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)	5	4	6	6	1	5	9	10	10	3
y (Salary in $1000)	32	34	35	32	32	38	43	37	40	33

In this setting we have Σx = 59, Σy = 356, Σx² = 429, Σy² = 12,804, and Σxy = 2176.

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

Conclusion

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

b. Reject the null hypothesis. There is insufficient evidence that ρ > 0.    

c. Fail to reject the null hypothesis. There is sufficient evidence that ρ > 0.

d. Fail to reject the null hypothesis. There is insufficient evidence that ρ > 0.

(e) If someone had x = 6 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 = 6 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 additional job change, the annual salary increases by an amount that falls within the confidence interval.For each less job change, the annual salary increases by an amount that falls within the confidence interval.    For each less job change, the annual salary increases by an amount that falls outside the confidence interval.For each additional job change, the annual salary increases by an amount that falls outside the confidence interval.

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	59	356	80.9	130.4	75.60
mean	5.90	35.60	SSxx	SSyy	SSxy

R² =    (Sxy)²/(Sx.Sy) =    0.5418

54.2% of variation is explained by y

...............

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

..................

e)

Predicted Y at X=   6   is                  
Ŷ =   30.08653   +   0.934487   *   6   =   35.69

................

f)

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    59.753
      
std error ,Se =    √(SSE/(n-2)) =    2.73

....................

g)

standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    0.865              
margin of error,E=t*Std error=t* S(ŷ) =   1.8595   *   0.8648   =   1.6081
                  
Confidence Lower Limit=Ŷ +E =    35.693   -   1.608   =   34.09
Confidence Upper Limit=Ŷ +E =   35.693   +   1.608   =   37.30

..................

h)

slope hypothesis test               tail=   1
Ho:   ß1=   0          
H1:   ß1 >   0          
n=   10              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    2.733   /√   80.90   =   0.3039
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    0.9345   /   0.3039   =   3.075

p-value =    0.0076

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 =    2.73296   /√   80.90   =   0.304
                  
margin of error ,E= t*std error =    1.860   *   0.304   =   0.565
estimated slope , ß^ =    0.9345              
                  
                  
lower confidence limit = estimated slope - margin of error =   0.9345   -   0.565   =   0.369
upper confidence limit=estimated slope + margin of error =   0.9345   +   0.565   =   1.500

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

..................

THANKS

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