Question

In: Statistics and Probability

Does prison really deter violent crime? Let x represent percent change in the rate of violent...

Does prison really deter violent crime? Let x represent percent change in the rate of violent crime and y represent percent change in the rate of imprisonment in the general U.S. population. For 7 recent years, the following data have been obtained.

x 5.9 5.7 4.1 5.2 6.2 6.5 11.1
y

−1.8

−4.1

−7.0

−4.0

3.6

−0.1

−4.4

A.given Σx = 44.7, Σy = −17.8,Σx2 = 315.05, Σy2 = 117.38, Σxy = −110.66, and r ≈ 0.065.

Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)

Σx =   
Σy =
Σx2 =
Σy2 =
Σxy =
r =


Find x, and y. Then find the equation of the least-squares line  = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

x =
y =
=    +   x

Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.)

r2 =
explained     %
unexplained     %


Considering the values of r and r2, does it make sense to use the least-squares line for prediction? Explain your answer. Select one

The correlation between the variables is so low that it makes sense to use the least-squares line for prediction.

The correlation between the variables is so high that it makes sense to use the least-squares line for prediction.     

The correlation between the variables is so high that it does not make sense to use the least-squares line for prediction.

The correlation between the variables is so low that it does not make sense to use the least-squares line for prediction.

Solutions

Expert Solution

n= 7.0000
X̅=ΣX/n 6.3857
Y̅=ΣY/n -2.5429
sx=(√(Σx2-(Σx)2/n)/(n-1))= 2.2214
sy=(√(Σy2-(Σy)2/n)/(n-1))= 3.4669
Cov=sxy=(ΣXY-(ΣXΣY)/n)/(n-1)= 0.5010
r=Cov/(Sx*Sy)= 0.065
slope= β̂1 =r*Sy/Sx= 0.1015
intercept= β̂0 ='y̅-β1x̅= -3.1911

b)

ΣX = 44.700
ΣY= -17.800
ΣX2 = 315.050
ΣY2 = 117.380
ΣXY = -110.660
r = 0.065

c)

X̅=ΣX/n = 6.39
Y̅=ΣY/n = -2.54
ŷ = -3.191+0.102x
coeff of determination r2 = 0.004
explained = 0.4%
unexplained= 99.6%

The correlation between the variables is so low that it does not make sense to use the least-squares line for prediction.


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