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	25	4	11	17	7	6
y	172	33	132	127	69	53

In this setting we have Σx = 70, Σy = 586, Σx² = 1136, Σy² = 71,796, and Σxy = 8844.

(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	=
ŷ	=	+   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 ρ is not zero at the 10% level of significance. (Round your test statistic to three decimal places and your P-value to four decimal places.)

t =
P-value =

Conclusion

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

(e) For a neighborhood with x = 19% 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 a 95% confidence interval for the change in crime rate when the percentage change in population is x = 19%. (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 10% level of significance. (Round your test statistic to three decimal places and your P-value to four decimal places.)

t =
P-value =

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 a 95% 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 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.  

  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

Answer a:

Here, Number of observatios, n = 6

Mean of x, = Σx / n = 70 / 6 = 11.67

Mean of y, = Σy / n = 586/6 = 97.67

Therefore, b = 45.8422 and a = -437.3085

x = 12.5 and y = 99

Now, Equation of the least squares line -

= -437.3085 + 45.8422x

Answer b:
The scatter diagram is given below -

Answer c:
The sample correlation coefficient, r = Cov.(x, y) / S.d.(x) S.d.(y) = 334.19 / (49.26 x 7.29) = 0.931

Coefficient of Determination, r2 = 0.867

Therefore, 86.7% of variation in y is explained by the least-squares model

Answer d:

The value of the test statistic is 5.106

and P-Value is 0.0069

Therefore, P-Value < 0.10

Conclusion:

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

(NOTE THAT: Since, nothing is mentioned, the first 4 parts of the question is attempted)