In: Statistics and Probability
The Vic 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 the number of police to be employed, the Ministry of Police asked that various regions of similar size throughout Vic to be surveyed to determine the relationship between the number of police employed and the number of crimes reported per day. Data collected is shown in the table.
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 |
Solution: a) The dependent and independent variables in this study are respectively Number of Crimes per day and Number of Police. The explanation for this choice is-- the number of crimes in an area will obviously depend upon the system of law in that place, i.e., the number of police provided in that specific area. Thus, in that way, number of police briefly explains the number of crimes occurring. Let x=independent variable and y=independent variable
b) Sx2 = sum(x*x) = 12742, Sy2 = sum(y*y) = 7160 and Sxy = sum(x*y) = 7545
c) The sample correlation coefficient is obtained by the formula
r = {n(sum (x*y)) - (sum (x) )(sum (y)}/[{\sqrt{[n*(sum (x*x)) - (sum (x))^2][n*(sum (y*y)) - (sum (y))^2]]
where n = number of observations, sqrt refers to square root function.
Here r = -0.9142646
d) An interpretation of the calculated value of the sample correlation coefficient in terms of the relation between number of police and number of crimes is stated as--
There is a strong negative correlation between the number of police and number of crimes, which means, as the number of police increases, no. of crimes will decrease.
e) The slope coefficient of the sample linear regression equation = b1 = {sum((x-xbar)*(y-ybar))} / sum(x-xbar)2
Here b1 = -1.01
f) An interpretation of the slope coefficient calculated in terms of the relation between number of police and number of crimes is--
For every increase in the number of police, there is a decrease of 1.01 units of crimes per day.