Statistics for Business and Economics (13e)
Slides by
Statistics
for
John
Business
and Economics (13e)
Loucks
Anderson,St.
Sweeney,
Williams, Camm, Cochran
Edward’s
University
© 2017 Cengage Learning
Slides by John Loucks
St. Edwards University
(Modifications/additions by Reid Kerr)
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
1
Statistics for Business and Economics (13e)
Chapter 9
Hypothesis Testing
• Developing Null and Alternative Hypotheses
• Type I and Type II Errors
• Population Mean: s Known
• Population Mean: s Unknown
• Population Proportion
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
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Statistics for Business and Economics (13e)
Hypothesis Testing
• Hypothesis testing can be used to determine whether a statement about the
value of a population parameter should or should not be rejected.
• The null hypothesis, denoted by H0 , is a tentative assumption about a
population parameter.
• The alternative hypothesis, denoted by Ha, is the opposite of what is stated in
the null hypothesis.
• The hypothesis testing procedure uses data from a sample to test the two
competing statements indicated by H0 and Ha.
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3
What is this about… really?
• For the last several weeks, we have been exploring the relationship between
samples and populations
• We take a sample to learn about a population
• BUT, we know that the statistics from the sample (e.g., mean) probably aren’t exactly
the same as the numbers for the population
• ‘Sampling’: we looked at the possible sample results we could get, relative to a
population
• ‘Confidence intervals’: a bit more applicable, we looked at how accurate our
sample results were likely to be as an estimate of the population
• Here, we consider the same issues, but ask a different question:
• With this sample, is there enough evidence to ‘prove’ that something is true?
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Statistics for Business and Economics (13e)
Developing Null and Alternative Hypotheses
• It is not always obvious how the null and alternative hypotheses should be
formulated.
• Care must be taken to structure the hypotheses appropriately so that the test
conclusion provides the information the researcher wants.
• The context of the situation is very important in determining how the
hypotheses should be stated.
• In some cases it is easier to identify the alternative hypothesis first. In other
cases the null is easier.
• Correct hypothesis formulation will take practice.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Developing Null and Alternative Hypotheses
• Alternative Hypothesis as a Research Hypothesis
• Many applications of hypothesis testing involve an attempt to gather
evidence in support of a research hypothesis.
• In such cases, it is often best to begin with the alternative hypothesis and
make it the conclusion that the researcher hopes to support.
• The conclusion that the research hypothesis is true is made if the sample
data provides sufficient evidence to show that the null hypothesis can be
rejected.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Developing Null and Alternative Hypotheses
• Alternative Hypothesis as a Research Hypothesis
• Example:
A new teaching method is developed that is believed to be better than
the current method.
• Alternative Hypothesis:
The new teaching method is better.
• Null Hypothesis:
The new method is no better than the old method.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Developing Null and Alternative Hypotheses
• Alternative Hypothesis as a Research Hypothesis
• Example:
A new sales force bonus plan is developed in an attempt to increase
sales.
• Alternative Hypothesis:
The new bonus plan will increase sales.
• Null Hypothesis:
The new bonus plan will not increase sales.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Developing Null and Alternative Hypotheses
• Alternative Hypothesis as a Research Hypothesis
• Example:
A new drug is developed with the goal of lowering blood pressure more
than the existing drug.
• Alternative Hypothesis:
The new drug lowers blood pressure more than the existing drug.
• Null Hypothesis:
The new drug does not lower blood pressure more than the existing
drug.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Developing Null and Alternative Hypotheses
• Null Hypothesis as an Assumption to be Challenged
• We might begin with a belief or assumption that a statement about the
value of a population parameter is true.
• We then use a hypothesis test to challenge the assumption and determine
if there is statistical evidence to conclude that the assumption is incorrect.
• In these situations, it is helpful to develop the null hypothesis first.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
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More simply…
• When doing hypothesis testing, we start with TWO hypotheses
• ALTERNATE (or ALTERNATIVE) hypothesis (H1)– What we are trying to establish
• NULL hypothesis (H0) – Our starting assumption – We start by assuming the
‘opposite’ of what we are trying to establish
• (By ‘opposite’, we mean it includes ALL cases that are not part of the ALTERNATE
hypothesis)
Why do we need two hypotheses?
