Transcript Document

Chapter 7
Hypothesis Testing with One
Sample
Larson/Farber 4th ed.
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Hypothesis Tests
Hypothesis test
• A process that uses sample statistics to test a claim
about the value of a population parameter.
• For example: An automobile manufacturer
advertises that its new hybrid car has a mean mileage
of 50 miles per gallon. To test this claim, a sample
would be taken. If the sample mean differs enough
from the advertised mean, you can decide the
advertisement is wrong.
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Stating a Hypothesis
Null hypothesis
• A statistical hypothesis
that contains a statement
of equality such as ≤, =,
or ≥.
• Denoted H0 read “H
subzero” or “H naught.”
Alternative hypothesis
• A statement of
inequality such as >, ≠,
or <.
• Must be true if H0 is
false.
• Denoted Ha read “H
sub-a.”
complementary
statements
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Example: Stating the Null and Alternative
Hypotheses
Write the claim as a mathematical sentence. State the null
and alternative hypotheses and identify which represents
the claim.
1. A university publicizes that the proportion of its
students who graduate in 4 years is 82%.
Solution:
H0: p = 0.82
Equality condition (Claim)
Ha: p ≠ 0.82
Complement of H0
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Example: Stating the Null and Alternative
Hypotheses
Write the claim as a mathematical sentence. State the null
and alternative hypotheses and identify which represents
the claim.
2. A water faucet manufacturer announces that the mean
flow rate of a certain type of faucet is less than 2.5
gallons per minute.
Solution:
H0: μ ≥ 2.5 gallons per minute
Ha: μ < 2.5 gallons per minute
Larson/Farber 4th ed.
Complement of Ha
Inequality
(Claim)
condition
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Types of Errors
Actual Truth of H0
Decision
Do not reject H0
Reject H0
H0 is true
Correct Decision
Type I Error
H0 is false
Type II Error
Correct Decision
• A type I error occurs if the null hypothesis is rejected
when it is true.
• A type II error occurs if the null hypothesis is not
rejected when it is false.
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Example: Identifying Type I and Type II
Errors
The USDA limit for salmonella contamination for
chicken is 20%. A meat inspector reports that the
chicken produced by a company exceeds the USDA
limit. You perform a hypothesis test to determine
whether the meat inspector’s claim is true. When will a
type I or type II error occur? Which is more serious?
(Source: United States Department of Agriculture)
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Solution: Identifying Type I and Type II
Errors
Let p represent the proportion of chicken that is
contaminated.
Hypotheses: H0: p ≤ 0.2
Ha: p > 0.2 (Claim)
Chicken meets
USDA limits.
H0: p ≤ 0.20
Chicken exceeds
USDA limits.
H0: p > 0.20
p
0.16
Larson/Farber 4th ed.
0.18
0.20
0.22
0.24
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Solution: Identifying Type I and Type II
Errors
Hypotheses: H0: p ≤ 0.2
Ha: p > 0.2 (Claim)
A type I error is rejecting H0 when it is true.
The actual proportion of contaminated chicken is less
than or equal to 0.2, but you decide to reject H0.
A type II error is failing to reject H0 when it is false.
The actual proportion of contaminated chicken is
greater than 0.2, but you do not reject H0.
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Solution: Identifying Type I and Type II
Errors
Hypotheses: H0: p ≤ 0.2
Ha: p > 0.2 (Claim)
• With a type I error, you might create a health scare
and hurt the sales of chicken producers who were
actually meeting the USDA limits.
• With a type II error, you could be allowing chicken
that exceeded the USDA contamination limit to be
sold to consumers.
• A type II error could result in sickness or even death.
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Level of Significance
Level of significance
• Your maximum allowable probability of making a
type I error.
 Denoted by , the lowercase Greek letter alpha.
• By setting the level of significance at a small value,
you are saying that you want the probability of
rejecting a true null hypothesis to be small.
