Question

Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profits per employee. The following is data about annual profits per employee (in units of 1 thousand dollars per employee) for representative companies in financial services. Assume σ ≈ 9.4 thousand dollars.

43.9	36.2	27.8	58.6	25.0	36.2	39.7	60.0	42.5	33.0	33.6
36.9	27.0	47.1	33.8	28.1	28.5	29.1	36.5	36.1	26.9	27.8
28.8	29.3	31.5	31.7	31.1	38.0	32.0	31.7	32.9	23.1	54.9
43.8	36.9	31.9	25.5	23.2	29.8	22.3	26.5	26.7

(a) Use a calculator or appropriate computer software to find x for the preceding data. (Round your answer to two decimal places.)
  thousand dollars

(b) Let us say that the preceding data are representative of the entire sector of (successful) financial services corporations. Find a 75% confidence interval for μ, the average annual profit per employee for all successful banks. (Round your answers to two decimal places.)

lower limit	thousand dollars
upper limit	thousand dollars

(c) Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful financial institutions? Explain by referring to the confidence interval you computed in part (b).

Yes. This confidence interval suggests that the bank profits are less than those of other financial institutions.Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.    No. This confidence interval suggests that the bank profits are less than those of other financial institutions.No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

(d) Suppose the annual profits are more than 40 thousand dollars per employee. As manager of the bank, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b).

No. This confidence interval suggests that the bank profits are higher than those of other financial institutions.No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.    Yes. This confidence interval suggests that the bank profits are higher than those of other financial institutions.Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

(e) Find a 90% confidence interval for μ, the average annual profit per employee for all successful banks. (Round your answers to two decimal places.)

lower limit	thousand dollars
upper limit	thousand dollars

(f) Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful financial institutions? Explain by referring to the confidence interval you computed in part (e).

Yes. This confidence interval suggests that the bank profits are less than those of other financial institutions.Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.    No. This confidence interval suggests that the bank profits are less than those of other financial institutions.No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

(g) Suppose the annual profits are more than 40 thousand dollars per employee. As manager of the bank, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (e).

No. This confidence interval suggests that the bank profits are higher than those of other financial institutions.No. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.    Yes. This confidence interval suggests that the bank profits are higher than those of other financial institutions.Yes. This confidence interval suggests that the bank profits do not differ from those of other financial institutions.

Answer 1

a) Sample Mean,    x̅ = ΣX/n =    33.95

b)

Level of Significance ,    α =    0.25          
'   '   '          
z value=   z α/2=   1.150   [Excel formula =NORMSINV(α/2) ]      
                  
Standard Error , SE = σ/√n =   9.4000   / √   42   =   1.4505
margin of error, E=Z*SE =   1.1503   *   1.4505   =   1.6685
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    33.95   -   1.668526   =   32.2815
Interval Upper Limit = x̅ + E =    33.95   -   1.668526   =   35.6185
75%   confidence interval is (   32.28   < µ <   35.62   )

c) Yes. This confidence interval suggests that the bank profits are less than those of other financial institutions.

d)  Yes. This confidence interval suggests that the bank profits are higher than those of other financial institutions.

e)

Level of Significance ,    α =    0.1          
'   '   '          
z value=   z α/2=   1.645   [Excel formula =NORMSINV(α/2) ]      
                  
Standard Error , SE = σ/√n =   9.4000   / √   42   =   1.4505
margin of error, E=Z*SE =   1.6449   *   1.4505   =   2.3858
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    33.95   -   2.385780   =   31.5642
Interval Upper Limit = x̅ + E =    33.95   -   2.385780   =   36.3358
90%   confidence interval is (   31.56   < µ <   36.34   )

f)  Yes. This confidence interval suggests that the bank profits are less than those of other financial institutions.

g) Yes. This confidence interval suggests that the bank profits are higher than those of other financial institutions