Transcript Document 7594410

SECTION 1
TEST OF A SINGLE PROPORTION
 If the data we are analyzing are nominal data,
the hypothesis might be a statement about:
proportion, , of population
members that have a certain characteristic
(one of the categories of the nominal
variable).
 The value of the
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SECTION 1
TEST OF A SINGLE PROPORTION
 For example, the hypothesis might be a
statement about the proportion of
 Students who are interested in graduate
school
 Vaccinated patients who remain cancer free
 CEOs who use computers as a major tool
 People who are unemployed
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SECTION 1
TEST OF A SINGLE PROPORTION
Two-Tail Tests of Proportions
 We will use precisely the same five steps that
we have been following for any hypothesis test:
 Step
1: Set up the null and alternative hypotheses
test
 Step 2: Pick the value of  and find the rejection
region.
 Step 3: Calculate the test statistic.
 Step 4: Decide whether or not to reject the null
hypothesis.
 Step 5: Interpret the statistical decision in terms of
the stated problem.
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SECTION 1
TEST OF A SINGLE PROPORTION
 Five-step hypothesis testing procedure is
identical to the one we have been using.
 The test statistic is the same as that used for a
two-tail test of proportions and the rejection
regions are the same as those used for one-tail
tests of the mean.
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SECTION 1
TEST OF A SINGLE PROPORTION
One-Tail Test of Proportions
 Step 1: Set up the null and alternative
hypotheses.
 There are two possible ways to set up a one-tail
test of proportions.
 Upper-Tail Test
  [a specific number]
 HA:  > [a specific number]
 Ho:
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TEST OF A SINGLE PROPORTION
 Lower-Tail Test
 Ho:
 [a specific number]
 HA:
< [a specific number]
 Step 2: Select the value of “” and find the
rejection region.
 Step 3: Calculate the test statistic.
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SECTION 1
TEST OF A SINGLE PROPORTION
 We have seen in the previous section that the
appropriate test statistic is
Z
p 
 (1   ) / n
 Steps 4 and 5 remain the same.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
P-level
 The value of the p-level represents a
decreasing index of the reliability of a result.
 The higher the p-level, the less we can believe
that the observed relation between variables in
the sample is a reliable indicator of the relation
between the respective variables in the
population.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 Specifically, the p-level represents the
probability of error that is involved in
accepting our observed result as valid, that is,
as "representative of the population."
 For example, the p-level of .05 (i.e.,1/20)
indicates that there is a 5% probability that the
relation between the variables found in our
sample is a "fluke."
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 In many areas of research, the p-level of .05 is
customarily treated as a "border-line
acceptable" error level.
P  value  P( Z  Z calculated ) for Upper  Tail Test
P  value  P( Z  Z calculated ) for Lower  Tail Test
P  value  2 P( Z  Z calculated ) for Two  Tail Test

For more info: http://www.fiu.edu/~howellip/P-value.pdf
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Problem 17 (Page 65 or 153)
 Fowle
Marketing Research Inc., bases
charges to a client on the assumption
that telephone surveys can be completed
in a mean time of 15 minutes or less. If a
longer mean survey time is necessary, a
premium rate is charged. Suppose a
sample of 35 surveys shows a sample
mean of 17 minutes and a sample
standard deviation of 4 minutes. Is the
premium rate justified?
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Problem 17 (Page 65 or 153)
 Formulate
the null and alternative
hypotheses for this application.
H 0 :   15
H a :   15
 Compute
Z
the value of the test statistic.
X 
s/ n

17  15
4 / 35
 2.96
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Problem 17 (Page 65 or 153)
 What
is the p -value?
P  value  P( Z  Z calculated )
P  value  P( Z  2.96)  1  P( Z  2.96)
P  value  1  0.9985
P  value  0.0015
 Using
0.01, what is your conclusion?
Since P-value= 0.0015 < = 0.01, we reject
H0, that means the premium rate is justified.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS

When you are comparing characteristics of two
different populations, you must have a sample from
each of the populations. These samples are usually
selected independently of each other.