And why do we assume the opposite of what we are trying to prove?
• A close parallel is the taken in our criminal courts
• Prosecution is trying to prove the defendant guilty
• Do we start by assuming them guilty?
• If we did, how much evidence is needed for us to continue to believe they are guilty?
• Instead, we start by presuming ‘innocent until proven guilty’
• I.e., assuming the opposite!
• Only if we have enough evidence that our starting assumption couldn’t possibly
be true (‘proof beyond reasonable doubt’) do we conclude they are guilty
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Hypothesis testing
• We take the same approach here:
• We start by ASSUMING THE OPPOSITE IS TRUE
• Null hypothesis (H0)
• If we have enough evidence to show the null is wrong, then the opposite (i.e., what
we are trying to ‘prove’, H1, the ‘alternative’ to the null) is right!
• In statistics, our hypotheses are numerical
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Statistics for Business and Economics (13e)
Developing Null and Alternative Hypotheses
• Null Hypothesis as an Assumption to be Challenged
• Example:
The label on a soft drink bottle states that it contains 67.6 fluid ounces.
• Null Hypothesis:
The label is correct. m > 67.6 ounces.
• Alternative Hypothesis:
The label is incorrect. m < 67.6 ounces.
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Statistics for Business and Economics (13e)
Summary of Forms for Null and Alternative Hypotheses
• The equality part of the hypotheses always appears in the null hypothesis.
• In general, a hypothesis test about the value of a population mean m must
take one of the following three forms (where m0 is the hypothesized value of
the population mean).
𝐻0 : 𝜇 ≥ 𝜇0
𝐻𝑎 : 𝜇 < 𝜇0
𝐻0 : 𝜇 ≤ 𝜇0
𝐻0 : 𝜇 = 𝜇0
𝐻𝑎 : 𝜇 > 𝜇0
𝐻𝑎 : 𝜇 ≠ 𝜇0
One-tailed
(lower-tail)
One-tailed
(upper-tail)
Two-tailed
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Statistics for Business and Economics (13e)
Null and Alternative Hypotheses
• Example: Metro EMS
A major west coast city provides one of the most comprehensive
emergency medical services in the world. Operating in a multiple hospital
system with approximately 20 mobile medical units, the service goal is to
respond to medical emergencies with a mean time of 12 minutes or less.
The director of medical services wants to formulate a hypothesis test that
could use a sample of emergency response times to determine whether or
not the service goal of 12 minutes or less is being achieved.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Null and Alternative Hypotheses
H0: m < 12
The emergency service is meeting the response goal;
no follow-up action is necessary.
Ha: m > 12
The emergency service is not meeting the response
goal; appropriate follow-up action is necessary.
where: m = mean response time for the population
of medical emergency requests
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Type I Error
• Because hypothesis tests are based on sample data, we must allow
for the possibility of errors.
• A Type I error is rejecting H0 when it is true.
• The probability of making a Type I error when the null hypothesis is
true as an equality is called the level of significance.
• Applications of hypothesis testing that only control for the Type I error
are often called significance tests.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Type II Error
• A Type II error is accepting H0 when it is false.
• It is difficult to control for the probability of making a Type II error.
• Statisticians avoid the risk of making a Type II error by using “do not reject
H0” and not “accept H0”.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Type I and Type II Errors
• You might find the following a bit easier to understand/remember
• (And, it’s terminology that is common used in the real world)
• If H1 (alternative) is what we are trying to ‘prove’:
• Type I error is a false positive
• We accepted H1 (considered it positive), when it wasn’t really true!
• Type II error is a false negative
• We didn’t accept H1 (considered it negative), when it was really true!
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Statistics for Business and Economics (13e)
Type I and Type II Errors
Population Condition
Conclusion
H0 True
(m < 12)
H0 False
(m > 12)
Accept H0
(Conclude m < 12)
Correct
Conclusion
Type II Error
Type I Error
Correct
Conclusion
Reject H0
(Conclude m > 12)
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otherwise on a password-protected website or school-approved learning management system for classroom use.