• Commonly used levels of significance:
  = 0.10
 = 0.05
 = 0.01
• P(type II error) = β (beta)
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P-values
P-value (or probability value)
• The probability, if the null hypothesis is true, of
obtaining a sample statistic with a value as extreme or
more extreme than the one determined from the
sample data.
• Depends on the nature of the test.
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Making a Decision
Decision Rule Based on P-value
• Compare the P-value with α.
 If P ≤ α, then reject H0.
 If P > α, then fail to reject H0.
Claim
Decision
Claim is H0
Claim is Ha
Reject H0
There is enough evidence to
reject the claim
There is enough evidence to
support the claim
Fail to reject H0
There is not enough evidence
to reject the claim
There is not enough evidence
to support the claim
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Example: Interpreting a Decision
You perform a hypothesis test for the following claim.
How should you interpret your decision if you reject
H0? If you fail to reject H0?
2. Ha (Claim): Consumer Reports states that the mean
stopping distance (on a dry surface) for a Honda
Civic is less than 136 feet.
Solution:
• The claim is represented by Ha.
• H0 is “the mean stopping distance…is greater than or
equal to 136 feet.”
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Solution: Interpreting a Decision
• If you reject H0 you should conclude “there is enough
evidence to support Consumer Reports’ claim that the
stopping distance for a Honda Civic is less than 136
feet.”
• If you fail to reject H0, you should conclude “there is
not enough evidence to support Consumer Reports’
claim that the stopping distance for a Honda Civic is
less than 136 feet.”
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Steps for Hypothesis Testing
1. State the claim mathematically and verbally. Identify
the null and alternative hypotheses.
H0: ? Ha: ?
2. Specify the level of significance. This sampling distribution
is based on the assumption
α= ?
that H0 is true.
3. Determine the standardized
sampling distribution and
draw its graph.
z
0
4. Calculate the test statistic
and its standardized value.
Add it to your sketch.
z
0
Test statistic
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Steps for Hypothesis Testing
5. Find the P-value.
6. Use the following decision rule.
Is the P-value less
than or equal to the
level of significance?
No
Fail to reject H0.
Yes
Reject H0.
7. Write a statement to interpret the decision in the
context of the original claim.
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Section 7.2
Hypothesis Testing for the Mean
(Large Samples)
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Using P-values to Make a Decision
Decision Rule Based on P-value
• To use a P-value to make a conclusion in a hypothesis
test, compare the P-value with α.
1. If P ≤ α, then reject H0.
2. If P > α, then fail to reject H0.
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Z-Test for a Mean μ
• Can be used when the population is normal and  is
known, or for any population when the sample size n
is at least 30.
• The test statistic is the sample mean x
• The standardized test statistic is z
x 
  standard error  
z
x
 n
n
• When n ≥ 30, the sample standard deviation s can be
substituted for σ.
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Using P-values for a z-Test for Mean μ
In Words
1. State the claim mathematically
and verbally. Identify the null
and alternative hypotheses.
In Symbols
State H0 and Ha.
2. Specify the level of significance.
Identify α.
3. Determine the standardized test
statistic.
z
4. Find the area that corresponds
to z.
Larson/Farber 4th ed.
x 
 n
Use Table 4 in
Appendix B.
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Using P-values for a z-Test for Mean μ
In Words
In Symbols
5. Find the P-value.
a. For a left-tailed test, P = (Area in left tail).
b. For a right-tailed test, P = (Area in right tail).
c. For a two-tailed test, P = 2(Area in tail of test
statistic).
Reject H0 if P-value
6. Make a decision to reject or
is less than or equal
fail to reject the null hypothesis.
to α. Otherwise, fail
to reject H0.
7. Interpret the decision in the
context of the original claim.
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Example: Hypothesis Testing Using Pvalues
In an advertisement, a pizza shop claims that its mean
delivery time is less than 30 minutes. A random
selection of 36 delivery times has a sample mean of
28.5 minutes and a standard deviation of 3.5 minutes. Is
there enough evidence to support the claim at α = 0.01?
