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
1985
194,957
1990
197,800
1992
199,532
2000
203,750
2005
206,561
2012
210,600
1985
1992
2000
2012

Answer 1

First make the line through (1990, 197,800) and (2005, 206,561).

Slope m = (206,561 − 197,800)/(2005 − 1990) = 8,761/15 ≈ 584.0667.

Linear model (point‑slope form):
y = 197,800 + 584.0667(x − 1990)
(or y ≈ 584.067x − 964,492.667).

Predicted vs actual and absolute differences:
- 1985: predicted = 197,800 + (−5)(584.0667) ≈ 194,879.7; actual = 194,957; |diff| ≈ 77.3
- 1990: predicted = 197,800; actual = 197,800; |diff| = 0
- 1992: predicted = 197,800 + 2(584.0667) ≈ 198,968.1; actual = 199,532; |diff| ≈ 563.9
- 2000: predicted ≈ 203,640.7; actual = 203,750; |diff| ≈ 109.3
- 2005: predicted = 206,561; actual = 206,561; |diff| = 0
- 2012: predicted ≈ 210,649.5; actual = 210,600; |diff| ≈ 49.5

The actual population was most different from the model in 1992 (about 564 people higher than predicted).