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. PLEASE HELP WITH QUESTIONS (F) - (I)

x	28	1	11	17	7	6
y	170	38	132	127	69	53

In this setting we have Σx = 70, Σy = 589, Σx² = 1280, Σy² = 71,467, and Σxy = 9210.

(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	= 11.67
y	= 98.17
b	= 5.0468
ŷ	= 39.2739	+5.0468  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 = .930
r2 =.865

What percentage of variation in y is explained by the least-squares model? (Round your answer to one decimal place.)
86.5 %

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

t = 5.057

Find or estimate the P-value of the test statistic.

0.005 < P-value < 0.010
Conclusion

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

(e) For a neighborhood with x = 11% 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.)
96.8959 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 = 11%. (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 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 decrease in population, the crime rate per 1,000 increases by an amount that falls outside the confidence interval.

Answer 1

f)

SSE =Syy-(Sxy)2/Sxx=

1845.820

Se =√(SSE/(n-2))=

21.482

g)

std error of confidence interval =				s*√(1+1/n+(x0-x̅)2/Sxx)=			23.2122
for 80 % confidence and 4degree of freedom critical t=							1.533
80% confidence interval =yo -/+ t*standard error=					(59.22,130.384)

lower limit =59.2

upper limit =130.4

h)

t = 5.057

0.005 < P-value < 0.010

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

i)

std error of slope sb1 =				s/√SSx=		0.9980
for 80 % confidence and -2degree of freedom critical t=						1.5330
80% confidence interval =b1 -/+ t*standard error=					(3.517,6.577)

lower limit =3.517

upper limit =6.577

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