Question

select the correct answer from each drop - down menu.
the table shows the population of a small but growing town from the time it was incorporated.

this data can be modeled with an exponential function. the equation of the curve of best fit is ( y = 1,495.3cdot1.1^{x} ). the correlation coefficient is 0.9997.
complete the sentences to describe the model.
based on the curve of best fit, the town had approximately people when it was incorporated, and the population grew around % each year.
there is association between time and the towns population because the correlation coefficient is.

Explanation:

Step1: Analyze the exponential function form

The general form of an exponential function is \(y = a\cdot b^{x}\), where \(a\) is the initial value (when \(x = 0\)) and \(b=1 + r\) ( \(r\) is the growth rate). For the function \(y = 1495.3\cdot1.1^{x}\), when \(x = 0\), \(y=a = 1495.3\approx1500\).

Step2: Calculate the growth rate

Since \(b = 1.1\), and \(b=1 + r\), then \(r=b - 1\). So \(r=1.1-1=0.1 = 10\%\).

Step3: Interpret the correlation coefficient

The correlation coefficient \(r = 0.9997\). A correlation coefficient close to \(1\) (in the range \(0.8\leq|r|\leq1\)) indicates a strong positive linear - like (in the context of the exponential model, a strong positive) association.

Answer:

Based on the curve of best fit, the town had approximately \(1500\) people when it was incorporated, and the population grew around \(10\%\) each year. There is a strong positive association between time and the town's population because the correlation coefficient is close to \(1\).