These steps are summarized here:

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SECTION 1HYPOTHESIS TEST OF THE MEAN SMALL
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• 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 t-cutoff"

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SECTION 1HYPOTHESIS TEST OF THE MEAN SMALL
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• 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 1HYPOTHESIS TEST OF THE MEAN SMALL
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• 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?² 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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SECTION 1?² 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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SECTION 1?² 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?² 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?² 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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SECTION 1?² 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 (?²) distribution.

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SECTION 1?² 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?² 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?² TEST OF A SINGLE VARIANCE

• n sample size
• s² the sample variance
• ?² the hypothesized value of the population
• variance under the null hypothesis

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SECTION 1?² 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 ?²
  distribution is not symmetric.

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SECTION 1?² 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 ?²lower and ?²upper 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?² 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
  two-sided 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?² 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 lt a specific number
• Ho ?²? a specific number
• HA ?² gt a specific number

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