Transcript Test Statistic

Welcome to MM207 – Statistics
Hypothesis Testing with One Sample
Chapter 7 7.1 – 7.4
Anthony J. Feduccia
Chapter Outline
• 7.1 Introduction to Hypothesis Testing
• 7.2 Hypothesis Testing for the Mean (Large
Samples)
• 7.3 Hypothesis Testing for the Mean (Small
Samples)
• 7.4 Hypothesis Testing for Proportions
• 0mit Section 7.5
Larson/Farber 4th ed.
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Guidelines -- Hypothesis Testing Steps:
•State H0 and Ha.
•Specify the level of significance alpha .
•Determine the test statistic, either z or t. Find the test statistic using the given
data.
•Find the P-value or the critical value(s) z0 or t0. Use the method specified in
the problem statement.
•Define the rejection region using either the P-value method or critical values
from the Normal distribution.
•Make a decision to reject or fail to reject the null hypothesis.
•Interpret the decision in the context of the original claim.
Section 7.1 Objectives
• State a null hypothesis and an alternative
hypothesis
• Identify type I and type I errors and interpret the
level of significance
• Determine whether to use a one-tailed or twotailed statistical test and find a p-value
• Make and interpret a decision based on the results
of a statistical test
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.
Larson/Farber 4th ed.
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Hypothesis Tests
Statistical hypothesis
• A statement, or claim, about a population
parameter.
• Need a pair of hypotheses
• one that represents the claim
• the other, its complement
• When one of these hypotheses is false, the
other must be true.
Larson/Farber 4th ed.
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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
Larson/Farber 4th ed.
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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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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.
3. A cereal company advertises that the mean weight of
the contents of its 20-ounce size cereal boxes is more
than 20 ounces.
Solution:
H0: μ ≤ 20 ounces
Ha: μ > 20 ounces
Complement of Ha
Inequality
(Claim)
condition
Larson/Farber 4th ed.
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1.a.
A candidate for mayor of a city claims to be favored by at least
half the voters.
Test this candidates claim. (4 points)
claim:
H0 :
Ha :
Test :
b.
The mean age of bus drivers in Chicago is 50.9 years.
Test the claim that the mean age differs from this.
(4 points)
claim:
H0 :
Ha :
Test :
Types of Errors
• No matter which hypothesis represents the claim, always begin
the hypothesis test assuming that the equality condition in
the null hypothesis is true.
• At the end of the test, one of two decisions will be made:
– reject the null hypothesis
– fail to reject the null hypothesis
• Because your decision is based on a sample, there is the
possibility of making the wrong decision.
Larson/Farber 4th ed.
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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.
Larson/Farber 4th ed.
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P-values
(see page 370)
P-value (or probability value)
• Find test statistic from the data. Either a z or a t.
• For a left-tail test: p value is area to the left of the test
statistic.
• For a right-tail test: p value is area to the right of the test
statistic (recall tables give area to the left)
• For a double-tail test: p value is twice the area to the left of
a negative test statistic or twice the area to the right of a
positive test statistic.
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Left-tailed Test
• The alternative hypothesis Ha contains the
less-than inequality symbol (<).
H0: μ  k
Ha: μ < k
P is the area to
the left of the
test statistic.
z
-3
-2
-1
Test
statistic
0
14
1
2
3
Right-tailed Test
• The alternative hypothesis Ha contains the
greater-than inequality symbol (>).
H0: μ ≤ k
Ha: μ > k
P is the area
to the right
of the test
statistic.
z
-3
Larson/Farber 4th ed.
-2
-1
0
15
1
2
Test
statistic
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Two-tailed Test
• The alternative hypothesis Ha contains the not
equal inequality symbol (≠). Each tail has an
area of ½P.
H0: μ = k
Ha: μ  k
P is twice the
area to the right
of the positive
test statistic.
P is twice the
area to the left of
the negative test
statistic.
z
-3
Larson/Farber 4th ed.
-2
-1
Test
statistic
0
16
1
2
Test
statistic
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Example Using P-Value
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  = 0.01? Use a P-value.
Larson/Farber 4th ed.
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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
•
•
•
•
H0: μ ≥ 30 min
Ha: μ < 30 min (claim)
 = 0.01
Test Statistic:
z
x 
 n
28.5  30

3.5 36
 2.57
• 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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An alternative approach to Hypothesis testing
Using Rejection regions (page 386)
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.
Larson/Farber 4th ed.
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The critical z based on alpha, 
Critical value of z has area of  (0.0500) to it’s left
•
•
•
•
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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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)
Larson/Farber 4th ed.
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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
•
•
•
•
•
Solution: Testing μ with a Small
Sample
H0: μ ≥ $23,900 (claim)•
Ha: μ < $23,900
α = 0.05
df = 14 – 1 = 13
•
Rejection Region:
0.05
-1.771 0
t
-3.026
Larson/Farber 4th ed.
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Test Statistic: use t
t
x 
s
n

23, 000  23,900
1113 14
 3.026
Decision: Reject H0
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
•A fast food outlet claims that the mean waiting time in line is less
than 4.9 minutes. A random sample of 60 customers yield a sample
mean of 4.8 minutes. From past studies it is know that the standard
deviation is 0.6 minutes. At alpha = 0.05, test the fast food outlet’s
claim
a.
Use the critical value z0 method from the normal distribution.
Known: u = 4.9, x bar = 4.8, sigma = 0.6;
alpha = 0.05; n =60
Ho:
Ha:
Test statistic:
Critical zo
Rejection Region:
Decision:
Use the P-value method.
  4.9, x  4.8,   0.6,   0.05, n  60
  4.9
:   4.9
•H0 :
Ha
Test
statistic
•:
claim
z
•P-value or critical z0 or t0.
x
/ n

4.8  4.9
0.6 / 60
 1.29
P-value = P(z < -1.29 ) = 0.0985
•Rejection Region: If P-value <= , then reject H0.
If P-value > , then fail to reject H0.
•Decision:  = 0.05 and P-value = 0.0985 and since 0.0985 > 0.05,
then we fail to reject H0.
•Interpretation:
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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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
Larson/Farber 4th ed.
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Use Normal z 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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