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	27	3	11	17	7	6
y	167	40	132	127	69	53

In this setting we have Σx = 71, Σy = 588, Σx² = 1233, Σy² = 70,612, and Σxy = 9041.

(e) For a neighborhood with x = 17% 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 = 17%. (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 > 0.2500.125 < P-value < 0.250     0.100 < P-value < 0.1250.075 < P-value < 0.1000.050 < P-value < 0.0750.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.0100.0005 < P-value < 0.005P-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 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.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

	X	Y	XY	X²	Y²
total sum	71.000	588.000	9041.000	1233	70612

sample size ,   n =   6          
here, x̅ =Σx/n =   11.8333   ,   ȳ = Σy/n =   98.00  
                  
SSxx =    Σx² - (Σx)²/n =   392.833          
SSxy=   Σxy - (Σx*Σy)/n =   2083.000          
SSyy =    Σy²-(Σy)²/n =   12988.000          
estimated slope , ß1 = SSxy/SSxx =   2083.000   /   392.833   =   5.3025
                  
intercept,   ß0 = y̅-ß1* x̄ =   35.2537          
                  
so, regression line is   Ŷ =   35.2537   +   5.3025   *x
----------------

e)Predicted Y at X=   17   is                  
Ŷ =   35.254   +   5.303   *   17   =   125.4

f)

      
SSE=   (Sx*Sy - S²xy)/Sx =    1942.8859
      
std error ,Se =    √(SSE/(n-2)) =    22.039

g)

Sample Size , n=   6                      
Degrees of Freedom,df=n-2 =   4                      
critical t Value=tα/2 =   1.533   [excel function: =t.inv.2t(α/2,df) ]                  
                          
X̅ =    11.83                      
Σ(x-x̅)² =Sxx   392.8                      
Standard Error of the Estimate,Se=   22.04                      
                          
Predicted Y at X=   17   is                  
Ŷ =   35.254   +   5.303   *   17   =   125.396
                          
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    10.675                      
margin of error,E=t*Std error=t* S(ŷ) =   1.5332   *   10.6752   =   16.3673      
                          
Confidence Lower Limit=Ŷ +E =    125.396   -   16.3673   =   109.0   
Confidence Upper Limit=Ŷ +E =   125.396   +   16.3673   =   141.8

h)

estimated std error of slope =Se(ß1) = Se/√Sxx =    22.039   /√   393   =   1.1120
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    5.3025   /   1.1120   =   4.769

i)

confidence interval for slope                  
α=   0.2              
t critical value=   t α/2 =    1.533   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    22.03909   /√   392.83   =   1.112
                  
margin of error ,E= t*std error =    1.533   *   1.112   =   1.705
estimated slope , ß^ =    5.3025              
                  
                  
lower confidence limit = estimated slope - margin of error =   5.3025   -   1.705   =   3.598
upper confidence limit=estimated slope + margin of error =   5.3025   +   1.705   =   7.007

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