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║ ███ ▄ ███ ║  CHAPTER 5
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║ ▄▄▄▄▄▄▄▄▄ ║  PARAMETRIC HYPOTHESIS TESTS
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  A wide variety of the commonly used parametric hypothesis tests
are available for use right on screen.  They provide quick and
easy solutions to the computation of various parametric tests
that are often very difficult to perform by hand or by use of a
calculator.  The computations are rapid and very accurate.  All
you need to do is enter a few items of information and you will
quickly have the answers you need. 


                    ONE-SAMPLE TEST OF MEANS

  When you choose the option to conduct a one-sample test of
means, you will be conducting a t-test of the hypothesis that a
sample mean significantly differs from 0 or from a known or
presumed population mean.  The program will ask you to enter the
sample size, the sample mean, the standard deviation computed
from the sample and corrected for bias, and the population mean.
If you enter a population mean of 0 you will be testing the
hypothesis that the sample mean departs from zero.  If you enter
any other population mean, Mu, you will be testing the hypothesis
that the Diff = Mn - Mu = 0.  You may enter raw or summary data
at the keyboard and you may conduct your hypothesis tests at the
90%, 95% or the 99% confidence level.

  The following are sample outputs from the one-sample test of
means where Mean is the sample mean, Mu is the population mean,
and SEM is the standard error of the mean.

      N = 7             SEM = 5.37568
   Mean = 27.57143        t = 5.12892
     Mu = 0.00000        p <= 0.00268
      99% Confidence Interval
     8.76192 <= Mn <= 46.38094

      N = 189           SEM = 0.80013
   Mean = 35.00000        t = 43.74277
     Mu = 0.00000        p <= 0.00000
      99% Confidence Interval
    32.93886 <= Mn <= 37.06114

      N = 300           SEM = 1.21244
   Mean = 47.00000        t = -0.82479
     Mu = 48.00000       p <= 0.20475
      95% Confidence Interval
   44.62363 <= Diff <= 49.37637


           TWO-SAMPLE TEST OF MEANS: DEPENDENT SAMPLES

  The two-sample test of means for dependent samples enables you
to compare two means that are obtained from the same subjects.
It is often called the "direct difference t-test of means".  If
you enter raw scores, the program will compute and report the
correlation for you.  If you enter summary data, you must know
the correlation between the two variables as well as the sample
means, standard deviations and the sample size.  The program
produces the difference between the two means, the standard error
of that difference, the t-test and its associated probability,
and one of three confidence intervals that you specify.

  The following are samples of outputs for the two-sample test of
means using dependent samples.

      GROUP I                   GROUP II
 N      = 34                   N      = 34
 Mean x = 87.00000             Mean x = 95.00000
 SDx    = 23.00000             SDx    = 28.00000
 SEDiff = 5.07323                r = 0.34000
    t   = -1.57691                p  <= 0.12070
 Mean 1 - Mean 2  = -8.00000

      99% Confidence Interval
   -21.95138 <= Diff <= 5.95138


          TWO-SAMPLE TEST OF MEANS: INDEPENDENT SAMPLES

  The two-sample test of means using independent samples is the
good old "t-test for mean differences".  You may enter raw or
summary data from the keyboard and the following are sample
outputs from this procedure.

      GROUP I                   GROUP II
 N      = 89                   N      = 114
 Mean x = 54.00000             Mean x = 61.00000
 SDx    = 28.00000             SDx    = 31.00000
 SEDiff = 4.20442
    t   = -1.66492                p  <= 0.04796
 Mean 1 - Mean 2  = -7.00000

      95% Confidence Interval
   -15.24066 <= Diff <= 1.24066


                    TEST PEARSON CORRELATIONS

  There are six tests of correlations that have identical inputs
and outputs.  These are the test of a Pearson r, Phi, Point-
Biserial, Spearman's Rho, partial correlations, and semi-partial
correlations.  For each of these tests, you need only enter the
sample size and the value of the correlation.  The program does
the rest and provides you with an F-ratio, degrees of freedom for
hypothesis (dfh), degrees of freedom for error (dfe), the
associated probability for the F-ratio and a confidence interval
for your correlation.

    N = 18        dfh =     1
    r = 0.46000   dfe =    16
    F =  4.29427   p <= 0.05229
      95% Confidence Interval
    -0.05003 <=  r  <= 0.77973


                      TEST PHI COEFFICIENT

  See the above explanation for "Test Pearson Correlation".  The
inputs and outputs are identical.  The only difference is that
you will enter your Phi coefficient instead of the Pearson r. 

