Question

The city council of Pine Bluffs is considering increasing the number of police in an effort to reduce crime. Before making a final decision, the council asked the chief of police to survey other cities of similar size to determine the relationship between the number of police and the number of crimes reported. The chief gathered the following sample information.

City Police Number of Crimes City Police Number of Crimes
Oxford 15 19 Holgate 17 10
Starksville 17 16 Carey 11 23
Danville 25 8 Whistler 9 27
Athens 28 10 Woodville 21 10

A. Determine the regression equation. (Round your answers to 4 decimal places. Negative amounts should be indicated by a minus sign.)
B. Estimate the number of crimes for a city with 20 police officers. (Round your answer to 4 decimal places.)
C. Interpret the regression equation.

Answer 1

Explanation:

Hello!

Given the variables

Y: Number of crimes committed for a city.

X: Number of police officers in a city.

a) You have to estimate the linear regression of the number of crimes in the function of the number of police officers per city.

The population model is

E(Yi)= α + βXi

Where α is the intercept and β is the slope of the population model.

To estimate the linear equation you have to estimate both of these values

The estimate of the intercept is "a" (some Authors call it b₀ and you might find the intercept in the model symbolized as β₀)

a= Y[bar] - b*X[bar]

And the estimate of the slope is "b"

`b= (sumXY-((sumX)(sumY))/(n) )/(sumX^2-((sumX)^2)/(n) )`

n=8; ∑X=143; ∑X²= 2855; ∑Y= 123; ∑Y²= 2239; ∑XY= 1913

X[bar]= ∑X/n= 143/8= 17.88

Y[bar]= ∑Y/n= 123/8= 15.38

`b= (1913-(143*123)/(8) )/(2855-((143)^2)/(8) )= -0.96`

a= 15.38-(-0.96)*17.88= 32.46

The estimated regression equation is ^Y= 32.46 - 0.96Xi

B) To estimate the number of crimes given a determined number of police officers, you have to replace the value of X in the regression equation:

^Y
`_(X=20)`=32.46-0.96*20= 13.26

The estimated number of crimes for a city with 20 police officers is 13.26.

C) See attachment for graph.

The slope of the estimated regression equation is negative this means that the linear regression is also negative. A negative linear regression means that when the variable X: number of police officers, increases, the variable Y: number of crimes committed, decreases.

I hope this helps!