Question

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 26 1 11 17 7 6 y 172 36 132 127 69 53 In this setting we have Σx = 68, Σy = 589, Σx2 = 1172, Σy2 = 72,003, and Σxy = 8920.

(e) For a neighborhood with x = 14% 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 Se. (Round your answer to three decimal places.)

Se =

(g) Find an 80% confidence interval for the change in crime rate when the percentage change in population is x = 14%. (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 β 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 β differs from 0.

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

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

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

(i) Find an 80% confidence interval for β 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 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.

Answer 1

e)

predicted value =

113.1

f)

Se =√(SSE/(n-2))=

20.177

g)

std error of confidence interval =				s*√(1+1/n+(x0-x̅)2/Sxx)=			21.9580
for 80 % confidence and 4degree of freedom critical t=							1.533
80% confidence interval =yo -/+ t*standard error=					(79.42,146.742)

lower limit =79.4

upper limit =146.7

h)

test statistic t =

r*(√(n-2)/(1-r2))=

5.553

0.005 < P-value < 0.010

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

i)

std error of confidence interval =				s*√(1+1/n+(x0-x̅)2/Sxx)=			21.9580
for 80 % confidence and 4degree of freedom critical t=							1.533
80% confidence interval =yo -/+ t*standard error=					(79.42,146.742)

lower limit=79.418

upper limit=146.742

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