Use a P-value.
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Solution: Hypothesis Testing Using Pvalues
•
•
•
•
H0: μ ≥ 30 min
Ha: μ < 30 min
 = 0.01
Test Statistic:
z
x 
 n
28.5  30

3.5 36
 2.57
Larson/Farber 4th ed.
• P-value
0.0051
-2.57
0
z
• Decision: 0.0051 < 0.01
Reject H0
At the 1% level of significance,
you have sufficient evidence to
conclude the mean delivery time
is less than 30 minutes.
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Rejection Regions and Critical Values
Rejection region (or critical region)
• The range of values for which the null hypothesis is
not probable.
• If a test statistic falls in this region, the null
hypothesis is rejected.
• A critical value z0 separates the rejection region from
the nonrejection region.
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Using Rejection Regions for a z-Test for a
Mean μ
In Words
1. State the claim mathematically
and verbally. Identify the null
and alternative hypotheses.
2. Specify the level of significance.
In Symbols
State H0 and Ha.
Identify .
3. Sketch the sampling distribution.
4. Determine the critical value(s).
Use Table 4 in
Appendix B.
5. Determine the rejection region(s).
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Using Rejection Regions for a z-Test for a
Mean μ
In Words
6. Find the standardized test
statistic.
7. Make a decision to reject or fail
to reject the null hypothesis.
8. Interpret the decision in the
context of the original claim.
Larson/Farber 4th ed.
In Symbols
x 
or if n  30
 n
use   s.
z
If z is in the rejection
region, reject H0.
Otherwise, fail to
reject H0.
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Example: Testing with Rejection Regions
Employees in a large accounting firm claim that the
mean salary of the firm’s accountants is less than that of
its competitor’s, which is $45,000. A random sample of
30 of the firm’s accountants has a mean salary of
$43,500 with a standard deviation of $5200. At
α = 0.05, test the employees’ claim.
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Solution: Testing with Rejection Regions
•
•
•
•
H0: μ ≥ $45,000
Ha: μ < $45,000
 = 0.05
Rejection Region:
• Test Statistic
x   43, 500  45, 000
z

 n
5200 30
0.05
-1.645 0
-1.58
Larson/Farber 4th ed.
z
 1.58
• Decision: Fail to reject H0
At the 5% level of significance,
there is not sufficient evidence
to support the employees’ claim
that the mean salary is less than
$45,000.
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Section 7.3
Hypothesis Testing for the Mean
(Small Samples)
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Finding Critical Values in a t-Distribution
1. Identify the level of significance α.
2. Identify the degrees of freedom d.f. = n – 1.
3. Find the critical value(s) using Table 5 in Appendix B in
the row with n – 1 degrees of freedom. If the hypothesis
test is
a. left-tailed, use “One Tail, α ” column with a negative
sign,
b. right-tailed, use “One Tail, α ” column with a
positive sign,
c. two-tailed, use “Two Tails, α ” column with a
negative and a positive sign.
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Example: Finding Critical Values for t
Find the critical value t0 for a left-tailed test given
α = 0.05 and n = 21.
Solution:
• The degrees of freedom are
d.f. = n – 1 = 21 – 1 = 20.
• Look at α = 0.05 in the
“One Tail, α” column.
• Because the test is lefttailed, the critical value is
negative.
Larson/Farber 4th ed.
0.05
-1.725 0
t
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t-Test for a Mean μ (n < 30, σ Unknown)
t-Test for a Mean
• A statistical test for a population mean.
• The t-test can be used when the population is normal
or nearly normal, σ is unknown, and n < 30.
• The test statistic is the sample mean x
• The standardized test statistic is t.
x 
t
s n
• The degrees of freedom are d.f. = n – 1.
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Using the t-Test for a Mean μ
(Small Sample)
In Words
1. State the claim mathematically
and verbally. Identify the null
and alternative hypotheses.
In Symbols
State H0 and Ha.
2. Specify the level of significance.
Identify α.
