Question

the table shows the population of center city in various years. use the data from 1990 and 2005 to create a linear model that predicts the population of the city (y) in a given year (x). in which year was the actual population of center city most different from the value predicted by this model?

year	city population
1990	197,800
1992	199,532
2000	203,750
2005	206,561
2012	210,600

options: 1985, 1992, 2000, 2012

Explanation:

Step1: Define variables and find slope

Let \( x \) be the year, \( y \) be the population. For 1990 (\( x_1 = 1990 \), \( y_1 = 197800 \)) and 2005 (\( x_2 = 2005 \), \( y_2 = 206561 \)). Slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{206561 - 197800}{2005 - 1990}=\frac{8761}{15}\approx584.07 \).

Step2: Find linear equation

Using point - slope form \( y - y_1=m(x - x_1) \), \( y - 197800 = 584.07(x - 1990) \). Simplify to \( y=584.07x-584.07\times1990 + 197800 \). Calculate \( 584.07\times1990=584.07\times(2000 - 10)=1168140-5840.7 = 1162299.3 \). Then \( y = 584.07x-1162299.3 + 197800=584.07x - 964499.3 \).

Step3: Predict population for each year

• 1985: \( x = 1985 \), \( y=584.07\times1985-964499.3\approx584.07\times(2000 - 15)-964499.3=(1168140-8761.05)-964499.3 = 1159378.95-964499.3 = 194879.65 \). Actual: 194957. Difference: \( |194957 - 194879.65| = 77.35 \).
• 1992: \( x = 1992 \), \( y=584.07\times1992-964499.3\approx584.07\times(1990 + 2)-964499.3=(584.07\times1990+1168.14)-964499.3=(1162299.3 + 1168.14)-964499.3=1163467.44 - 964499.3 = 198968.14 \). Actual: 199532. Difference: \( |199532 - 198968.14| = 563.86 \).
• 2000: \( x = 2000 \), \( y=584.07\times2000-964499.3 = 1168140-964499.3 = 203640.7 \). Actual: 203750. Difference: \( |203750 - 203640.7| = 109.3 \).
• 2012: \( x = 2012 \), \( y=584.07\times2012-964499.3\approx584.07\times(2000 + 12)-964499.3=(1168140+7008.84)-964499.3=1175148.84 - 964499.3 = 210649.54 \). Actual: 210600. Difference: \( |210600 - 210649.54| = 49.54 \).

Step4: Compare differences

Compare 77.35, 563.86, 109.3, 49.54. The largest difference is 563.86 which corresponds to the year 1992.

Answer:

1992