Transcript Chapter 9:Statistical Inference

Chapter 9:
Statistical Inference:
Hypothesis Testing for
Single Populations
Learning Objectives
LO1
Develop both one- and two-tailed null and alternative
hypotheses that can be tested in a business setting by
examining the rejection and non-rejection regions in light of
Type I and Type II errors.
LO2
Reach a statistical conclusion in hypothesis testing problems
about a population mean with a known population standard
deviation using the z statistic.
LO3
Reach a statistical conclusion in hypothesis testing problems
about a population mean with an unknown population
standard deviation using the t statistic.
Learning Objectives
LO4
Reach a statistical conclusion in hypothesis testing problems
about a population proportion using the z statistic.
LO5
Reach a statistical conclusion in hypothesis testing problems
about a population variance using the chi-square statistic.
LO6
Solve for possible Type II errors when failing to reject the null
hypothesis.
Importance of Hypothesis Testing in Decision
Making
• It is one of the most important mechanisms for decision
making
• It reduces the risk of intuitive guessing by using evidence
based decision making
• Hypothesis testing is central to inferential statistics
• With this tool business researchers can structure problems in
order to be able to use statistical evidence to test various
theories about business phenomena
• It is widely used in business to determine if production
processes are functioning efficiently and effectively (or are out
of control); or if new management approaches are making a
significantly positive difference
Types of Hypotheses
• Research Hypothesis
– A statement of what the researcher believes will be the outcome of an
experiment or a study
• Statistical Hypotheses
– A formal structure used to scientifically test the research hypothesis
• Substantive Hypotheses
– Conclusion reached on the basis of the data obtained and used in the
study.
– It may merely be a statistical outcome strongly and logically suggested
by the data available and the technique (models) used.
– Thus, a statistically significant difference may not imply a material, or
substantive difference for a business situation.
•
Warning
– Since what is mathematically significant may not be so in the business sense,
business decision makers must exercise care in interpreting statistical results.
Example of Research Hypotheses
• Older workers are more loyal to a company.
• Companies with more than $1 billion of assets spend a higher
percentage of their annual budget on advertising than do
companies with less than $1 billion of assets.
• The price of scrap metal is a good indicator of the industrial
production index six months later.
HTAB System to Test Hypotheses
Statistical Procedures for Hypothesis Testing
Step 1. Establish a null and an alternative hypothesis.
Step 2. Determine the appropriate statistical test.
Step 3. Set the value of alpha, the level of significance.
Step 4. Establish the decision rule.
Step 5. Gather sample data.
Step 6. Analyze the data.
Step 7. Reach a statistical conclusion.
Step 8. Make a business decision.
Type I and Type II Errors
• Type I Error
– Committed by a decision to reject a true null hypothesis
– This can happen when by chance the sample is not representative of
the population
– The probability of committing a Type I error is called α (alpha), the
level of significance.
• Type II Error
– Committed by failing to reject a false null hypothesis
– In this case the null hypothesis is really false but a decision is made
not to reject it.
– The probability of committing a Type II error is called (beta).
Decision Table
for Hypothesis Testing
Statistical Hypotheses
Step 1: Establish a null and an alternative hypothesis.
– A Null hypothesis: The null condition assumed to be true.
– An Alternative Hypothesis: Declares something new is
happening.
– Notation
• Null:
H0
• Alternative: Ha
Null and Alternative Hypotheses
• The Null and Alternative Hypotheses are mutually exclusive.
Only one of them can be true.
• The Null and Alternative Hypotheses are collectively
exhaustive. They are stated to include all possibilities. (An
abbreviated form of the null hypothesis is often used.)
• The Null Hypothesis is assumed to be true.
• The burden of proof falls on the Alternative Hypothesis.
Algebraic Expressions of the Null Hypothesis
• Note that the Null Hypothesis (H0) always has an equal sign in
the algebraic form of the expression
• The Alternative Hypothesis (Ha) always has the algebraic sign of
less than, greater than or not equal
Examples:
H0: µ = 2
Ha: µ ≠ 2
or
H0: µ ≥ 2
Ha: µ ˂ 2
or
H0: µ ≤ 2
Ha: µ > 2
Null and Alternative Hypotheses: Example
• A manufacturer is filling 2 kg packages with flour.
• The company wants the package contents to average 2
kilograms.
Null and Alternative Hypotheses: Example
• Because of an increased marketing effort, company officials
believe the company’s market share is now greater than 18%,
and the officials would like to prove it.
HTAB Paradigm: Task 2
Rejection and Nonrejection Regions
• Conceptually and graphically, statistical outcomes that result
in the rejection of the null hypothesis lie in what is termed the
rejection region.
• Statistical outcomes that fail to result in the rejection of the
null hypothesis lie in what is termed the nonrejection region.
Possible Rejection and Nonrejection Regions
• There are three possibilities which can be stipulated in the
alternative hypothesis.
• The three possibilities are: >, <, or ≠.
• The rejection regions for these possibilities, if a standard
normal distribution is used for the test statistic, are shown on
the next slide.