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General Concept
• We have been talking about the idea of ‘enough evidence’ for awhile, but what is
statistical evidence, and what is ‘enough’?
• Remember, we are trying to answer questions about the population, based on
our sample
• Evidence is based on the concepts we have been learning over the past few weeks
• If the population mean is ___, what is the probability that we get a sample in
range ____?
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A simple example
• Suppose that I had an opaque bag, and told you the bag was full of M&Ms
• I claimed that the M&Ms were 50% red and 50% green
• Now, suppose that I asked you to put your hand into the bag, and take out 20
randomly selected M&Ms
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A simple example
• If you got 10 red and 10 green, would you believe that the bag was 50/50?
Would that provide much evidence that the bag was not 50/50?
• If the bag was really 50/50, it seems very likely that you could get 10 red/10 green
• High probability that it could happen
• Not much evidence that it isn’t true
• If you got 19 red and 1 green, would you believe that the bag was 50/50? Would
that provide much evidence that the bag was not 50/50?
• If the bag was really 50/50, it seems very unlikely that you could get 19 red/1 green
• Very low probability that it could happen
• Strong evidence that it isn’t true
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Framing this as a hypothesis test
• Suppose that I had an opaque bag, and told you the bag was full of M&Ms
• I claimed that the M&Ms were 50% red and 50% green
• You thought I might be lying, so wanted to prove otherwise
• H1: Bag is not 50/50
• (More formally, pred ≠ 0.5)
• H0: Bag IS 50/50
• (pred = 0.5)
• Now, suppose that I asked you to put your hand into the bag, and take out 20
randomly selected M&Ms
• Take a sample, and see whether the result we got was likely or unlikely, if the null
was true
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Statistics for Business and Economics (13e)
p-Value Approach to One-Tailed Hypothesis Testing
• The p-value is the probability, computed using the test statistic, that
measures the support (or lack of support) provided by the sample for the
null hypothesis.
• I.e., it tells us, if the null was true, what is the probability we could have
gotten a sample result like the one we got!
• If the p-value is less than or equal to the level of significance , the value of the
test statistic is in the rejection region.
• Reject H0 if the p-value <  .
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Suggested Guidelines for Interpreting p-Values
• Less than .01
Overwhelming evidence to conclude Ha is true.
• Between .01 and .05
Strong evidence to conclude Ha is true.
• Between .05 and .10
Weak evidence to conclude Ha is true.
• Greater than .10
Insufficient evidence to conclude Ha is true.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
27
Statistics for Business and Economics (13e)
Lower-Tailed Test About a Population Mean: s Known
• p-Value Approach
Sampling
Distribution of
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
 = .10
p-value
= .0721
p-Value <  ,
so reject H0.
z
z=
-1.46
z = 0
-1.28
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otherwise on a password-protected website or school-approved learning management system for classroom use.
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An Example
• You suspect that the average time spent on homework is less than 9 hours per
week. Based on some past data (?), you know that the population standard
deviation is 2.1. You take a sample of 40 students, and get a sample mean of 7.2.
Can you conclude, with 90% confidence, that the population average is less than
9 hours?
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An Example
• You suspect that the average time spent on homework is less than 9 hours per
week. Based on some past data (?), you know that the population standard
deviation is 2.1. You take a sample of 40 students, and get a sample mean of 7.2.
Can you conclude, with 90% confidence, that the population average is less than
9 hours?
H1: 𝜇 < 9
H0: 𝜇 >= 9
𝜎 = 2.1
𝑥 = 7.2
Confidence = .90
 = .10
n = 40
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Example
• If the null was really true (i.e., the population mean was really 9), what
is the probability that I could have gotten a sample mean at least as
extreme as 7.2?
Sampling
Distribution of
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
p-value
z
7.2
9
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P-value calculation
• Exactly the same as the calculation for sampling
• Two step method:
• Step 1:
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
7.2 − 9
𝑧=
2.1 40
𝑧 ≅ −5.421
• Step 2:
• Table lookup or norm.s.dist (remembering ‘less than’)
• p-value ≅ 0.000000030
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P-value calculation
• One step method:
• NORM.DIST(7.2, 9, 2.1/SQRT(40), TRUE) ≅ 0.000000030
• What does it mean?
