Chapter 9: Introduction to Hypothesis Testing

400 CHAPTER 9 • INTRODUCTION TO HYPOTHESIS TESTING
9.2 Hypothesis Tests for Proportions
So far, this chapter has focused on hypothesis tests about a single population mean. Although
many decision problems involve a test of a population mean, there are also cases in which the
parameter of interest is the population proportion. For example, a production manager might
consider the proportion of defective items produced on an assembly line in order to determine
whether the line should be restructured. Likewise, a life insurance salesperson’s performance
assessment might include the proportion of existing clients who renew their policies.
Business
Application
Requirement
Testing a Hypothesis About a Single Population Proportion
The basic concepts of hypothesis testing for proportions are the same as for means.
1. The null and alternative hypotheses are stated in terms of a population parameter,
now instead of , and the sample statistic becomes p instead of x.
2. The null hypothesis should be a statement concerning the parameter that includes the
equality.
3. The significance level of the hypothesis determines the size of the rejection region.
4. The test can be one- or two-tailed, depending on how the alternative hypothesis is
formulated.
FIRST AMERICAN BANK AND TITLE The internal auditors at First American Bank and
Title Company routinely test the bank’s system of internal controls. Recently, the audit
manager examined the documentation on the bank’s 22,500 outstanding automobile loans.
The bank’s procedures require that the file on each auto loan account contain certain specific
documentation, such as a list of applicant assets, statement of monthly income, list of
liabilities, and certificate of automobile insurance. If an account contains all the required
documentation, then it complies with bank procedures.
The audit manager has established a 1% noncompliance rate as the bank’s standard. If
more than 1% of the 22,500 loans do not have appropriate documentation, then the internal
controls are not effective and the bank needs to improve the situation. The audit staff does
not have enough time to examine all 22,500 files to determine the true population
noncompliance rate. As a result, the audit staff selects a random sample of 600 files, examines
them, and determines the number of files not in compliance with bank documentation
requirements. The sample findings will tell the manager if the bank is exceeding the 1%
noncompliance rate for the population of all 22,500 loan files. The manager will not act
unless the noncompliance rate exceeds 1%. The default position is that the internal controls
are effective. Thus, the null and alternative hypotheses are
H 0
: 0.01 (Internal controls are effective.)
H A
: 0.01 (Internal controls are not effective.)
Suppose the sample of 600 accounts uncovered 9 files with inadequate loan documentation.
The question is whether 9 out of 600 is sufficient to conclude that the bank has a
problem. To answer this question statistically, we need to recall a lesson from Chapter 7.
The sample size, n, is large such that n 5 and n(1 ) 5. 6
If this requirement is satisfied, the sampling distribution is approximately normal with
mean and standard deviation ( 1)/
n .
The bank’s auditors have a general policy of performing these tests with a significance
level of
0.02
6 A paper published in Statistical Science by L. Brown et. al. entitled “Interval Estimation for a Binomial
Proportion” in 2001, pp. 101–133, suggests that the requirement should be n 15 and n(1 − ) 15.
However, most sources still use the 5 limit.