In order to control costs, a company wishes to study the amount of money its sal
ID: 3352879 • Letter: I
Question
In order to control costs, a company wishes to study the amount of money its sales force spends entertaining clients. The following is a random sample of six entertainment expenses (dinner costs for four people) from expense reports submitted by members of the sales force.
$357 $332 $309 $345 $325 $339
a. Calculate sample mean, s^2, and s for the expense data. In addition, show that the two different formulas for caluating s^2 give the same result
b. Assuming that thew distribution of entertainment expenses is approximately normally distributed, calulate estimates of tolerance intervals containing 68.26 percent, 95.44 percent, and 99.73 percent of all entertainment expenses by the sales force.
c. If a member of the sales force submits an entertainment expense (dinner cost for four) of $390 should this expense be considered unusually high (and possibly worthy of investigation by the company)? explain your answer.
d. Compute and interpret the z-score for each of the six entertainment expenses.
Explanation / Answer
357 332 309 345 325 339
a)Mean = bar x = (357+332+309+345+325+339)/6 =334.5
s^2 = [sum((x – bar x)^2)] / n
x
357
332
309
345
325
339
x - bar x
22.5
-2.5
-25.5
10.5
-9.5
4.5
(x - bar x)^2
506.25
6.25
650.25
110.25
90.25
20.25
Thus, s^2 = 1383.5/6 = 230.58
Thus, s = sqrt(230.58) = 15.18
Alternate formula is,
S^2 = E(x^2) – (E(x))^2
= (357^2+332^2+..+339^2)/6 – (334.5)^2
= 230.58
Hence it’s same by any of the above two formulas
b)
For tolerance level 68.26 percent calculate z for which Probability is (1- 0.6828)/2 (as both sided) from table and calculate confident interval range
For 68.28% z = -1 and z = +1 (for P = (1-.6828)/2 and (1+0.6828)/2)
Thus, Range = z.s +mean
= (-1)15.18 + 334.5 to (1)15.18 +334.5
= 319.32 to 349.68
Similarly solve others using the same formula by changing value of z according to tolerance percent using table
c) use the formula for z, If value of z is greater than 3 or less than -3 then unexpected value
z = (x – ) / s, Here, X = 390, =334.5 and s=15.18
d) simply use formula for z and solve
z = (x – ) / s. use this formula 6 times for all values of x
x
357
332
309
345
325
339
x - bar x
22.5
-2.5
-25.5
10.5
-9.5
4.5
(x - bar x)^2
506.25
6.25
650.25
110.25
90.25
20.25
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