Question

In order to properly manage expenses, the company investigates the amount of money spent by its sales office. The below numbers are related to six randomly selected receipts provided by the staff.

$147 $124 $93 $158 $164 $171

a) Calculate x ̅ , s2 and s for the expense data.

b) Assume that the distribution of expenses is approximately normally distributed. Calculate estimates of tolerance intervals containing 68.26 percent, 95.44 percent, and 99.73 percent of all expenses by the sales office.

c) If a member of the sales office submits a receipt with the amount of $190, should this expense be considered unusually high? Explain your answer.

d) Compute and interpret the z-score for each of the six expenses.

Answer 1

Observation Table:

	Xi	(Xi-)	(Xi-)2
	147	4.17	17.36111
	124	-18.83	354.6944
	93	-49.83	2483.361
	158	15.17	230.0278
	164	21.17	448.0278
	171	28.17	793.3611
Total	857		4326.83

a)

S² = 29.42² = 865.37

b)

68.26% expenses would fall within 1 standard deviation from the mean:

= 142.83 1*(29.42) = ($113.42, $172.25)

95.44% expenses would fall within 2 standard deviation from the mean:

= 142.83 2*(29.42) = ($84,$201.67)

99.73% expenses would fall within 1 standard deviation from the mean:

= 142.83 3*(29.42) = ($54.6 , $231.1)

C)

No, An expense of $190 is considered unusually high, because it lies in the interval of 3 standard deviation of the mean.

d)

Z score for 147 is,

Z = (X - ) /

Z = (147 - 142.83) / 29.42

Z = 0.14

An expense of $147 is 0.14 standard deviation above the average expense.

Z score for 124 is,

Z = (X - ) /

Z = (124 - 142.83) / 29.42

Z = -0.64

An expense of $124 is 0.64 standard deviation below the average expense.

Z score for 93 is,

Z = (X - ) /

Z = (93 - 142.83) / 29.42

Z = -1.69

An expense of $93 is 1.69 standard deviation below the average expense.

Z score for 158 is,

Z = (X - ) /

Z = (158 - 142.83) / 29.42

Z = 0.52

An expense of $158 is 0.52 standard deviation above the average expense.

Z score for 164 is,

Z = (X - ) /

Z = (164 - 142.83) / 29.42

Z = 0.72

An expense of $164 is 0.72 standard deviation above the average expense.

Z score for 171 is,

Z = (X - ) /

Z = (171 - 142.83) / 29.42

Z = 0.96

An expense of $171 is 0.96 standard deviation above the average expense.