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

x 31 5 11 17 7 6 y 172 32 132 127 69 53

In this setting we have x = 77, y = 585, x² = 1481, y² = 71,731, and xy = 9904.

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

x =

y =

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 coefficient r and the coefficient of determination. (Round your answers to three decimal places.)

r =

r² =

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

%

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

t =

Find or estimate the P-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 that differs from 0.

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

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

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

(e) For a neighborhood with x = 24% 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) Find S_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 is x = 24%. (Round your answers to one decimal place.)

lower limit crimes per 1000 residents upper limit crimes per 1000 residents

(h) Test the claim that the slope of 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 the P-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 that differs from 0. Reject the null hypothesis, there is insufficient evidence that differs from 0.Fail to reject the null hypothesis, there is sufficient evidence that differs from 0. Fail to reject the null hypothesis, there is insufficient evidence that differs from 0.

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

lower limit

upper limit

Interpretation

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

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 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 within the confidence interval.