Week 7
October 13-17
Three Mini-Lectures
QMM 510
Fall 2014
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Chapter 8
Confidence Interval
For a Proportion ()
ML 7.1
A proportion is a mean of data whose only values are 0 or 1.
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Sample of 100 Binary Outcomes
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Number of "successes"
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1's
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8-2
Chapter 8
Confidence Interval for a Proportion ()
Applying the CLT
• The distribution of a sample proportion p = x/n is symmetric if  = .50.
• The distribution of p approaches normal as n increases, for any .
8-3
Chapter 8
Confidence Interval for a Proportion ()
When Is It Safe to Assume Normality of p?
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Rule of Thumb: The sample proportion p = x/n may be assumed to be
normal if both n  10 and n(1)  10.
Sample size to assume
normality:
Rule: It is safe to assume
normality of p = x/n if we have at
least 10 “successes” and 10
“failures” in the sample.
Table 8.9
8-4
Chapter 8
Confidence Interval for a Proportion ()
Confidence Interval for 
•
The confidence interval for the unknown  (assuming a large sample) is
based on the sample proportion p = x/n.
8-5
Chapter 8
Confidence Interval for a Proportion ()
Example: Auditing
8-6
N = population size; n = sample size
• The FPCF narrows the confidence interval somewhat.
• When the sample is small relative to the population, the FPCF has little
effect. If n/N < .05, it is reasonable to omit it (FPCF  1 ).
8-7
Chapter 8
Estimating from Finite Population
ML 7-2
Chapter 8
Sample Size Determination
Sample Size to Estimate m
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8-8
To estimate a population mean with a precision of + E (allowable error),
you would need a sample of size n. Now,
How to Estimate s?
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Method 1: Take a Preliminary Sample
Take a small preliminary sample and use the sample s in place of s in the
sample size formula.
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Method 2: Assume Uniform Population
Estimate rough upper and lower limits a and b and set s = [(b  a)/12]½.
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Method 3: Assume Normal Population
Estimate rough upper and lower limits a and b and set s = (b  a)/4.
This assumes normality with most of the data within m ± 2s so the
range is 4s.
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Method 4: Poisson Arrivals
In the special case when m is a Poisson arrival rate, then s =  m .
8-9
Chapter 8
Sample Size Determination for a Mean
Using MegaStat
For example, how large a sample is needed to estimate the population mean
age of college professors with 95 percent confidence and precision of ± 2
years, assuming a range of 25 to 70 years (i.e., 2 years allowable error)? To
estimate σ, we assume a uniform distribution of ages from 25 to 70:
(70  25) 2
s
 13
12
8-10
z 2s 2 (1.96) 2 (13) 2
n 2 
 163
E
22
Chapter 8
Sample Size Determination for a Mean
8-11
•
To estimate a population proportion with a precision of ± E (allowable
error), you would need a sample of size n.
•
Since  is a number between 0 and 1, the allowable error E is also
between 0 and 1.
Chapter 8
Sample Size Determination for a Mean
How to Estimate ?
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Method 1: Assume that  = .50
This conservative method ensures the desired precision. However, the
sample may end up being larger than necessary.
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Method 2: Take a Preliminary Sample
Take a small preliminary sample and use the sample p in place of  in the
sample size formula.
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Method 3: Use a Prior Sample or Historical Data
How often are such samples available?  might be different enough to
make it a questionable assumption.
8-12
Chapter 8
Sample Size Determination for a Mean
Using MegaStat
For example, how large a sample is needed to estimate the population
proportion with 95 percent confidence and precision of ± .02 (i.e., 2%
allowable error)?.
z 2 (1   ) (1.96) 2 (.50)(1  .50)
n
8-13
E2

(.02) 2
 2401
Chapter 8
Sample Size Determination for a Mean
ML 7-3
Learning Objectives
LO9-1: List the steps in testing hypotheses.
LO9-2: Explain the difference between H0 and H1.
LO9-3: Define Type I error, Type II error, and power.
LO9-4: Formulate a null and alternative hypothesis for μ or π.
9-14
Chapter 9
One-Sample Hypothesis Tests
Chapter 9
Logic of Hypothesis Testing
9-15
Chapter 9
Logic of Hypothesis Testing
LO9-2: Explain the difference between H0 and H1.
State the Hypothesis
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Hypotheses are a pair of mutually exclusive, collectively exhaustive
statements about some fact about a population.
One statement or the other must be true, but they cannot both be
true.
H0: Null hypothesis
H1: Alternative hypothesis
These two statements are hypotheses because the truth is
unknown.
