When an audit must be conducted that involves a tedious examination of a large inventory, the audit may be very costly and time consuming if each item in the inventory must be examined. In such situations, the auditor frequently obtains a random sample of items from the complete inventory and uses the results of an audit of the sampled items to check the validity of the company's financial statement. A large company’s financial statement claims an inventory that averages $600 per item. The following data are the auditor’s assessment of a random sample of 75 items from the company’s inventory. The values resulting from the audit are rounded to the nearest dollar. 303 547 1,368 493 984 507 148 2,546 738 83 2 135 274 74 1,472 399 1,784 71 751 136 571 147 282 2,039 1,909 748 188 548 1 280 102 618 129 1,324 1,428 469 102 454 1,059 939 303 600 234 514 17 551 293 1,395 7 28 2 973 506 511 812 1,290 685 447 11 35 252 1,526 464 5 67 99 67 259 7 67 248 3,215 3 33 41 Estimate the mean value of an item in the inventory using a 95% confidence interval. Is there substantial evidence (a 5 .01) that the mean value of an item in the inventory is less than $600? What is the target population for the above inferences? Would normal distribution–based procedures be appropriate for answering the above questions?

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Description: 5.6 When an audit must be conducted that involves a tedious examination of a large inventory, the audit may be very costly and time consuming if each item in the inventory must be examined. In such situations, the auditor frequently obtains a random

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5.6 When an audit must be conducted that involves a tedious examination of a large inventory, the audit may be very costly and time consuming if each item in the inventory must be examined. In such situations, the auditor frequently obtains a random sample of items from the complete inventory and uses the results of an audit of the sampled items to check the validity of the company's financial statement. A large company’s financial statement claims an inventory that averages $600 per item. The following data are the auditor’s assessment of a random sample of 75 items from the company’s inventory. The values resulting from the audit are rounded to the nearest dollar.

303	547	1,368	493	984	507	148	2,546	738	83	2	135	274	74	1,472
399	1,784	71	751	136	571	147	282	2,039	1,909	748	188	548	1	280
102	618	129	1,324	1,428	469	102	454	1,059	939	303	600	234	514	17
551	293	1,395	7	28	2	973	506	511	812	1,290	685	447	11	35
252	1,526	464	5	67	99	67	259	7	67	248	3,215	3	33	41

Estimate the mean value of an item in the inventory using a 95% confidence interval.
Is there substantial evidence (a 5 .01) that the mean value of an item in the inventory is less than $600?
What is the target population for the above inferences?
Would normal distribution–based procedures be appropriate for answering the above questions?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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