Question

Problem 2 – 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	35	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

∑?? = 344 ∑?? = 245 ∑??² = 12742 ∑??²= 7135 ∑??? ? = 7509

a) State the dependent and independent variables. Briefly explain your selection of the dependent and independent variables. (1 mark) ​

​ ​ b) Calculate Sx², Sy² and Sxy using the information given above. Display working. (1.5 marks) ​

​ c) Calculate the sample correlation coefficient. Display working. (1 mark) ​

d) Provide an interpretation of the calculated value of the sample correlation coefficient in terms of the relation between number of police and number of crimes. (1 mark) ​ ​

e) Calculate the slope coefficient of the sample linear regression equation. Display working. (0.5 marks) ​

​ f) Provide an interpretation of the slope coefficient you calculated in terms of the relation between number of police and number of crimes. (1.5 marks)

​ g) Calculate the intercept coefficient of the sample linear regression equation. Display working. (0.5 marks)

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

​ i) State the estimated sample linear regression equation. (1 mark) ​ ​

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

​ 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. ​ ​

​ l) Calculate the coefficient of determination for the regression line. Display working. (0.5 marks) ​

​ m) Provide an interpretation of the calculated coefficient of determination in terms of the relation between number of police and number of crimes. (1.5 marks)

Answer 1

Observation table:

region	no. of police (X)	no. of crimes/ day (Y)	X2	Y2	XY
1	35	28	1225	784	980
2	44	14	1936	196	616
3	36	12	1296	144	432
4	48	9	2304	81	432
5	49	15	2401	225	735
6	24	36	576	1296	864
7	32	28	1024	784	896
8	20	42	400	1764	840
9	25	30	625	900	750
10	32	31	1024	961	992
	345	245	12811	7135	7537