Question

A sociologist is interested in the relation between x = number of job changes and y = annual salary (in thousands of dollars) for people living in the Nashville area. A random sample of 10 people employed in Nashville provided the following information.

x (number of job changes)	6	3	5	6	1	5	9	10	10	3
y (Salary in $1000)	38	34	32	32	32	38	43	37	40	33

In this setting we have Σx = 58, Σy = 359, Σx² = 422, Σy² = 13,023, and Σxy = 2160.
(e) If someone had x = 7 job changes, what does the least-squares line predict for y, the annual salary? (Round your answer to two decimal places.)
thousand dollars

(f) Find S_e. (Round your answer to two decimal places.)
S_e =

(g) Find a 90% confidence interval for the annual salary of an individual with x = 7 job changes. (Round your answers to two decimal places.)

lower limit	thousand dollars
upper limit	thousand dollars

(h) Test the claim that the slope β of the population least-squares line is positive at the 5% 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 > 0.Reject the null hypothesis. There is insufficient evidence that β > 0.    Fail to reject the null hypothesis. There is sufficient evidence that β > 0.Fail to reject the null hypothesis. There is insufficient evidence that β > 0.

(i) Find a 90% confidence interval for β and interpret its meaning. (Round your answers to three decimal places.)

lower limit
upper limit

Interpretation

For each less job change, the annual salary increases by an amount that falls within the confidence interval.For each additional job change, the annual salary increases by an amount that falls within the confidence interval.    For each additional job change, the annual salary increases by an amount that falls outside the confidence interval.For each less job change, the annual salary increases by an amount that falls outside the confidence interval.

Answer 1

e)

Predicted Y at X=   7   is                  
Ŷ =   30.629   +   0.909   *   7   =   36.99

f)

std error ,Se =    √(SSE/(n-2)) =    2.83

g)

X Value=   7                      
Confidence Level=   90%                      
                          
                          
Sample Size , n=   10                      
Degrees of Freedom,df=n-2 =   8                      
critical t Value=tα/2 =   1.860   [excel function: =t.inv.2t(α/2,df) ]                  
                          
X̅ =    5.80                      
Σ(x-x̅)² =Sxx   85.600000                      
Standard Error of the Estimate,Se=   2.83                      
                          
Predicted Y at X=   7   is                  
Ŷ =   30.629   +   0.909   *   7   =   36.991
                          
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    0.968                      
margin of error,E=t*Std error=t* S(ŷ) =   1.8595   *   0.9682   =   1.8003      
                          
Confidence Lower Limit=Ŷ +E =    36.991   -   1.8003   =   35.19   
Confidence Upper Limit=Ŷ +E =   36.991   +   1.8003   =   38.79

h)

  estimated std error of slope =Se(ß1) = Se/√Sxx =    2.833   /√   86   =   0.3062
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    0.9089   /   0.3062   =   2.969
                  
0.005 < P-value < 0.010

Reject the null hypothesis. There is sufficient evidence that β > 0

i)

confidence interval for slope                  
α=   0.1              
t critical value=   t α/2 =    1.860   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    2.83261   /√   85.60   =   0.306
                  
margin of error ,E= t*std error =    1.860   *   0.306   =   0.569
estimated slope , ß^ =    0.9089              
                  
                  
lower confidence limit = estimated slope - margin of error =   0.9089   -   0.569   =   0.340
upper confidence limit=estimated slope + margin of error =   0.9089   +   0.569   =   1.478

For each additional job change, the annual salary increases by an amount that falls within the confidence interval