3. Identify the degrees of freedom
and sketch the sampling
distribution.
d.f. = n – 1.
4. Determine any critical value(s).
Use Table 5 in
Appendix B.
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Using the t-Test for a Mean μ
(Small Sample)
In Words
5. Determine any rejection
region(s).
6. Find the standardized test
statistic.
7. Make a decision to reject or
fail to reject the null
hypothesis.
8. Interpret the decision in the
context of the original claim.
Larson/Farber 4th ed.
In Symbols
x 
t
s n
If t is in the rejection
region, reject H0.
Otherwise, fail to
reject H0.
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Example: Testing μ with a Small Sample
A used car dealer says that the mean price of a 2005
Honda Pilot LX is at least $23,900. You suspect this
claim is incorrect and find that a random sample of 14
similar vehicles has a mean price of $23,000 and a
standard deviation of $1113. Is there enough evidence to
reject the dealer’s claim at α = 0.05? Assume the
population is normally distributed. (Adapted from Kelley
Blue Book)
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Solution: Testing μ with a Small Sample
•
•
•
•
•
H0: μ ≥ $23,900
Ha: μ < $23,900
α = 0.05
df = 14 – 1 = 13
Rejection Region:
• Test Statistic:
t
-3.026
Larson/Farber 4th ed.
s
n

23, 000  23, 900
1113
14
 3.026
• Decision: Reject H0
0.05
-1.771 0
x
t
At the 0.05 level of
significance, there is enough
evidence to reject the claim
that the mean price of a 2005
Honda Pilot LX is at least
$23,900
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Section 7.4
Hypothesis Testing for Proportions
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z-Test for a Population Proportion
z-Test for a Population Proportion
• A statistical test for a population proportion.
• Can be used when a binomial distribution is given
such that np ≥ 5 and nq ≥ 5.
• The test statistic is the sample proportion p̂ .
• The standardized test statistic is z.
z
Larson/Farber 4th ed.
pˆ   pˆ
 pˆ
pˆ  p

pq n
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Using a z-Test for a Proportion p
Verify that np ≥ 5 and nq ≥ 5
In Words
1. State the claim mathematically
and verbally. Identify the null
and alternative hypotheses.
2. Specify the level of significance.
In Symbols
State H0 and Ha.
Identify α.
3. Sketch the sampling distribution.
4. Determine any critical value(s).
Larson/Farber 4th ed.
Use Table 5 in
Appendix B.
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Using a z-Test for a Proportion p
In Words
In Symbols
5. Determine any rejection
region(s).
6. Find the standardized test
statistic.
7. Make a decision to reject or
fail to reject the null
hypothesis.
8. Interpret the decision in the
context of the original claim.
Larson/Farber 4th ed.
p̂  p
z
pq n
If z is in the rejection
region, reject H0.
Otherwise, fail to
reject H0.
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Example: Hypothesis Test for
Proportions
Zogby International claims that 45% of people in the
United States support making cigarettes illegal within
the next 5 to 10 years. You decide to test this claim and
ask a random sample of 200 people in the United States
whether they support making cigarettes illegal within the
next 5 to 10 years. Of the 200 people, 49% support this
law. At α = 0.05 is there enough evidence to reject the
claim?
Solution:
• Verify that np ≥ 5 and nq ≥ 5.
np = 200(0.45) = 90 and nq = 200(0.55) = 110
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Solution: Hypothesis Test for Proportions
•
•
•
•
• Test Statistic
pˆ  p
0.49  0.45
z

pq n
(0.45)(0.55) 200
H0: p = 0.45
Ha: p ≠ 0.45
 = 0.05
Rejection Region:
0.025
-1.96
0.025
0
1.96
1.14
Larson/Farber 4th ed.
z
 1.14
• Decision: Fail to reject H0
At the 5% level of significance,
there is not enough evidence to
reject the claim that 45% of
people in the U.S. support
making cigarettes illegal within
the next 5 to 10 years.
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