Possible Rejection and Nonrejection Regions
Rejection region
for hypothesis
that involve the
standard
normal
distribution and
the > symbol
(right–tailed
test)
Possible Rejection and Nonrejection Regions
Rejection region
for hypothesis
that involve the
standard normal
distribution and
the < symbol
(left–tailed test)
Possible Rejection and Nonrejection Regions
Rejection region
for hypothesis
that involve the
standard normal
distribution and
the  symbol
(two–tailed test)
Testing Hypotheses about a Population Mean
Using the z Statistic ( Known)
Example:
•A survey of chartered accountants, done more than 15
years ago, found that their average salary was $74,914. An
accounting researcher would like to test whether this
average has changed over the years. A sample of 112 CAs
produced a mean salary of $78,695. Assume that the
population standard deviation of salaries  = $14,530.
Testing Hypotheses about a Population Mean
Using the z Statistic ( Known)
• Step 1: Hypothesize
• Step 2: Test
Testing Hypotheses about a Population Mean
Using the z Statistic ( Known)
• Step 3: Specify the Type I error rate
–  = 0.05  z/2 = 1.96
• Step 4: Establish the decision rule
– Reject H0 if the test statistic < -1.96 or it the test
statistic > 1.96.
Testing Hypotheses about a Population Mean
Using the z Statistic ( Known)
• Step 5: Gather sample data
x-bar = $78,695, n = 112,  = $14,530,
hypothesized  = $74,914.
• Step 6: Compute the test statistic.
Testing Hypotheses about a Population Mean
Using the z Statistic ( Known)
• Step 7: Reach a statistical conclusion
– Since z = 2.75 > 1.96, reject H0.
• Step 8: Business decision
– Statistically, the researcher has enough evidence to reject
the figure of $74,914 as the true average salary for CAs. In
addition, based on the evidence gathered, it may suggest
that the average has increased over the 10-year period.
Testing Hypotheses about a Population Mean Using the z
Statistic ( Known) from a Finite Population
• Test statistic:
Using the p-Value to Test Hypotheses
• Another way to reach a statistical conclusion in hypothesis
testing problems is to use the p-value, sometimes referred to
as the observed significance level.
p-value < ,  reject H0
p-value  ,  do not reject H0
Using the p-Value to Test Hypotheses
• One should be careful when using p-values from statistical
software outputs.
• Both Minitab and Excel report the actual p-values for
hypothesis tests.
• Minitab doubles the p-value for a two-tailed test so you can
compare with .
• Excel does not double the p-value for a two-tailed test. So
when using the p-value from Excel, you may multiply the value
by 2 and then compare with .
Demonstration Problem: Minitab
Using the p-Value to Test Hypotheses
Critical Value Method to Test Hypotheses
• The critical value method determines the critical mean value
required for z to be in the rejection region and uses it to test
the hypotheses.
Critical Value Method to Test Hypotheses
• For the previous example,
Critical Value Method to Test Hypotheses
• Thus, a sample mean greater than $77,605 or less than
$72,223 will result in the rejection of the null hypothesis.
• This method is particularly attractive in industrial settings
where standards can be set ahead of time and then quality
control technicians can gather data and compare actual
measurements of products to specifications.
Testing Hypotheses about a Population Mean
Using the t Statistic ( Unknown)
• In this case, the test statistic will be
Two-tailed Test:  Unknown,  = .05
Example:
Masses in Kilograms of a Sample of 20 Plates
22.6
27.0
26.2
25.8
22.2 23.2
26.6 28.1
25.3 23.1
30.4 28.6
27.4
26.9
24.2
23.5
24.5
24.9
26.1
23.6
Two-tailed Test:  Unknown,  = .05
-2.093 < t= 1.04 <2.093,
therefor the statistical
conclusion is to fail to reject
the null hypothesis. It
appears that the machine is
not out of control.
MINITAB Computer Printout
for the Machine Plate Example
Machine Plate Example: Excel
(Part 1)
Machine Plate Example: Excel
(Part 2)
z Test of Population Proportion
z Test of Population Proportion
• A manufacturer believes exactly 8% of its products contain at
least one minor flaw. Suppose the company wants to test this
belief. A sample of 200 products resulted in 33 items have at
least one minor flaw. Use a probability of a Type I error of
0.10.
Testing Hypotheses about a Proportion:
Manufacturer Example (Part 2)
Minitab Computer Printout
for the Minor Flaw Example
Using the Critical Value Method
Testing Hypotheses About a Variance
• The test statistic for this test is
Testing Hypotheses About a Variance:
Demonstration Problem 9.4
A small business has 37 employees. Because of the uncertain demand for
its product, the company usually pays overtime in any given week. The
company assumed that about 50 total hours of overtime per week are
required and that the variance on this figure is about 25. Company officials
want to know whether the variance of overtime hours has changed. Given
here is a sample of 16 weeks of overtime data (in hours per week). Assume
hours of overtime are normally distributed. Use these data to test the null
hypothesis that the variance of overtime data is 25. Let α = 0.10.
