Transcript Chapter 9 第九章

HYPOTHESIS TEST
假设检验
Instruction
In this chapter you will learn how to
formulate hypotheses about a population
mean and a population proportion.
Through the analysis of sample data, you
will be able to determine whether a
hypothesis should or should not be
rejected.
9.1 DEVELOPING NULL AND ALTERNATIVE
HYPOTHESIS
In some application it may not be obvious how
the null and alternative hypothesis should be
formulated. We must be careful to make them
properly.
There will be three types of situations in which
hypothesis testing procedures are commonly
employed.
Testing Research Hypothesis
There is an automobile model
example on page 325 to illustrate the
testing research hypothesis.
Testing the validity of a
claim
there is a soft drinks example on the
page 325 to illustrate the testing the
validity of a claim.
Testing in decision-making
situation
There is an example of shipment on
the page 326 to illustrate the testing in
decision-making situation.
Summary
of Forms for Null and Alternative
Hypotheses
H μ≥µ
H µ <µ
0:
0
a:
0
H µ ≤µ
H µ >µ
0:
0
a:
0
Hµ
Hµ
0:
a:
µ
≠µ
=
0
0
9.2 type I and type II errors
population condition
----------------------------H0 True
Ha True
Accept H0
Correct
Conclusion
Type II
Error
Reject H0
Type I
Error
Correct
Conclusion
Instructions
If the sample data are consistent with
the null hypothesis H0, we will follow the
practice of concluding “do not reject H0.”
This conclusion is preferred over “accept
H0.” because the conclusion to accept H0
puts us at risk of making a Type II error.
9.3 One-tailed test about a population
mean: large-sample case
There is a Hilltop Coffee example to
illustrate the one-tailed test about
population mean on page 330.
The null and alternative hypotheses are
as follows:
H0:µ≥3
Ha:µ<3
The test statistic
In the large-sample case with the population
standard deviation σassumed known, the test
statistic is given by
公式:
x  0
z
/ n
Summary
large-sample (n≥30) hypothesis test about a
population mean for a one-tailed test of the form
H0:µ≥µ0
Ha:µ< µ0
Test Statistic:σ known
x  0
z
/ n
Test statistic:σEstimated by s
x  0
z
s/ n
Rejection Rule
Using test statistic: Reject H0 if z<-zα
Using p-value: Reject H0 if p-value<α
Steps of Hypothesis Testing
Steps of Hypothesis Testing
1. Develop the null and alternative hypotheses.
2. Specify the level of significance α.
3. Select the statistic that will be used to test the
hypothesis.
Using the test statistic
4. Use the level of significance to determine the critical
value for the test statistic and state the rejection rule for H0.
5. Collect the sample data and compute the value of the
test statistic.
6. Use the value of the test statistic and the rejection rule
to determine whether to reject H0.
Using the p-value
4. Collect the sample data and compute the
value of the test statistic.
5. Use the value of the test statistic to
compute the p-value.
6. Reject H0 if p-value<α
9.4 Two-tailed test about a population
mean: large-sample case
Two-tailed tests differ from one-tailed
hypothesis tests in that two-tailed tests
have rejection regions in both the lower
and upper tails of the sampling
distribution.
example
There is an example of golf balls mean
distance at page 339.
The null and alternative hypotheses are
as follows:
H0 :μ=280
Ha :μ≠280
The test statistic
With a sample size of 36 golf balls used each
time the quality control test is performed, we
have a large-sample case. As s result the test
statistic is
x  0
z
/ n
Summary: Two-tailed tests about a
population mean
Large-sample (n≥30) hypothesis test about a
population mean for a two-tailed test of the
form
H0:μ=μ0
Ha:μ≠μ0
Test statistic:σassumed known
x  0
z
/ n
Test statistic:σ estimated by s
x  0
z
s/ n
Rejection rule
Using test statistic: Reject H0 if
z<-zα/2 or if z>zα/2
Using p-value: Reject H0 if pvalue<α
Relationship between interval
estimation and hypothesis testing
The details about the differences
between these two approaches are at
page 342-343.
A confidence interval approach to
testing a hypothesis of the form
H0:μ=μ0
Ha:μ≠μ0
1.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 μ. If σis assumed known,
compute the interval estimate by using
xz