• If the null was true (i.e., the population mean was 9), there is only a 0.000000030
probability that we could have gotten a sample as extreme as the one we got
• That’s pretty unlikely—the null is probably wrong
• Formal conclusion:
• p-value is less than alpha
• Therefore, we reject the null
• (I.e., we have enough evidence to support our H1), that the population mean is less than 9.
33
Statistics for Business and Economics (13e)
Upper-Tailed Test About a Population Mean: s Known
• p-Value Approach
Sampling
Distribution of
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
 = .04
p-Value
 .011
(p-Value <  ,
so reject H0.)
z
0
z =
1.75
z=
2.29
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otherwise on a password-protected website or school-approved learning management system for classroom use.
34
Upper Tailed on the Right
• Again, exactly the same as in the sampling cases
• Because we need the right tail (i.e., a ‘greater than’ case), we use 1-area from the
table/formula
• In Excel, 1-norm.dist
35
Example
• The success of your new product depends on people in your target market having
an income of at least $47,000. The population standard deviation is $7,827. In a
sample of 40 people, you find an average income of $48,123. Can we conclude,
with 95% confidence, that the average income in the population is greater than
$47,000?
H1: 𝜇 > 47000
H0: 𝜇 <= 47000
𝜎 = 7827
𝑥 = 48123
Confidence = .95
 = .05
n = 40
36
P-value calculation
• One step method:
• 1-NORM.DIST(48123, 47000, 7827/SQRT(40), TRUE) ≅ 0.182089016
• What does it mean?
• If the null was true (i.e., the population mean was 47000), there is only a
0.182089016 probability that we could have gotten a sample as extreme as the one
we got
• That’s pretty likely—the null could easily be true!
• Formal conclusion:
• p-value is greater than alpha
• Therefore, we don’t reject the null
• (I.e., we do not have enough evidence to support our H1)
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Statistics for Business and Economics (13e)
Critical Value Approach to One-Tailed Hypothesis Testing
• The test statistic z has a standard normal probability distribution.
• We can use the standard normal probability distribution table to find the zvalue with an area of  in the lower (or upper) tail of the distribution.
• The value of the test statistic that established the boundary of the rejection
region is called the critical value for the test.
• The rejection rule is:
• Lower tail: Reject H0 if z < -z
• Upper tail: Reject H0 if z > z
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
38
Critical Value Method (simply)
• In p-value method, we find the probability of getting the sample, then see how it
compares to alpha
• In the critical value method, we:
• First convert our alpha to a z-score, to map out the rejection region
• I.e., a sample result in the rejection region would be more extreme than our
alpha
• Compute the z-score for our sample
• If it falls in the rejection region, our we have enough evidence to reject the null
39
Statistics for Business and Economics (13e)
Lower-Tailed Test About a Population Mean: s Known
• Critical Value Approach
Sampling
Distribution of
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
Reject H0
 1
Do Not Reject H0
z
-z = -1.28
0
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otherwise on a password-protected website or school-approved learning management system for classroom use.
40
Same Example, but the Critical value
• You suspect that the average time spent on homework is less than 9 hours per
week. Based on some past data (?), you know that the population standard
deviation is 2.1. You take a sample of 40 students, and get a sample mean of 7.2.
Can you conclude, with 90% confidence, that the population average is less than
9 hours?
H1: 𝜇 < 9
H0: 𝜇 >= 9
𝜎 = 2.1
𝑥 = 7.2
Confidence = .90
 = .10
n = 40
41
Same Example: Critical Value approach
• Alpha = 0.1, so we find the zvalue that applies to a left-tail
area of 0.1
• Can use table, or norm.s.inv,
or remember common values
Reject H0
 1
Do Not Reject H0
z
-z = -1.28
0
42
Same Example: Critical Value approach
• So, if we get a z-score of -1.28 or less for our sample, we have enough evidence
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
7.2 − 9
𝑧=
2.1 40
𝑧 ≅ −5.421
• Because the sample falls in the rejection region:
• We have enough evidence to reject the null
• (I.e., we have enough evidence to support our H1)
43
Statistics for Business and Economics (13e)
Upper-Tailed Test About a Population Mean: s Known
• Critical Value Approach
Sampling
Distribution of
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
Reject H0
.05
Do Not Reject H0
z
0
z = 1.645
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
44
Statistics for Business and Economics (13e)
Steps of Hypothesis Testing
Step 1. Develop the null and alternative hypotheses.
