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

a) 1985
b) 1992
c) 2000
d) 2012

Solution-

Taking years as input variable and population of that city as output variable,

x = years after 1985

y = population in thousand

*The table is in the attachment.

Linear regression is a way to model the relationship between two variables. The equation has the form Y=a+bX, where Y is the dependent variable (that’s the variable that goes on the Y axis), X is the independent variable (i.e. it is plotted on the X axis), b is the slope of the line and a is the y-intercept.

Putting the values from the table,

Then calculating a, b,

The equation of the best fit line would be,

To know in which year the actual population of Center City was most different from the value predicted by this model, we have to calculate the residues. The point with max residue will be the year, where the above mentioned condition happens.

Residue = Actual value - Predicted value

Taking the residue values on y-axis and x value on x-axis, residue plot was generated.

Residue values of these years are,

1990 = -0.271,

1992 = 0.322,

2000 = -0.012,

2005 = -0.047,

2012 = 0.008,

Therefore, it is cleared that in the year of 1992, as the residue value is greatest, the actual population of Center City is most different from the value predicted by this model.

ratio is 2 : 9

the answer is three

13.5 : 3