2
n
If σis estimated by s, compute the interval
estimate by using
xz

2
s
n
2. If the confidence interval contains the
hypothesized value μ0, do not reject H0.
otherwise, reject H0.
9.5 Test about a population mean:
small-sample case
The procedures for hypothesis test
about a population mean discussed in
section 9.3 and 9.4 were based on the
central limit theorem and large-sample
theory.
In section 9.5, we consider tests about
a population mean using a small sample
(n<30).
Assumption
The small-sample case require the
assumption that the population has a
normal probability distribution.
If this assumption is not appropriate,
the best alternative is to increase the
sample size to n≥30 and rely on the largesample hypothesis testing procedures.
Test statistic
If the population standard deviation σ is
assumed known:
z
x  0

n
If the population standard deviation σis
estimated by the sample standard deviation s:
x
t
s
n
0
There is an example of the international
air transport association on page 348.
p-values and the t distribution is on
page 348-349.
Two-tailed test is similar to one-tailed
test. It is on the page 349-350.
9.6 Test about a population
proportion
Three forms for a hypothesis test about
a population proportion are as follows:
H0:p≥p0 H0:p≤p0
H0:p=p0
Ha :p<p0
Ha :p>p0
Ha:p≠p0
The first two forms are one-tailed tests,
and the third form is two-tailed test.
If both np and n(1-p) are greater than or
equal to 5. The sampling p distribution of can
be approximated by a normal probability
distribution , the following test statistic can be
used.
Test statistic for test about a population
proportion:
z
where
p  p0
p
p0 (1  p0 )
p 
n
9.7 hypothesis testing and decision
making
There is an example of shipment of
batteries on page 360.
When we do not reject H0, action we
take may risk Type ІІ error . In section 9.8
and 9.9 we explain how to compute the
probability of making a Type ІІ error and
how the sample size can be adjusted to
help control the probability of making a
Type ІІ error.
9.8 calculating the probability of
type ІІ errors
There is an example on page 360-362.
suppose α=.05 and a mean life of
µ=112 hours.
probability of a Type ІІ error when µ=112
β=.0091
112
116.71
Accept H0
There is a table to show the probability of making a Type ІІ error
for a variety of values of µ less than 120. Note that as µ increases
toward 120, the probability of making a Type ІІ error increases
toward an upper bound of .95.
value of µ
公式
112
114
115
116.17
117
118
119.999
2.36
1.36
.86
.00
-.15
-.65
-.1.645
p ability of
a Type ІІ error(β)
.0091
.0869
.1949
.5000
.5596
.7422
.9500
power
(1-β)
.9909
.9131
.8051
.5000
.4404
.2578
.0500
Power curve for the lot-acceptance
hypothesis test

1.00

.80

.60

.40

.20
112
115
118
120
summary
Procedure to compute the probability of making a Type ІІ error:
1. Formulate the null and alternative hypothesis
2. Use the level of significance αto establish a rejection rule based
on the test statistic.
3. Using the rejection rule, solve for the value of the sample mean
that identifies the rejection region for the test.
4. Use the results from step3 to state the values of the sample mean
that lead to the acceptance of H0. It also defines the acceptance
region for the test.
5. Using the sampling distribution of x for any value of µ from the
alternative hypothesis, and the acceptance region from step4,
compute the probability that the sample mean will be in the
acceptance region. This probability is the probability of making a
Type ІІ error at the chosen value of µ
9.9 determining the sample size for a
hypothesis test about a population mean
Recommended sample size for a one-tailed hypothesis test about a
population mean
n
( z  z  )
2

(0   a ) 2
2
Where
zα=z value providing an area of αin the tail of a standard normal
distribution
zβ=z value providing an area of βin the tail of a standard normal
distribution
σ=the population standard deviation
µ0 =the value of the population mean in the null hypothesis
µa =the value of the population mean used for the Type ІІ error