Step 2. Specify the level of significance .
Step 3. Collect the sample data and compute the value of the test statistic.
p-Value Approach
Step 4. Use the value of the test statistic to compute the p-value.
Step 5. Reject H0 if p-value < .
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
45
Statistics for Business and Economics (13e)
Steps of Hypothesis Testing
Critical Value Approach
Step 4. Use the level of significance  to determine the critical value and the
rejection rule.
Step 5. Use the value of the test statistic and the rejection rule to determine
whether to reject H0.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
46
Statistics for Business and Economics (13e)
One-Tailed Tests About a Population Mean: s Known
• Example: Metro EMS
The response times for a random sample of 40 medical emergencies were
tabulated. The sample mean is 13.25 minutes. The population standard
deviation is believed to be 3.2 minutes.
The EMS director wants to perform a hypothesis test, with a .05 level of
significance, to determine whether the service goal of 12 minutes or less is
being achieved.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
47
Statistics for Business and Economics (13e)
One-Tailed Tests About a Population Mean: s Known
• p -Value and Critical Value Approaches
1. Develop the hypotheses.
H0: m < 12
Ha: m > 12
2. Specify the level of significance.
 = .05
3. Compute the value of the test statistic.
𝑧=
𝑥−𝜇0
𝜎 𝑛
=
13.25−12
= 2.47
3.2/ 40
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
48
Statistics for Business and Economics (13e)
One-Tailed Tests About a Population Mean: s Known
• p –Value Approach
4. Compute the p –value.
For z = 2.47, cumulative probability = .9932.
p-value = 1 - .9932 = .0068
5. Determine whether to reject H0.
Because p-value = .0068 <  = .05, we reject H0.
There is sufficient statistical evidence to infer that Metro EMS is not
meeting the response goal of 12 minutes.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
49
Statistics for Business and Economics (13e)
One-Tailed Tests About a Population Mean: s Known
• p –Value Approach
Sampling
Distribution of
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
 = .05
p-value

(p-Value < ,
so reject H0.)
z
0
z =
1.645
z=
2.47
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50
Statistics for Business and Economics (13e)
One-Tailed Tests About a Population Mean: s Known
• Critical Value Approach
4. Determine the critical value and rejection rule.
For  = .05, z.05 = 1.645
Reject H0 if z > 1.645
5. Determine whether to reject H0.
Because 2.47 > 1.645, we reject H0.
There is sufficient statistical evidence to infer that Metro
EMS is not meeting the response goal of 12 minutes.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
51
Statistics for Business and Economics (13e)
p-Value Approach to Two-Tailed Hypothesis Testing
• Compute the p-value using the following three steps:
1. Compute the value of the test statistic z.
2. If z is in the upper tail (z > 0), compute the probability that z is greater
than or equal to the value of the test statistic. If z is in the lower tail
(z < 0), compute the probability that z is less than or equal to the value
of the test statistic.
3. Double the tail area obtained in step 2 to obtain the p-value.
• The rejection rule: Reject H0 if the p-value <  .
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otherwise on a password-protected website or school-approved learning management system for classroom use.
52
Statistics for Business and Economics (13e)
Critical Value Approach to Two-Tailed Hypothesis Testing
• The critical values will occur in both the lower and upper tails of the standard
normal curve.
• Use the standard normal probability distribution table to find z/2 (the z-value
with an area of /2 in the upper tail of the distribution).
• The rejection rule is: Reject H0 if z < -z/2 or z > z/2.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
53
Statistics for Business and Economics (13e)
Two-Tailed Tests About a Population Mean: s Known
• Example: Glow Toothpaste
The production line for Glow toothpaste is designed to fill tubes with a
mean weight of 6 oz. Periodically, a sample of 30 tubes will be selected in
order to check the filling process.
