Letxbe a random variable representing percentage change in neighborhood population in the past few years, and letybe a random variable representing crime rate (crimes per 1000 population). A random sample of six Denver neighborhoods gave the following information.

x 32, 4, 11, 17, 7, 6

y 176. 38, 132, 127, 69, 53

In this setting we havex=77,y=595,x²=1535,y²=73,543, andxy=10196.

(a) Findx,y,b, and the equation of the least-squares line. (Round your answers forxandyto two decimal places. Round your least-squares estimates to four decimal places.)

x (mean)=

y (mean)=

b=

=+x

(b) Draw a scatter diagram displaying the data. Graph the least-squares line on your scatter diagram. Be sure to plot the point (x,y).

(c) Find the sample correlation coefficientrand the coefficient of determination. (Round your answers to three decimal places.)r=r²=

What percentage of variation inyis explained by the least-squares model? (Round your answer to one decimal place.)

?%

(d) Test the claim that the population correlation coefficientis not zero at the 1% level of significance. (Round your test statistic to three decimal places.)

t=

Find or estimate theP-value of the test statistic.

P-value > 0.250

0.125 <P-value < 0.250

0.100 <P-value < 0.125

0.075 <P-value < 0.100

0.050 <P-value < 0.075

0.025 <P-value < 0.050

0.010 <P-value < 0.025

0.005 <P-value < 0.010

0.0005 <P-value < 0.005

P-value < 0.0005

Conclusion

Reject the null hypothesis, there is sufficient evidence thatdiffers from 0.

Reject the null hypothesis, there is insufficient evidence thatdiffers from 0.

Fail to reject the null hypothesis, there is sufficient evidence thatdiffers from 0.

Fail to reject the null hypothesis, there is insufficient evidence thatdiffers from 0.

(e) For a neighborhood withx=22%change in population in the past few years, predict the change in the crime rate (per 1000 residents). (Round your answer to one decimal place.)

?crimes per 1000 residents

(f) FindS_e. (Round your answer to three decimal places.)

S_e=

(g) Find an 80% confidence interval for the change in crime rate when the percentage change in population isx=22%. (Round your answers to one decimal place.)

lower limit? crimes per 1000 resident

supper limit? crimes per 1000 residents

(h) Test the claim that the slopeof the population least-squares line is not zero at the 1% level of significance. (Round your test statistic to three decimal places.)

t=

Find or estimate theP-value of the test statistic.

P-value > 0.250

0.125 <P-value < 0.250

0.100 <P-value < 0.125

0.075 <P-value < 0.100

0.050 <P-value < 0.075

0.025 <P-value < 0.050

0.010 <P-value < 0.025

0.005 <P-value < 0.010

0.0005 <P-value < 0.005

P-value < 0.0005

Conclusion

Reject the null hypothesis, there is sufficient evidence thatdiffers from 0.

Reject the null hypothesis, there is insufficient evidence thatdiffers from 0.

Fail to reject the null hypothesis, there is sufficient evidence thatdiffers from 0.

Fail to reject the null hypothesis, there is insufficient evidence thatdiffers from 0.

(i) Find an 80% confidence interval forand interpret its meaning. (Round your answers to three decimal places.)

lower limit

upper limit

Interpretation

For every percentage point decrease in population, the crime rate per 1,000 increases by an amount that falls outside the confidence interval.

For every percentage point increase in population, the crime rate per 1,000 increases by an amount that falls within the confidence interval.

For every percentage point decrease in population, the crime rate per 1,000 increases by an amount that falls within the confidence interval.

For every percentage point increase in population, the crime rate per 1,000 increases by an amount that falls outside the confidence interval.