Question

the table shows the population of a small town from 1996 to 2004. p is measured in thousands of people.

determine the regression model (round to the hundredths place) and then use the model to determine the population of the small town in 2007.

the population of the small town in 2007 is

Explanation:

Step1: Set \( t = 0 \) for 1996

Let \( t = 0 \) correspond to 1996. Then the years 1996 - 2004 correspond to \( t=0,1,\cdots,8 \).

Step2: Use quadratic regression formula \( P = at^{2}+bt + c \)

Using a calculator or software for quadratic regression with the data points \((t,P)\) where \( t = 0,1,\cdots,8 \) and \( P=22.8,25.0,\cdots,26.1 \).
We get \( a=-0.27\), \( b = 2.59\), \( c=22.8\). So the regression model is \( P=- 0.27t^{2}+2.59t + 22.8 \)

Step3: Find \( t \) for 2007

For 2007, since \( t = 0\) is 1996, then \( t=2007 - 1996=11 \)

Step4: Substitute \( t = 11 \) into the regression model

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Answer:

\(18.62\)