Quality assurance procedures call for the continuation of the filling
process if the sample results are consistent with the assumption that the
mean filling weight for the population of toothpaste tubes is 6 oz.; otherwise
the process will be adjusted.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
54
Statistics for Business and Economics (13e)
Two-Tailed Tests About a Population Mean: s Known
• Example: Glow Toothpaste
Assume that a sample of 30 toothpaste tubes provides a sample mean of
6.1 oz. The population standard deviation is believed to be 0.2 oz.
Perform a hypothesis test, at the .03 level of significance, to help
determine whether the filling process should continue operating or be
stopped and corrected.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
55
Statistics for Business and Economics (13e)
Two-Tailed Tests About a Population Mean: s Known
• p –Value and Critical Value Approaches
1. Determine the hypotheses.
𝐻0 : 𝜇 = 6
𝐻𝑎 : 𝜇 ≠ 6
 = .03
2. Specify the level of significance.
3. Compute the value of the test statistic.
𝑧=
𝑥−𝜇0
𝜎 𝑛
=
6.1−6
2/ 30
=
2.74
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otherwise on a password-protected website or school-approved learning management system for classroom use.
56
Statistics for Business and Economics (13e)
Two-Tailed Tests About a Population Mean: s Known
• p –Value Approach
4. Compute the p –value.
For z = 2.74, cumulative probability = .9969
p-value = 2(1 - .9969) = .0062
5. Determine whether to reject H0.
Because p-value = .0062 <  = .03, we reject H0.
There is sufficient statistical evidence to infer that the alternative hypothesis
is true (i.e. the mean filling weight is not 6 ounces).
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
57
Statistics for Business and Economics (13e)
Two-Tailed Tests About a Population Mean: s Known
• p-Value Approach
1/2
p-value
= .0031
1/2
p-value
= .0031
/2 =
/2 =
.015
.015
z
z = -2.74
-z/2 = -2.17
6
z/2 = 2.17
z = 2.74
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
58
Statistics for Business and Economics (13e)
Two-Tailed Tests About a Population Mean: s Known
• Critical Value Approach
4. Determine the critical value and rejection rule.
For /2 = .03/2 = .015, z.015 = 2.17
Reject H0 if z < -2.17 or z > 2.17
5. Determine whether to reject H0.
Because 2.74 > 2.17, we reject H0.
There is sufficient statistical evidence to infer that the alternative
hypothesis is true (i.e. the mean filling weight is not 6 ounces).
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
59
Statistics for Business and Economics (13e)
Two-Tailed Tests About a Population Mean: s Known
• Critical Value Approach
Sampling
Distribution of
𝑥 − 𝜇0
𝑧=
𝜎 𝑛
Reject H0
Reject H0
/2 = .015
Do Not Reject H0
/2 = .015
z
-2.17
0
2.17
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
60
Statistics for Business and Economics (13e)
Confidence Interval Approach to
Two-Tailed Tests About a Population Mean
• Select a simple random sample from the population and use the value of
the sample mean 𝑥 to develop the confidence interval for the population
mean m. (Confidence intervals are covered in Chapter 8.)
• If the confidence interval contains the hypothesized value m0, do not
reject H0. Otherwise, reject H0. (Actually, H0 should be rejected if m0
happens to be equal to one of the end points of the confidence
interval.)
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otherwise on a password-protected website or school-approved learning management system for classroom use.
61
Statistics for Business and Economics (13e)
Confidence Interval Approach to
Two-Tailed Tests About a Population Mean
• The 97% confidence interval for m is
𝜎
𝑥 ± 𝑧𝛼/2
= 6.1 ± 2.17 .2 30 = 6.1 ± .07924
𝑛
or 6.02076 to 6.17924
• Because the hypothesized value for the population mean, m0 = 6, is not in
this interval, the hypothesis-testing conclusion is that the null hypothesis,
H0: m = 6, can be rejected.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
62
Statistics for Business and Economics (13e)
Tests About a Population Mean: s Unknown
• Test Statistic:
𝑥 − 𝜇0
𝑡=
𝑠 𝑛
• This test statistic has a t distribution with n - 1 degrees of freedom.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
63
Statistics for Business and Economics (13e)
Tests About a Population Mean: s Unknown
• Rejection Rule: p -Value Approach
Reject H0 if p –value < 
• Rejection Rule: Critical Value Approach
H0: m > m0
Reject H0 if t < -t
H0: m < m0
Reject H0 if t > t
H0: m = m0
Reject H0 if t < - t/2 or t > t/2
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otherwise on a password-protected website or school-approved learning management system for classroom use.
