Homework:1,3,7,17,23,26,31,33,39,41,47,49,55,61,67,69,100,107,109ab

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Title: Homework:1,3,7,17,23,26,31,33,39,41,47,49,55,61,67,69,100,107,109ab

1
CHAPTER 8

Homework1,3,7,17,23,26,31,33,39,41,47,49,55,61,67
,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.

2

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.
3
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.
4
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.
5

(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?

(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?

(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.

8
(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.
9

(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
P-value.
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.

(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
null-hypothesis 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.

Sec 8.2 Large Sample Test for a population mean
For a large sample, usually the sample size gt
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.

Large sample test for a population mean
(a) Alternative Hypothesis
(i) Two-Tailed Test Ha m ¹ m0.
(ii) Right-Tailed Test Ha m gt m0.
(iii) Left-Tailed Test Ha m lt m0.
(b) Null Hypothesis
(i) Two-Tailed Test Ha m m0.
(ii) Right-Tailed Test Ha m m0.
(iii) Left-Tailed Test Ha m ³ m0

(c) Test Statistic
If the population standard deviation is
unknown, we can use the sample standard deviation
to replace it, i.e.

14
(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 gt Za/2 or z lt -Za/2,
(ii) Right-Tailed Test z gt Z a, (iii)
Left-Tailed Test z lt -Za.
15

(e) P-Value of this test
(i) Two-Tailed Test P-value 2 P(z gt
Zc),
(ii) Right-Tailed Test P-value P (z gt Zc),
(iii) Left-Tailed Test P-value P(z lt 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).

(EX 8.3) (Basic)
A sample of n35 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.

(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.

(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.

(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.

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).

(a). Alternative Hypothesis
(i) Two-Tailed Test Ha m ¹ m0.
(ii) Right-Tailed Test Ha m gt m0.
(iii) Left-Tailed Test Ha m lt m0.
(b) Null-Hypothesis
(i) Two-Tailed Test Ha m m0.
(ii) Right-Tailed Test Ha m m0.
(iii) Left-Tailed Test Ha m ³ m0

(d) Rejection Region of the test A size ? test
has the following rejection region (i).
Two-Tailed Test t gt ta/2,n-1 or t lt
-ta/2,n-1, (ii). Right-Tailed Test t gt
ta,n-1, (iii). Left-Tailed Test t lt -ta,n-1.
23

(e). P-Value of this test
(i). Two-Tailed Test P-value2P( t gt tc )
(ii).Right-Tailed Test P-value P(t gt tc)
(iii).Left-Tailed Test P-value P(t lt tc)
The null hypothesis is rejected if and only if
the P-
value is less than ?, and this test reaches the
same
conclusion and the test based on the rejection
region
in (d).

(EX 8.7) (Basic)
The test statistics for testing a right-tailed
test with a sample of n15 observations has the
value tc1.82.
(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).

(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

(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.

(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.

(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.

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.

(a). Alternative hypothesis
(i) Two-Tailed Test Ha p ¹ p0.
(ii) Right-Tailed Test Ha p gt p0.
(iii) Left-Tailed Test Ha p lt p0.
(b) Null Hypothesis
(i) Two-Tailed Test Ha p p0.
(ii) Right-Tailed Test Ha p p0.
(iii) Left-Tailed Test Ha p ³ p0.

(c) Test Statistics
Under the assumption that the null hypothesis
is true, the population standard deviation should
be estimated by

(d) Assumption
The sample size is large enough, i.e. the
interval 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 ? test
has the following rejection region (i).
Two-Tailed Test z lt -Za/2 or z gt Za/2 ,
(ii). Right-Tailed Test z gt Za, (iii).
Left-Tailed Test z lt -Za.
33

(f) P-Value of this test
(i). Two-Tailed Test P-value2P(z gt Zc)
(ii). Right-Tailed Test P-valueP(z gt Zc)
(iii). Left-Tailed Test P-valueP(z lt Zc)
A size ? test rejects the null hypothesis if
and only if the p-value is less than ?. And the
conclusion of this test is the same as the test
in (d).

(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 n100 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?

(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?

(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,