    N = 67        dfh =     1
    r = 0.18000   dfe =    65
    F =  2.17652   p <= 0.14120
      95% Confidence Interval
    -0.06791 <=  r  <= 0.40698


                 TEST POINT-BISERIAL CORRELATION

  See the above explanation for "Test Pearson Correlation".  The
inputs and outputs are identical.  The only difference is that
you will enter your Point-Biserial correlation instead of the
Pearson r. 

    N = 219       dfh =     1
    r = 0.17000   dfe =   217
    F =  6.45793   p <= 0.00552
      95% Confidence Interval
     0.03829 <=  r  <= 0.29591


                        TEST SPEARMAN RHO

  See the above explanation for "Test Pearson Correlation".  The
inputs and outputs are identical.  The only difference is that
you will enter your Spearman Rho instead of the Pearson r. 

    N = 189       dfh =     1
    r = 0.67000   dfe =   187
    F = 152.32136   p <= 0.00000
      95% Confidence Interval
     0.58302 <=  r  <= 0.74179


                    TEST PARTIAL CORRELATION

  See the above explanation for "Test Pearson Correlation".  The
inputs and outputs are identical.  The only difference is that
you will enter your partial correlation instead of the Pearson r.

    N = 118       dfh =     1
    r = 0.23000   dfe =   116
    F =  6.47915   p <= 0.01178
      95% Confidence Interval
     0.04951 <=  r  <= 0.39594


                  TEST SEMI-PARTIAL CORRELATION

  See the above explanation for "Test Pearson Correlation".  The
inputs and outputs are identical.  The only difference is that
you will enter your semi-partial correlation instead of the
Pearson r. 

    N = 56        dfh =     1
    r = 0.54000   dfe =    54
    F = 22.22812   p <= 0.00009
      95% Confidence Interval
     0.31801 <=  r  <= 0.70586


                   TEST MULTIPLE CORRELATIONS

  The test of a multiple correlation is used to test the null
hypothesis, H: R = 0.  In order to use it, you must enter the
sample size, the multiple correlation, R, and the number of
independent variables, IVs, used to obtain R.  The program will
produce the F-ratio, degrees of freedom for hypothesis (dfh),
degrees of freedom for error (dfe), the probability associated
with the F-ratio, and the confidence interval that you specify.
The program then produces the same information for the shrunken
multiple correlation, and the following are sample outputs from
the program.

    N = 137       dfh =     3
    r = 0.43000   dfe =   133
    F = 10.05672   p <= 0.00004
    IV's = 3
      95% Confidence Interval
     0.27989 <=  r  <= 0.55959

      SHRUNKEN CORRELATION
    N = 137       dfh =     3
    r = 0.40806   dfe =   133
    F = 10.05672   p <= 0.00004
    IV's = 3
      95% Confidence Interval
     0.25518 <=  r  <= 0.54104


                       TEST CHANGE IN R^2

  The ability to test a change in a squared multiple correlation,
R^2, can be extremely important.  In order to do that, you will
need to enter the degrees of freedom for hypothesis, dfh, the
degrees of freedom for error, dfe, the change in R^2 that you
wish to test, and the overall R^2 for the multiple regression
model.  In the first example shown below, the value of dfh = 1
for the test of an increment in R^2 due to the addition of only
one variable.  In the second example, the change in R^2 was based
on the addition of three variables so the value of dfh = 3.

     dfh        = 1
     dfe        = 218
     R^2 Change = 0.14000
     R^2 Total  = 0.47000
     F-ratio    = 57.5849
     p         <= 0.48394

     dfh        = 3
     dfe        = 321
     R^2 Change = 0.26000
     R^2 Total  = 0.37000
     F-ratio    = 44.1587
     p         <= 0.00006


                 ONE-SAMPLE TEST OF PROPORTIONS

  The one-sample test of proportions is much like the one-sample
test of means.  If you enter a population proportion of 0 you
will be testing the hypothesis that your sample proportion is
equal to 0.  If you enter any other value for the population
proportion, you will be testing the hypothesis that Diff = SP-PP
= 0.  These features are illustrated by the following sample
outputs. 

      SP = 0.14000          PP = 0.00000
       N = 89              SEP = 0.03678
       t = 2.00000          p <= 0.00051
     95% Confidence Interval for Proportion
           0.00000 <= SP <= 0.28000

      SP = 0.23000          PP = 0.27000
       N = 121             SEP = 0.03826
       t = 1.98000          p <= 0.29819
     95% Confidence Interval for Proportion
           0.19000 <= Diff <= 0.27000


        TWO-SAMPLE TEST OF PROPORTIONS: DEPENDENT SAMPLES

  The two-sample test of proportions for dependent samples
enables you to test the hypothesis, Ho: p1 = p2.  You need only
enter the two proportions and the sample size in order to obtain
the output illustrated below.

