Transcript Part 2 - Angelfire

SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE

These steps are summarized here:
 Step
1: Use the column of the t table that
corresponds to the value of  you have
selected.
 Step 2: Find the number of degrees of freedom
by calculating n-1 and use that row of the t
table.
 Step 3: For upper-tail tests the desired t cutoff
value is found at the intersection of that row and
column. For lower-tail tests, place a negative
sign in front of the value to get the value of tcutoff"
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SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
Degrees of Freedom
 Degrees of Freedom were first introduced by
Fisher. The degrees of freedom of a set of
observations are the number of values which
could be assigned arbitrarily within the
specification of the system.
 For example, in a sample of size n grouped into
k intervals, there are k-1 degrees of freedom,
because k-1 frequencies are specified while the
other one is specified by the total size n.
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SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
Summary of Tests of the Mean:
 There were two major differences from the tests
of  when  is known:
 a t test statistic was used instead of Z and
 the cutoff values for the rejection region were
found by using the t table instead of the Z
table.
 The rejection regions are summarized next.
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SECTION 1
χ² TEST OF A SINGLE VARIANCE
 Hypothesis tests of the population variance, ²,
follow the same basic steps that were used to
do a hypothesis test of the population mean
and the population proportion.
 There are two types of situations for which you
might be interested in doing a hypothesis test
of the population variance.
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χ² TEST OF A SINGLE VARIANCE
 First, you may wish to see if a manufacturing
process is running to the specified standard
deviation.
 Second, to perform some other statistical
analysis of the data, you may need to know the
population variance. In this case you would use
sample data to test to see if the population
variance is, in fact, a particular value.
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χ² TEST OF A SINGLE VARIANCE
Two-Tail Hypothesis Test of the Variance
 To learn how to do a two-sided test of the
variance we will complete the five steps of the
hypothesis testing procedure for the tissue
manufacturer.
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SECTION 1
χ² TEST OF A SINGLE VARIANCE
 The next step in the hypothesis testing
procedure is to select a value for  and find the
rejection region.
 To do this we need to know what test statistic
will be used in step 3 of the procedure. This is
the main difference between a hypothesis test
of the population mean, , and the population
variance, ².
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SECTION 1
χ² TEST OF A SINGLE VARIANCE
 In testing means; we used the sample mean as
the basis for our decision to reject or fail to
reject the null hypothesis.
 Because the Central Limit Theorem told us that
X-bar has an approximately normal distribution
for sufficiently large sample sizes, the
appropriate test statistic for a large-sample test
of the mean is a Z statistic and thus the
rejection region is determined by Z values.
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χ² TEST OF A SINGLE VARIANCE
 If we are testing the population variance then
the sample variance, s², will be used as the
basis for deciding between H0 and HA.
 Relying once again on the mathematical
statisticians to do the theoretical work, we learn
that the sampling distribution associated with
the sample variance, s², is called the chi-square
(χ²) distribution.
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χ² TEST OF A SINGLE VARIANCE
 The rejection region will be determined by
critical values from the chi-square distribution.
 All example of a chi-square distribution is
shown in Figure 12.3.
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SECTION 1
χ² TEST OF A SINGLE VARIANCE
 The chi-square distribution, just like any
distribution, describes how the random variable
behaves.
 If the variable being studied is assumed to be
normally distributed, then the statistic to test
whether or not the population variance is equal
to a particular value is calculated as follows:
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SECTION 1
χ² TEST OF A SINGLE VARIANCE
 
2
(n  1) s

2
2
n = sample size
s² = the sample variance
² = the hypothesized value of the population
variance under the null hypothesis
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χ² TEST OF A SINGLE VARIANCE
 Like the t distribution, the shape of the chi-
square distribution is determined by the
number of degrees of freedom. This test
statistic has n -1 degrees of freedom.
 Unlike the Z and t distributions, the χ²
distribution is not symmetric.
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χ² TEST OF A SINGLE VARIANCE
 Since this is a two-sided rejection region, the
area in each tail must be /2. A typical rejection
region is shown as the shaded region in Figure
12.4.
 The specific values for χ²lower and χ²upper must be
found from the chi-square table. A portion of
this table is shown in Table 12.1.
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SECTION 1
χ² TEST OF A SINGLE VARIANCE
One-Sided Tests of the Variance
 If you are testing the population variance, most
of the time you are interested in doing a twosided test.
 However, it is possible to do a one-sided test of
the variance. The only change in the procedure
needed to complete a one-sided test of the
variance is in step 2.
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SECTION 1
χ² TEST OF A SINGLE VARIANCE
 A one-sided rejection region is used in this
case. The rejection regions for the two
possibilities are shown in Figure 12.5.
 Ho: ²  [a specific number]
HA: ² < [a specific number]
 Ho: ²
[a specific number]
HA: ² > [a specific number]
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