In other words, the selection of one sample should
not have any effect on the selection of the second
sample.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 We will label all of the parameters of one
population with a subscript 1 and all the
parameters of the second population with a
subscript 2.
 It does not matter which population you label 1
or 2. The populations and samples are shown
in Figure 13.1.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 Consider the question posed about whether
men or women spend more money on frozen
foods.
 We could label the population of males as
population 1 and the population of females as
population 2.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 If we do this then the parameters and statistics
corresponding to the male population will be
identified with a subscript 1 and those
describing the female population will carry a
subscript 2.
 For this example, we would select a sample of
men shoppers and a separate sample of
women shoppers.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 We would ask all members of the sample how
much money they spent on frozen foods in the
past week. It is not necessary that the sample
sizes be equal, but if possible it is desirable to
have both sample sizes (n1, n2) greater than or
equal to 30.
 The reason for this stems from the fact that the
Central Limit Theorem generally applies when
the sample size is 30 or greater.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 Remember that we developed the Z test
statistic based on the knowledge that the
sample mean, X (X-bar), has a normal
distribution.
 However, often a single sample is selected and
a qualitative variable is used to identify two
populations for comparison. For the food
shopper example, we might select one sample
of shoppers and then record the gender of the
respondent as part of the data.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 This means that the data can then be divided
into the two comparison-populations after the
data have been collected.
 If at the same time you collect data on the age
of the person as "under 40" or "40 and over"
then you can also compare the average frozen
food expenditure for younger buyers to that of
older buyers.
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SECTION 1
COLLECTING DATA FROM TWO POPULATIONS
 Clearly, spending differences that are
identified by gender or age could be of great
assistance in developing a marketing strategy.
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SECTION 1
HYPOTHESIS TEST OF THE DIFFERENCE IN TWO
POPULATION MEANS

Regardless of the particular case, each hypothesis test
will follow the five-step procedure
 Step 1: Set up the null and alternative hypotheses.
 Step 2: Pick the value of  and find the rejection
region.
 Step 3: Calculate the test statistic and the p value.
 Step 4: Decide whether or not to reject the null
hypothesis.
 Step 5: Interpret the statistical decision in terms of
the stated problem.
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SECTION 1
LARGE-SAMPLE TESTS OF THE DIFFERENCE IN
TWO POPULATION MEANS
Large-Sample Tests of Two Means with Known
Standard Deviations
 The basic test concerning two population
means occurs when we want to know whether
the two samples come from populations with
equal means and we assume that the population
standard deviations are known.
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SECTION 1
LARGE-SAMPLE TESTS OF THE DIFFERENCE IN
TWO POPULATION MEANS
 The first step of the procedure is to construct the
null and alternative hypotheses. As with tests of a
single mean, there are three different ways to set
up the hypothesis test.
 Notice that the equals sign is always part of the
null hypothesis and hypotheses are statements
about the relationship between the size of the
mean of population 1 and the mean of population
2.
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SECTION 1
LARGE-SAMPLE TESTS OF THE DIFFERENCE IN
TWO POPULATION MEANS
 The tests do not give you information about the
value of the means, only about how the value of
1 compares to the value of 2.
 You can see that there are two ways to state each
of the hypotheses.
 For each setup shown above the first way of
writing the test makes a statement about the
relative value of 1 to 2.
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SECTION 1
LARGE-SAMPLE TESTS OF THE DIFFERENCE IN
TWO POPULATION MEANS
 The second way of writing the same test makes a
statement about the value of the difference, 1-2.
 They are equivalent to each other. Look at the
two-sided test.
 Clearly, if 1=2 then the difference between them
must be zero. It is also possible to test for
differences of values other than zero.
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SECTION 1
LARGE-SAMPLE TESTS OF THE DIFFERENCE IN
TWO POPULATION MEANS
Large-Sample Tests of Two Means with Unknown
Standard Deviations
 The large-sample test for the difference between
two population means requires that both of the
sample sizes be greater than 30.
 Since the sample sizes are large, each individual
sample standard deviation is a good estimate of
the corresponding unknown population standard
deviation.
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SECTION 1
LARGE-SAMPLE TESTS OF THE DIFFERENCE IN
TWO POPULATION MEANS
 So, we simply use each of the sample standard
deviations in the formula instead of the
corresponding values of σ.
 The test statistic becomes
Z
(X1  X 2 )  d
s12 / n1  s 22 / n 2
 The rejection region depends on  and whether
the test is one-sided or two-sided.
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