To create a linear model that predicts the population [tex]\( y \)[/tex] of Center City in a given year [tex]\( x \)[/tex], we will use the data from the years 1990 and 2005. Let's break this down step-by-step:
1. Identify the given data for the years 1990 and 2005:
- Year 1990: Population is 197,800
- Year 2005: Population is 206,561
2. Determine the slope [tex]\( m \)[/tex] of the linear model:
The slope of the line connecting the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the values:
[tex]\[
m = \frac{206,561 - 197,800}{2005 - 1990} = \frac{8,761}{15} \approx 584.0667
\][/tex]
3. Find the y-intercept [tex]\( b \)[/tex] of the linear model:
To find the y-intercept, use the point-slope form of the line equation:
[tex]\[
b = y_1 - m \cdot x_1
\][/tex]
Substituting the values from the year 1990:
[tex]\[
b = 197,800 - 584.0667 \cdot 1990 \approx 197,800 - 1,162,292.6667 \approx -964,492.6667
\][/tex]
4. Write the linear equation:
The linear model predicting the population [tex]\( y \)[/tex] in a given year [tex]\( x \)[/tex] is:
[tex]\[
y = 584.0667 \cdot x - 964,492.6667
\][/tex]
5. Predict the populations using the linear model for the given years:
- 1985: [tex]\( y = 584.0667 \cdot 1985 - 964,492.6667 \approx 194,879.67 \)[/tex]
- 1990: [tex]\( y = 197,800 \)[/tex]
- 1992: [tex]\( y = 584.0667 \cdot 1992 - 964,492.6667 \approx 198,968.13 \)[/tex]
- 2000: [tex]\( y = 584.0667 \cdot 2000 - 964,492.6667 \approx 203,640.67 \)[/tex]
- 2005: [tex]\( y = 206,561 \)[/tex]
- 2012: [tex]\( y = 584.0667 \cdot 2012 - 964,492.6667 \approx 210,649.47 \)[/tex]
6. Calculate the differences between the actual and predicted populations:
- 1985: [tex]\( \left| 194,957 - 194,879.67 \right| \approx 77.33 \)[/tex]
- 1990: [tex]\( \left| 197,800 - 197,800 \right| = 0 \)[/tex]
- 1992: [tex]\( \left| 199,532 - 198,968.13 \right| \approx 563.87 \)[/tex]
- 2000: [tex]\( \left| 203,750 - 203,640.67 \right| \approx 109.33 \)[/tex]
- 2005: [tex]\( \left| 206,561 - 206,561 \right| = 0 \)[/tex]
- 2012: [tex]\( \left| 210,600 - 210,649.47 \right| \approx 49.47 \)[/tex]
7. Identify the year with the maximum difference between the actual and predicted population:
By examining the differences, we see the maximum difference occurs in the year 1992, with a difference of approximately 563.87.
Therefore, the actual population of Center City was most different from the value predicted by the model in the year 1992.
