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CHAPTER 8
Homework:1,3,7,17,23,26,31,33,39,41,47,49,55,61,
67,69,100,107,109ab
Sec 8.1: Elements of a hypothesis Testing:
(1) Set up hypotheses
A hypothesis is simply a statement about a
population parameter, e.g. the population mean.
There are two types of hypotheses -- the null
hypothesis and alternative hypothesis.
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A NULL HYPOTHESIS
is a hypothesis to be tested. Typically, we believe that
null-hypothesis is true unless the data provide enough
evident that it is false.
AN ALTERNATIVE HYPOTHESIS
is a hypothesis that contradicts the null-hypothesis. If
the null hypothesis is rejected by a test, then we
believe the alternative hypothesis is true.
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Remember: If the null hypothesis is not rejected by a
test, we can not infer that the null hypothesis is true.
That is, a hypotheses test can only prove (with a
confidence) that the alternative may be true, but never
the null.
Because of this special feature of a hypotheses testing
procedure, the alternative hypothesis is usually set up
as a hypothesis that is hoped to be shown to be true by
the test.
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TWO-TAILED ALTERNATIVE
If the alternative states that a population parameter is
different from a specific value. The corresponding test
is called a two-tailed test.
RIGHT-TAILED ALTERNATIVE
If the alternative states that a population parameter is
greater than a specific value. The corresponding test is
called a right-tailed test.
LEFT-TAILED ALTERNATIVE
If the alternative states that a population parameter is
less than a specific value. The corresponding test is
called a left-tailed test.
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(EX 8.1) (Basic-- Set up the hypotheses)
The R.R Bowker company of New-York
collects information on the retail prices of books.
In 1986, the mean retail price of all hardcover
history books was $28.44. Suppose you want to
know whether the mean retail price of this kind of
books is higher than $28.44 this year. Can you set
up a test answering your problem?
(a). Determine the alternative hypothesis.
(b). Determine the null hypothesis.
(c). What type of hypothesis it is?
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(EX 8.2) (Basic -- Set up the hypotheses)
High airline occupancy rates on scheduled
flights are essentially to profitability. Suppose that
a scheduled flight must average at least 60%
occupancy rate to be profitable. We know that the
occupancy rate of the Sunday morning flight from
Orlando to New-York City is only 54%. Before
the company decides to close this scheduled flight,
they ask you to set up a test helping them to make
their decision.
(a). What should be the alternative hypothesis if
the company's goal is to close this flight?
(b). What is the null hypothesis?
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(2) Compute the test statistic
We already discussed that there are several
different types of statistics to measure the central
tendency of a population. Also, there are several
different test statistics for testing about a
population mean.
For example, there are z-statistic and t-statistic.
Which one should be used depends on assumptions
requied by these tests, as in the construction of
confidence intervals.
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(3) Decide the rejection region of the test
Based on the test statistic and a given confidence
level, we can determine the rejection region, the
acceptance region, and the critical value of the test.
Rejection region is the region in which we can
reject the null-hypothesis when the test statistics falls
in this region. Acceptance region is simply the
complement of the rejection region.
Critical value is the value (or values) on the
boundary of the rejection region and acceptance
region.
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(4) p-value and hypotheses testing
As an alternative approach to the
rejection/acceptance-region approach, we can
calculate a probability related to the test statistic,
called P-value, and base our decision of
rejection/acceptance on the magnitude of the Pvalue.
P-value is the probability to observe a value of the
test statistic as extreme as the one observed, if the
null hypothesis is true. So a small P-value
indicates that the null hypothesis is not true and
hence should be rejected.
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(5) Two possible errors in hypotheses
testing, and the size/significance level of a
test
There are two types of error which will occur in a
statistical test of hypotheses.
Type I error occurs when you reject a nullhypothesis while it is true.
Type II error occurs when you fail to reject a false
null-hypothesis.
The probability of making type I error is called the
size or significant level () of the test, often denoted
as a.
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Sec 8.2 Large Sample Test for a population mean
For a large sample, usually the sample size > 30,
the central limiting theorem ensures that the
sample mean is at least approximately normally
distributed for a wide range of sampled
populations. Also, the sample variance provides a
good estimation for the unknown population
variance. Therefore, we can use the standard
normal z test statistic to complete our test.
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Large sample test for a population mean
(a) Alternative Hypothesis:
(i) Two-Tailed Test: Ha: m  m0.
(ii) Right-Tailed Test: Ha: m > m0.
(iii) Left-Tailed Test: Ha: m < m0.
(b) Null Hypothesis:
(i) Two-Tailed Test: Ha: m = m0.
(ii) Right-Tailed Test: Ha: m  m0.
