Question

Problem 3 – Should more police be employed?

The Victorian Government is considering increasing the number of police employed in a region in Victoria in an effort to reduce crime. Before making the final decision on number of police to be employed, the Ministry of Police asked that various regions of similar size throughout Victoria to be surveyed to determine the relationship between the number of police employed and the number of crimes reported per day. The data collected is shown in the table below.

Region	Number of Police	Number of Crimes per day
1	34	28
2	44	14
3	36	12
4	48	9
5	49	15
6	24	36
7	32	28
8	20	42
9	25	30
10	32	31

1. h) Provide an interpretation of the intercept coefficient you calculated in terms of the relation between number of police and number of crimes.

2. i) State the estimated sample linear regression equation.

3. j) Predict the number of crimes per day if 45 polices are employed. Display working. Comment on the validity of this prediction

4. k) Conduct a test on the slope coefficient to see if a negative relation exists between the two variables. Use a 1% level of significance. Display working of the six steps hypothesis test. The t test-statistic has been calculated. It equals -6.06.

5. l) Calculate the coefficient of determination for the regression line. Display working.

6. m) Provide an interpretation of the calculated coefficient of determination in terms of the relation between number of police and number of crimes.

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	344.00	245.00	908.40	1132.50	-919.00
mean	34.40	24.50	SSxx	SSyy	SSxy

sample size ,   n =   10          
here, x̅ = Σx / n=   34.400   ,     ȳ = Σy/n =   24.500  
                  
SSxx =    Σ(x-x̅)² =    908.4000          
SSxy=   Σ(x-x̅)(y-ȳ) =   -919.0          
                  
estimated slope , ß1 = SSxy/SSxx =   -919.0   /   908.400   =   -1.01167
                  
intercept,   ß0 = y̅-ß1* x̄ =   59.30141          
                  
so, regression line is   Ŷ =   59.3014   +   -1.012   *x

h) when number of police employed is 0 then estimated crime per day is 59.3014

i)  Ŷ =   59.3014   +   -1.012   *x

j)

Predicted Y at X=   45   is          
Ŷ =   59.3014   +   -1.0117   *45=   13.77631

k)

Ho:   ß1=   0          
H1:   ß1<   0          
n=   10              
alpha =   0.01              
estimated std error of slope =Se(ß1) = Se/√Sxx =    5.035   /√   908.40   =   0.1670
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    -1.0117   /   0.1670   =   -6.0564
                  
t-critical value=    -2.8965   [Excel function: =T.INV(α,df) ]          
Degree of freedom ,df = n-2=   8              
p-value =    0.0002              
decison :    p-value<α , reject Ho              

l)

R² =    (SSxy)²/(SSx.SSy) =    0.8209   

m)

about 82.09%of variation in observation of number of crimes ia explained by variable number of police