Question

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.

Answer 1

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.