(iii) Left-Tailed Test: Ha: m  m0
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(c) Test Statistic
x  m0
Zc =
x
If the population standard deviation is
unknown, we can use the sample standard
deviation to replace it, i.e.
x  m0
Zc =
sx
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(d) Rejection Region of the test
If it is required that the size of the test is a , then
the rejection region is given by
(i) Two-Tailed Test: z > Za/2 or z < -Za/2,
(ii) Right-Tailed Test: z > Z a,
(iii) Left-Tailed Test:
z < -Za.
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(e) P-Value of this test
(i) Two-Tailed Test: P-value = 2 * P(z > |Zc|),
(ii) Right-Tailed Test: P-value = P (z > Zc),
(iii) Left-Tailed Test: P-value = P(z < Zc).
If it is required that the size of the test should be a,
then the null hypothesis is rejected if and only if the
P-value is smaller than a.
The conclusion, either rejection or acceptance, of
this procedure is exactly the same as the test based
on the rejection region in (d).
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(EX 8.3) (Basic)
A sample of n=35 observations from a long
tail population produced a mean equal to 2.4 and
standard deviation equal to 0.29. Suppose that
your research project is to show that the
population mean exceeds 2.3.
(a). Give the null and the alternative hypotheses of
the test.
(b). Find the test statistics and the p-value.
(c). State the assumption you need.
(d). Locate the rejection region of this test at 0.05
level and make your decision at 0.05 level.
(e). Describe what types of error are possible in
this decision process.
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(EX 8.4) (Basic)
Refer to example 8.3. Suppose that your
research goal is to show that the population mean
is less than 2.9.
(a). Give the null and alternative hypotheses of the
test.
(b). Locate the rejection region of this test at 0.05
level.
(c). Make your decision.
(d). Describe possible erros in your decision.
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(EX 8.5) (Basic)
Refer to example 8.3. Suppose that your
research goal is to show that the population mean
differs from 2.45.
(a). Give the null and alternative hypotheses of the
test.
(b). Locate the rejection region of this test at 0.05
level.
(c). Make your decision.
(d). Describe the types of error possible in this
decision process.
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(EX 8.6) (Intermediate)
A drug manufacturer claimed that the mean
potency of one of its antibiotics was 0.8. A sample
of n = 100 capsules were tested and produced a
sample mean equal to 0.797 with a standard
deviation equal to 0.008. Do the data present
sufficient evident to refute the manufacture's
claim?
(a). Give the null and alternative hypotheses of the
test and find the test statistics.
(b). State the assumption you need.
(c). Find the p-value of this test and locate the
rejection region of this test at 0.05 level.
(d). Make your decision and describe what type of
error possible in your decision.
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Sec 8.3: Small Sample Test for one
Population Mean
If we can assume the population we are interested
in has a normal distribution, then we test the
hypotheses using the t statistic, irrespective the
size of the sample (whether it is small or large).
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(a). Alternative Hypothesis:
(i) Two-Tailed Test: Ha: m  m0.
(ii) Right-Tailed Test: Ha: m > m0.
(iii) Left-Tailed Test: Ha: m < m0.
(b) Null-Hypothesis:
(i) Two-Tailed Test: Ha: m = m0.
(ii) Right-Tailed Test: Ha: m  m0.
(iii) Left-Tailed Test: Ha: m  m0
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(c) Test Statistics
x  m0
tc =
.
Sx
(d) Rejection Region of the test
A size a test has the following rejection region:
(i). Two-Tailed Test: t > ta/2,n-1 or t < -ta/2,n-1,
(ii). Right-Tailed Test: t > ta,n-1,
(iii). Left-Tailed Test: t < -ta,n-1.
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(e). P-Value of this test:
(i). Two-Tailed Test: P-value=2*P( t > |tc |)
(ii).Right-Tailed Test: P-value = P(t > tc)
(iii).Left-Tailed Test: P-value = P(t < tc)
The null hypothesis is rejected if and only if the Pvalue is less than a, and this test reaches the same
conclusion and the test based on the rejection region
in (d).
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(EX 8.7) (Basic)
The test statistics for testing a right-tailed test
with a sample of n=15 observations has the value
tc=1.82.
(a). State the assumptions you need. What are
the degrees of freedom for this statistics?
(b). Find the p-value of this test.
(c). Give the rejection region of the test at 0.05
level and make your decision.
(d). Give the rejection region of the test at
0.01 level and make your decision.
(e). Describe what types of errors can possibly
be made in (c) and (d).
