In: Statistics and Probability
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.
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.