \(m\) is the slope and \(c\) is the y - intercept. For a decreasing linear relationship, \(m<0\).

Step3: Evaluate the linear models

For \(y=-0.85x + 10.89\) and \(y=-10.89x + 0.85\), we need to estimate the slope and y - intercept from the scatter - plot. The y - intercept is the value of \(y\) when \(x = 0\). Looking at the scatter - plot, the y - intercept seems to be around 10. Also, the slope seems to be around \(- 0.85\) rather than \(-10.89\) (a slope of \(-10.89\) would be a very steep line and does not match the scatter - plot's trend).

Answer:

\(y=-0.85x + 10.89\)