64
Statistics for Business and Economics (13e)
p -Values and the t Distribution
• The format of the t distribution table provided in most statistics textbooks does
not have sufficient detail to determine the exact p-value for a hypothesis test.
• However, we can still use the t distribution table to identify a range for the pvalue.
• An advantage of computer software packages is that the computer output will
provide the p-value for the t distribution.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
65
Statistics for Business and Economics (13e)
Example: Highway Patrol
• One-Tailed Test About a Population Mean: s Unknown
A State Highway Patrol periodically samples vehicle speeds at various
locations on a particular roadway. The sample of vehicle speeds is used to
test the hypothesis H0: m < 65.
The locations where H0 is rejected are deemed the best locations for radar
traps. At Location F, a sample of 64 vehicles shows a mean speed of 66.2 mph
with a standard deviation of 4.2 mph. Use a = .05 to test the hypothesis.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
66
Statistics for Business and Economics (13e)
One-Tailed Test About a Population Mean: s Unknown
• p –Value and Critical Value Approaches
1. Determine the hypotheses.
H0: m < 65
Ha: m > 65
2. Specify the level of significance.
 = .05
3. Compute the value of the test statistic.
𝑡=
𝑥−𝜇0
𝑠 𝑛
=
66.2−65
4.2/ 64
= 2.286
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
67
Statistics for Business and Economics (13e)
One-Tailed Test About a Population Mean: s Unknown
• p –Value Approach
4. Compute the p –value.
For t = 2.286, the p-value must be less than .025
(for t = 1.998) and greater than .01 (for t = 2.387).
.01 < p–value < .025
5. Determine whether to reject H0.
Because p-value <  = .05, we reject H0.
We are at least 95% confident that the mean speed of vehicles at
Location F is greater than 65 mph.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
68
Statistics for Business and Economics (13e)
One-Tailed Test About a Population Mean: s Unknown
• Critical Value Approach
4. Determine the critical value and rejection rule.
For  = .05 and d.f. = 64 – 1 = 63, t.05 = 1.669
Reject H0 if t > 1.669
5. Determine whether to reject H0.
Because 2.286 > 1.669, we reject H0.
We are at least 95% confident that the mean speed of vehicles at
Location F is greater than 65 mph. Location F is a good candidate
for a radar trap.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
69
Statistics for Business and Economics (13e)
One-Tailed Test About a Population Mean: s Unknown
Reject H0
()
p-value (p-Value <  ,
so reject H0.)
< .025
Do Not Reject H0
0
t =
1.669
t
t=
2.286
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otherwise on a password-protected website or school-approved learning management system for classroom use.
70
Statistics for Business and Economics (13e)
A Summary of Forms for Null and Alternative Hypotheses
About a Population Proportion
• The equality part of the hypotheses always appears in the null hypothesis.
• In general, a hypothesis test about the value of a population proportion p
must take one of the following three forms (where p0 is the hypothesized
value of the population proportion).
𝐻0 : 𝑝 ≥ 𝑝0
𝐻𝑎 : 𝑝 < 𝑝0
𝐻0 : 𝑝 ≤ 𝑝0
𝐻0 : 𝑝 = 𝑝0
𝐻𝑎 : 𝑝 > 𝑝0
𝐻𝑎 : 𝑝 ≠ 𝑝0
One-tailed
(lower tail)
One-tailed
(upper tail)
Two-tailed
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otherwise on a password-protected website or school-approved learning management system for classroom use.