57 56 52 44
46 53 44 44
48 51 55 48
63 53 51 50
Testing Hypotheses About a Variance:
Demonstration Problem 9.4
• Step 1:
• Step 2: Test statistic
Testing Hypotheses About a Variance:
Demonstration Problem 9.5
• Step 3: Because this is a two-tailed test,  = 0.10 and /2 =
0.05.
• Step 4: The degrees of freedom are 16 – 1 = 15. The two
critical chi-square values are 2(1 – 0.05), 15 = 2 0.95, 15 =
7.26093 and 2 0.05, 15 = 24.9958.
• Step 5: The data are listed in the text & previous slide.
• Step 6: The sample variance is s2 = 28.06. The observed chisquare value is calculated as 2 = 16.84.
Testing Hypotheses About a Variance:
Demonstration Problem 9.4
• Step 7: The observed chi-square value is in the non-rejection
region because
• 2 0.05, 15 = 7.26094 < 2observed = 16.84 < 2 0.05, 15 = 24.9958
• Step 8: This result indicates to the company managers that
the variance of weekly overtime hours is about what they
expected.
Solving for Type II Errors
• A Type II error can be committed only when the researcher
fails to reject the null hypothesis and the null hypothesis is
false.
• A Type II error, β, varies with possible values of the
alternative parameter.
• Often, when the null hypothesis is false, the value of the
alternative mean is unknown, so the researcher will
compute the probability of committing Type II errors for
several possible values. How can the probability of
committing a Type II error be computed for a specific
alternative value of the mean?
Solving for Type II Errors for
Small Machine Parts Example
• Suppose a test is conducted on the following hypotheses: H0:
= 12 ounces vs. Ha: < 12 g when the sample size is 60 with
mean of 11.985 and a standard deviation of 0.10g.
• The first step in determining the probability of a Type II error:
calculate a critical value for the sample mean for this case.
• For  =0.05, the critical value for the sample mean is
Graphic for Type II Error:
Machine Parts Example
Solving for Type II Errors:
Small Machine Parts
• The decision to accept or reject
the null hypothesis is being
made on the assumption that the
critical region is to the left of z = 1.645 where µ = 12, σ = 10.
• However, the distribution of the
population from which the
samples come may be different.
Solving for Type II Errors:
Small Machine Parts
• Assume that the alternative mean
µ = 11.99 . Since a Type II error, ,
varies with possible values of the
alternative parameter, then for an
alternative mean of 11.99 (< 12)
the corresponding z-value is -0.85
• The probability of committing a
Type II error is all the area to the
right of 11.979 in distribution (b),
or 0.3023 + 0.5000 = 0.8023.
• Hence, there is an 80.23% chance
of committing a Type II error if the
alternative mean is 11.99 g.
Type II Error When the Difference Between the Alternative Mean
and the Hypothesized Mean Increases
Demonstration Problem 9.5.
• Let the alternative mean now
be 11.96, to the left of
11.979.
• Note that the new Critical Z1
is 1.47 and that β = 0.0798 <
the previous value of β =
0.8023.
• Thus the probability of
committing a type II error is
only 0.0708.
Demonstration Problem 9.6
Proportions Example
• Suppose you are conducting a two-tailed hypothesis test of
proportions. The null hypothesis is that the population
proportion is 0.40. The alternative hypothesis is that the
population proportion is not 0.40. A random sample of 250
produces a sample proportion of 0.44. With alpha of 0.05, the
table z value for α/2 is ±1.96. The observed z from the
sample information is:
The critical values are 0.34 on the lower
end and 0.46 on the upper end.
Demonstration Problem 9.6
Proportions Example
• Suppose the alternative
population proportion is really
0.36. What is the probability of
committing a Type II error?
• Solving for the area between
• and p1 = 0.36 yields:
• The area to the right of -0.66 is
0.2454 + 0.4995 = 0.7449, the
probability of committing a type
II error.
Operating Characteristic and
Power Curve
• Because the probability of committing a Type II error changes
for each possible different value of the alternative parameter,
it is best to examine a series of possible alternative values.
• The power of a test is (1 - ) It is the probability of rejecting
the null hypothesis when it is false.
• The series of values of (1-β) resulting from changes in the
alternative value of the parameter are called the power curve
• The series of values of  resulting from the changes in the
alternate value of the parameters are called the operating
curve.
Operating Characteristic and Power Curves
for Small Parts Machines Example
Note that as the alternative means get closer to the hypothesized
mean of µ =12g that the probability of a type II error is greater: it is
difficult to discriminate between distributions with means of 12 and
11.999. The reverse is true when the alternative mean is vey
different, for example 11.95. The calculations below show power
curves and operating curves for µ = 12g and α = 0.05.
Operating Characteristic Curve Small Parts
Machines Problem
0.9
0.8
Type II Error
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
11.95
11.96
11.97
11.98
Alternative Means
11.99
12.00
The Power Curve for Small Parts Machines
Problem
1.0
Power
0.8
0.6
0.4
0.2
0.0
11.95
11.96
11.97
11.98
Alternative Means
11.99
12.00
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