3. In one of the Boston city parks, there has been a problem with muggings in th

Question

3. In one of the Boston city parks, there has been a problem with muggings in the summer months. A police cadre took a random sample of 10 days ( out of a 90-day summer) and compiled the following data. For each day, x represents the number of police ocers on duty and y represents the number of reported muggings on that day.
x 10 15 16 1 4 6 18 12 14 7 y 5 2 1 9 7 8 1 5 3 6

x = 103, y = 47, x2 = 1347, y2 = 295, xy = 343

a) Calculate the sample correlation coecient.

b) Find the equation of the regression line relating y to x.

c) Predict the number of muggings for 20 police ocers on duty.

In one of the Boston city parks, there has been a problem with muggings in the summer months. A police cadre took a random sample of 10 days ( out of a 90-day summer) and compiled the following data. For each day, x represents the number of police ocers on duty and y represents the number of reported muggings on that day.
x 10 15 16 1 4 6 18 12 14 7 y 5 2 1 9 7 8 1 5 3 6

x = 103, y = 47, x2 = 1347, y2 = 295, xy = 343

a) Calculate the sample correlation coecient.

b) Find the equation of the regression line relating y to x.

d) Test at 1% level of signicance, the claim that the slope of the regression line between the number of police ocers and number of muggings is positive. Interpret the result in the context of the problem.

Explanation / Answer

a) correlation coefficient is given by

r=xy/n*sqrt(x^2/n)*sqrt(y^2/n) ##n=10

=343/10*sqrt(134.7)*sqrt(29.5)

r =0.54 ##correlation coefficient

b) regression equation y on x is given by

Y = a + bX # a :- y intercept

#b: slope

Two normal equations are

y=na + bx

xy = ax + bx^2

Put the values we get

10a + 103b=47

103a +1347b=343

By solving this simultaneous equation

We get

a=9.77; b=-0.49

Therefore, the regression equation is

Y= 9.77 - 0.49X

c) when X=20

Y = 9.77 -0.49*20

=-0.03

Y ~= 0

Therefore when 20 police officers are on duty the number of

mugging is Zero.

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