71
Statistics for Business and Economics (13e)
Tests About a Population Proportion
𝑝 − 𝑝0
• Test Statistic: 𝑧 =
𝜎𝑝
where:
𝜎𝑝 =
𝑝0 1 − 𝑝0
𝑛
assuming np > 5 and n(1 – p) > 5
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otherwise on a password-protected website or school-approved learning management system for classroom use.
72
Example
• In Toronto, 46% of your customers are repeat buyers. You believe that, in Vancouver, the
rate of repeat buyers is less than this. You do a survey of 200 people, and find that
44.6% are repeat buyers. With 99% confidence, can we conclude that the rate is lower in
Vancouver than Toronto?
H1: p < 0.46
H0: p >= 0.46
n = 200
𝑝 ̅= 0.446
Confidence = 0.99
alpha = 0.01
(Note that the text often uses a 0 subscript for the assumed population proportion/mean
(e.g., p0). The 0 simply indicates that it is from the null hypothesis
Example
Sampling
Distribution of
𝑝 − 𝑝0
𝑧=
𝜎𝑝
p-value
z
0.446
0.46
74
Example
• First, test for sufficient sample size:
• n*p = 200 * .46 = 92
• n*(1-p) = 200 * (1-.46) = 108
• Next, find the standard deviation of the sampling distribution:
𝜎𝑝 =
𝑝0 1 − 𝑝0
𝑛
𝜎𝑝 =
0.46 1−0.46
200
≅ 0.0352
75
Example (Continued)
• Using the standard deviation, calculate the z-score
𝑝 − 𝑝0
𝑧=
𝜎𝑝
𝑧=
0.446−0.46
0.0352
= -0.397253048
• With our z-score, we can use the table or norm.s.dist to calculate the probability
(p-value)
• Remembering that because this is a ‘less than’ question, we use the table
results or norm.s.dist directly
• p-value = Norm.s.dist(-0.397253048, true) ≅ 0.34559
• Because our p-value is not less than alpha, we do not have enough evidence
76
Statistics for Business and Economics (13e)
Tests About a Population Proportion
• Rejection Rule: p –Value Approach
Reject H0 if p –value < 
• Rejection Rule: Critical Value Approach
H0: p < p0
Reject H0 if z > z
H0: p > p0
Reject H0 if z < -z
H0: p = p0
Reject H0 if z < -z/2 or z > z/2
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
77
Statistics for Business and Economics (13e)
Two-Tailed Test About a Population Proportion
• Example: National Safety Council (NSC)
For a Christmas and New Year’s week, the National Safety Council
estimated that 500 people would be killed and 25,000 injured on the nation’s
roads. The NSC claimed that 50% of the accidents would be caused by drunk
driving.
A sample of 120 accidents showed that 67 were caused by drunk driving.
Use these data to test the NSC’s claim with  = .05.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
78
Statistics for Business and Economics (13e)
Two-Tailed Test About a Population Proportion
• p –Value and Critical Value Approaches
𝐻0 : 𝑝 = .5 and 𝐻𝑎 : 𝑝 ≠ .5
1. Determine the hypotheses.
2. Specify the level of significance.
 = .05
3. Compute the value of the test statistic.
𝜎𝑝 =
𝑝0 1−𝑝0
𝑛
𝑧=
𝑝−𝑝0
𝜎𝑝
=
=
.5 1−.5
120
67
120
−.5
.045644
= .045644
= 1.28
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
79
Statistics for Business and Economics (13e)
Two-Tailed Test About a Population Proportion
• p-Value Approach
4. Compute the p -value.
For z = 1.28, cumulative probability = .8997
p-value = 2(1 - .8997) = .2006
5. Determine whether to reject H0.
Because p-value = .2006 >  = .05,
we cannot reject H0.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
80
Statistics for Business and Economics (13e)
Two-Tailed Test About a Population Proportion
• Critical Value Approach
4. Determine the critical values and rejection rule.
For /2 = .05/2 = .025, z.025 = 1.96
Reject H0 if z < -1.96 or z > 1.96
5. Determine whether to reject H0.
Because 1.278 > -1.96 and < 1.96,
we cannot reject H0.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
81
Statistics for Business and Economics (13e)
End of Chapter 9
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
82