Question

the number of newly reported crime cases in a county in new york state is shown in the accompanying table, where x represents the number of years since 2011, and y represents number of new cases. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. using this equation, estimate the calendar year in which the number of new cases would reach 1141.
years since 2011 (x) new cases (y)
0 814
1 825
2 854
3 869
4 947
copy values for calculator
open statistics calculator
answer
regression equation:
final answer:

Explanation:

Step1: Input data into statistics calculator

Using a statistics calculator, input the \(x\)-values (\(0,1,2,3,4\)) and \(y\)-values (\(814,825,854,869,947\)).

Step2: Find the linear regression equation

The linear regression equation has the form \(y = ax + b\). After calculation, \(a\approx38.20\) and \(b = 809.40\). So the equation is \(y=38.20x + 809.40\).

Step3: Solve for \(x\) when \(y = 1141\)

Set \(y = 1141\) in the equation \(1141=38.20x + 809.40\).
Subtract \(809.40\) from both sides: \(1141−809.40=38.20x\), so \(331.60 = 38.20x\).
Divide both sides by \(38.20\): \(x=\frac{331.60}{38.20}\approx8.68\).
Since \(x\) is the number of years since 2011, the year is \(2011 + 9=2020\) (rounding \(8.68\) up to the next whole - number year as we are looking for when the cases reach the value).

Answer:

Regression Equation: \(y = 38.20x+809.40\)
Final Answer: \(2020\)