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╔═══════════╗
║ ███ ▄ ███ ║ CHAPTER 5
║ ▀▀▀▀▀▀▀▀▀ ║
║ ▄▄▄▄▄▄▄▄▄ ║ PARAMETRIC HYPOTHESIS TESTS
║ ███ ▀ ███ ║
╚═══════════╝
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