9-16
Chapter 9
Logic of Hypothesis Testing
State the Hypothesis
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Efforts will be made to reject the null hypothesis.
If H0 is rejected, we tentatively conclude H1 to be the case.
H0 is sometimes called the maintained hypothesis.
H1 is called the action alternative because action may be required if we
reject H0 in favor of H1.
Can Hypotheses Be Proved?
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We cannot accept a null hypothesis; we can only fail to reject it.
Role of Evidence
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The null hypothesis is assumed true and a contradiction is sought.
9-17
LO9-3: Define Type I error, Type II error, and power.
Types of Error
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9-18
Type I error: Rejecting the null hypothesis when it is true. This occurs
with probability a (level of significance). Also called a false positive.
Type II error: Failure to reject the null hypothesis when it is false. This
occurs with probability b. Also called a false negative.
Chapter 9
Logic of Hypothesis Testing
Probability of Type I and Type II Errors
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9-19
If we choose a = .05, we expect to commit a Type I error about
5 times in 100.
b cannot be chosen in advance because it depends on a and
the sample size.
A small b is desirable, other things being equal.
Chapter 9
Logic of Hypothesis Testing
Power of a Test
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9-20
A low b risk means high power.
Larger samples lead to increased power.
Chapter 9
Logic of Hypothesis Testing
Relationship between a and b
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9-21
Both a small a and a small b are desirable.
For a given type of test and fixed sample size, there is a trade-off
between a and b.
The larger critical value needed to reduce a risk makes it harder to
reject H0, thereby increasing b risk.
Both a and b can be reduced simultaneously only by increasing the
sample size.
Chapter 9
Logic of Hypothesis Testing
Chapter 9
Logic of Hypothesis Testing
Consequences of Type I and Type II Errors
• The consequences of these two errors are quite different, and the
are borne by different parties.
costs
• Example: Type I error is convicting an innocent defendant, so the costs
are borne by the defendant. Type II error is failing to convict a guilty
defendant, so the costs are borne by society if the guilty person returns
to the streets.
• Firms are increasingly wary of Type II error (failing to recall a product as
soon as sample evidence begins to indicate potential problems.)
9-22
LO9-4: Formulate a null and alternative hypothesis for μ or π.
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9-23
Chapter 9
Statistical Hypothesis Testing
A statistical hypothesis is a statement about the value of a population parameter.
A hypothesis test is a decision between two competing mutually exclusive and
collectively exhaustive hypotheses about the value of the parameter.
When testing a mean we can choose between three tests.
One-Tailed and Two-Tailed Tests
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The direction of the test is indicated by H1:
> indicates a right-tailed test
< indicates a left-tailed test
≠ indicates a two-tailed test
9-24
Chapter 9
Statistical Hypothesis Testing
Decision Rule
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9-25
A test statistic shows how far the sample estimate is from its expected
value, in terms of its own standard error.
The decision rule uses the known sampling distribution of the test
statistic to establish the critical value that divides the sampling
distribution into two regions.
Reject H0 if the test statistic lies in the rejection region.
Chapter 9
Statistical Hypothesis Testing
Decision Rule for Two-Tailed Test
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9-26
Reject H0 if the test statistic < left-tail critical value or if the test
statistic > right-tail critical value.
Chapter 9
Statistical Hypothesis Testing
When to use a One- or Two-Sided Test
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9-27
Chapter 9
Statistical Hypothesis Testing
A two-sided hypothesis test (i.e., µ ≠ µ0) is used when direction (< or >)
is of no interest to the decision maker.
A one-sided hypothesis test is used when
- the consequences of rejecting H0 are
asymmetric, or
- where one tail of the distribution is of special
importance to the researcher.
Rejection in a two-sided test guarantees rejection in a one-sided test,
other things being equal.
Chapter 9
Statistical Hypothesis Testing
Decision Rule for Left-Tailed Test
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Reject H0 if the test statistic < left-tail critical value.
Figure 9.2
9-28
Chapter 9
Statistical Hypothesis Testing
Decision Rule for Right-Tailed Test
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Reject H0 if the test statistic > right-tail critical value.
9-29
Type I Error
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9-30
also called a false positive
A reasonably small level of significance a is desirable, other things
being equal.
Chosen in advance, common choices for a are .10, .05, .025, .01, and
.005 (i.e., 10%, 5%, 2.5%, 1%, and .5%).
The a risk is the area under the tail(s) of the sampling distribution.
In a two-sided test, the a risk is split with a/2 in each tail since there
are two ways to reject H0.
Chapter 9
Statistical Hypothesis Testing