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(EX 8.8) (Basic)
A manufacturer of gunpowder has developed a
new powder that is designed to produce a muzzle
velocity of 3000 feet per second. Eight shells are
loaded with the charge and the muzzle velocities
measured. The resulting velocities are shown in
the following table. Does this set of data provide
enough information to claim that the muzzle
velocity are less than 3000.
Muzzle Velocities(feet per second)
3005 2925 2995 2935
3005 2965 2935 2905
Note:
x = 2958.75 and s = 39.26
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(a). Give the null and alternative hypotheses of the
test.
(b). Find the test statistics.
(c). State the assumptions you need.
(d). Find the p-value of this test.
(e). Locate the rejection region of this test at 0.05
level.
(f). Make your decision.
(g). Describe what types of error are possible in
this type of decision.
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(EX 8.9) (Applications)
"Lake Champlain Found to be Polluted by
PCBs," reports the New York Times(June 16,
1985). PCBs, a group of chemicals used for years
as an insulator in some electrical equipment, have
been found to cause cancer in laboratory animals
and are suspected of similar effects on humans.
Although the federal level of tolerance of PCBs in
fish is two PPM, a sampling of 15 American eels
in Lake Champlain gave PCB readings ranging
from 4.05 to 19.49 PPM with a mean value and
standard deviation of 9.84 and 3.86, respectively.
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(a). Give the null and alternative hypotheses of the
test.
(b). Find the test statistics.
(c). State the assumptions you need.
(d). Find the p-value of this test.
(e). Locate the rejection region of this test at 0.10
level.
(f). Make your decision.
(g). Describe what types of error can possible be
made in this type of decision.
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Sec 8.4 Large Sample Test for
a population proportion
The properties of a sample proportion were
discussed in the previous chapter. For a
sufficiently large sample, the sampling
distribution of the sample proportion is
approximately normal.
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(a). Alternative hypothesis
(i) Two-Tailed Test: Ha: p  p0.
(ii) Right-Tailed Test: Ha: p > p0.
(iii) Left-Tailed Test: Ha: p < p0.
(b) Null Hypothesis:
(i) Two-Tailed Test: Ha: p = p0.
(ii) Right-Tailed Test: Ha: p  p0.
(iii) Left-Tailed Test: Ha: p  p0.
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(c) Test Statistics:
p  p0
Zc =
 p0
Under the assumption that the null hypothesis
is true, the population standard deviation should
be estimated by
 p0 =
p0 (1  p0 )
.
n
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(d) Assumption
The sample size is large enough, i.e. the interval
(p 0  3 *  p0 , p0  3 *  p0 ) must be contained
in the interval (0 , 1), so that the sampling
distribution of the test statistic can be
approximated by a normal distribution .
(e) Reject Region of the test
A size a test has the following rejection region
(i). Two-Tailed Test: z < -Za/2 or z > Za/2 ,
(ii). Right-Tailed Test: z > Za,
(iii). Left-Tailed Test: z < -Za.
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(f) P-Value of this test
(i). Two-Tailed Test: P-value=2*P(z > |Zc|)
(ii). Right-Tailed Test: P-value=P(z > Zc)
(iii). Left-Tailed Test: P-value=P(z < Zc)
A size a test rejects the null hypothesis if and only
if the p-value is less than a. And the conclusion of
this test is the same as the test in (d).
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(EX 8.10) (Basic)
Regardless of age, about 20% of American
adults participate in fitness activities at least twice
a week. However, the fitness activities are
different among the students in UCF. In a local
survey of n=100 students randomly selected from
UCF, a total of 27 students indicated that they
participated in a fitness activity at least twice a
week. Does this data indicates that the UCF
students’ participation rate differs significantly
from the 20% national average at a = 0.10?
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(EX 8.11) (Basic -- Large sample test)
A random sample of n = 1000
observations from a binomial population
produced x = 279.
(a). If your research hypothesis is that p is less
than 0.3, what should you choose for your
alternative hypothesis and null hypothesis?
(b). Does your alternative hypothesis in part (a)
imply a one or two tailed test? Explain.
(c). Find the test statistics.
(d). Does the data set provide sufficient evidence
to indicate that p is less than 0.3 at a = 0.05?
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(EX 8.12) (Applications)
More than ever before, Americans are working
at two jobs, according to a Labor Department
survey reported in the Wall Street Journal
(November 7, 1994). According to the survey, the
proportion of employed Americans holding two or
more jobs is 7.2% compared to 6.2% in 1989.
Assume that the current survey was based on a
random sample of 950 employed Americans. If
you wish to show that the proportion of Americans
holding two or more jobs is greater than the 1989
figure,
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(a). State the null and alternative hypotheses to be
tested.
(b). Locate the rejection region for 0.01 level.
(c). Conduct the test and state your conclusion.
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