          GROUP I                   GROUP II
     Prop   = 0.42000              Prop   = 0.38000
     N      = 134                  N      = 134
     Difference = 0.04000
     t = -0.51769    p <= 0.30234


       TWO-SAMPLE TEST OF PROPORTIONS: INDEPENDENT SAMPLES

  The two-sample test of proportions for independent samples
enables you to test the hypothesis, Ho: p1 = p2.  You need only
enter the two proportions and the sample size in order to obtain
the output illustrated below.

          GROUP I                   GROUP II
     Prop   = 0.38000              Prop   = 0.31000
     N      = 89                   N      = 112
     Std Dev = 0.05025            Std Dev = 0.04479
     Difference     = 0.07000    SEDiff = 0.06732
     t = 1.03989     p <= 0.14920
      95% Confidence Interval for Difference
            -0.06194 <= Diff <= 0.20194


             ONE-SAMPLE TEST OF A STANDARD DEVIATION

  This test is analagous to the one-sample test of means except
that it is conducted for standard deviations.

      N = 67           Mean = 35
     SD = 17.00000      PSD = 21.00000
      t = -2.72373     SESD = 1.46858
     p <= 0.00816
         95% Confidence Interval
      -6.93715 <= Diff <= -1.06285


         TWO-SAMPLE TEST OF VARIANCES: DEPENDENT SAMPLES

  This test is analagous to the two-sample test of means using
dependent samples except that the test is applied to variances.

          GROUP I                   GROUP II
     N      = 36                   N      = 36
     Var X  = 127.00000            Var X  = 138.00000
     SD  X  = 11.26943             SD  X  = 11.74734
     Difference = Var1 - Var2  = -11.00000
     SEDiff = 43.72152                t    = -0.25159
        r   = 0.27000                p   <= 0.79825
              95% Confidence Interval
          -100.27934 <= Diff <= 78.27934


        TWO-SAMPLE TEST OF VARIANCES: INDEPENDENT SAMPLES

  This test is analagous to the two-sample test of means using
independent samples except that the test is applied to variances.

          GROUP I                   GROUP II
     N      = 168                  N      = 236
     Var x  = 489.00000            Var x  = 512.00000
         F  = 1.04703               p     <= 0.18867


            TEST TWO CORRELATIONS: DEPENDENT SAMPLES

  Often one has a correlation between X and Y and for X and Z
from the same sample.  In such cases it is desirable to test the
hypothesis that the two correlations are equal, i.e.  that r(xy)
= r(xz).  This procedure enables you to do that.  You need only
enter the sample size and the two correlations.  However, you
must also enter the correlation, r(yz), between the variables Y
and Z.  Remember, the test is on the hypothesis, Ho: r(xy) =
r(xz).  The following is a sample of the output from this
procedure.

 N      = 79             r(xy)  = 0.47000
 r(xz)  = 0.52000        r(yz)  = 0.23000
 Diff   = r(xy) - r(xz)     = -0.05000
 z      = -0.44074           p <= 0.66481
      95% Confidence Interval
    -2.46105 <= Diff <= 1.57223


           TEST TWO CORRELATIONS: INDEPENDENT SAMPLES

  Often one has two correlations for the same variables but from
independent samples.  In such cases it is often desirable to test
the hypothesis that the two correlations are equal.  This
procedure enables you to do that.  You need only enter the sample
size and the correlation for each of the two groups or samples
and the following is a sample of the output from this procedure.

          GROUP I               GROUP II
        N   = 89                  N   = 131
        r   = 0.46000             r   = 0.37000
        Z   = 0.49731             Z   = 0.38842
 Difference = 0.10889
        t   = 0.78096             p  <= 0.21741
              95% Confidence Interval
            -0.16439 <= Diff <= 0.38217


                         OMNIBUS R TEST

  The Omnibus R Test enables you to enter an entire correlation
matrix from the keyboard and then test the hypothesis that there
are no significant correlations within the entire matrix.  In
other words, one is never justified in analyzing a correlation
matrix, using multivariate procedures for example, unless one can
first reject the null hypothesis, Ho: R = 0.  The Omnibus R Test
provides such a test and the following is a sample of the output
from this procedure for 4x4 correlation matrix.

 N    = 367             df   = 6
 Chi-Square  = 571.82834
          p <= 0.00000


                         END OF CHAPTER
