COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE
Instructor: Dr. John J. Kerbs, Associate Professor Joint Ph.D. in
Social Work and Sociology Chapter 10: t Test for Two Independent
Samples 2013 - - DO NOT CITE, QUOTE, REPRODUCE, OR DISSEMINATE
WITHOUT WRITTEN PERMISSION FROM THE AUTHOR: Dr. John J. Kerbs can
be emailed for permission at [email protected] Slide 2
Independent-Measures Designs Allows researchers to evaluate the
mean difference between two populations using data from two
separate samples. Allows researchers to evaluate the mean
difference between two populations using data from two separate
samples. The identifying characteristic of the independent-
measures or between-subjects design is the existence of two
separate or independent samples. The identifying characteristic of
the independent- measures or between-subjects design is the
existence of two separate or independent samples. Thus, an
independent-measures design can be used to test for mean
differences between two distinct populations (such as men versus
women) or between two different treatment conditions (such as drug
versus no-drug). Thus, an independent-measures design can be used
to test for mean differences between two distinct populations (such
as men versus women) or between two different treatment conditions
(such as drug versus no-drug). Slide 3 Teaching Method A vs. B
Slide 4 Independent-Measures Designs (cont'd.) The
independent-measures design is used in situations where a
researcher has no prior knowledge about either of the two
populations (or treatments) being compared. The
independent-measures design is used in situations where a
researcher has no prior knowledge about either of the two
populations (or treatments) being compared. In particular, the
population means and standard deviations are all unknown. In
particular, the population means and standard deviations are all
unknown. Because the population variances are not known, these
values must be estimated from the sample data. Because the
population variances are not known, these values must be estimated
from the sample data. Slide 5 The t Statistic for an Independent-
Measures Research Design As with all hypothesis tests, the general
purpose of the independent-measures t test is to determine whether
the sample mean difference obtained in a research study indicates a
real mean difference between the two populations (or treatments) or
whether the obtained difference is simply the result of sampling
error. As with all hypothesis tests, the general purpose of the
independent-measures t test is to determine whether the sample mean
difference obtained in a research study indicates a real mean
difference between the two populations (or treatments) or whether
the obtained difference is simply the result of sampling error.
Remember, if two samples are taken from the same population and are
given exactly the same treatment, there will still be some
difference between the sample means (i.e., this difference is
called sampling error). Remember, if two samples are taken from the
same population and are given exactly the same treatment, there
will still be some difference between the sample means (i.e., this
difference is called sampling error). The hypothesis test provides
a standardized, formal procedure for determining whether the mean
difference obtained in a research study is significantly greater
than can be explained by sampling error The hypothesis test
provides a standardized, formal procedure for determining whether
the mean difference obtained in a research study is significantly
greater than can be explained by sampling error Slide 6 Two
Population Distributions Slide 7 Hypothesis Tests and Effect Size
with the Independent Measures t Statistic To prepare the data for
analysis, the first step is to compute the sample mean and SS (or
s, or s 2 ) for each of the two samples. To prepare the data for
analysis, the first step is to compute the sample mean and SS (or
s, or s 2 ) for each of the two samples. The hypothesis test
follows the same four-step procedure outlined in Chapters 8 and 9.
The hypothesis test follows the same four-step procedure outlined
in Chapters 8 and 9. Slide 8 Elements of a t statistics:
Single-Sample & Independent Measures NOTE: This alternative
formula on the bottom left for pooled variance works when you have
sample variances for the first and second samples (s 2 ) and not
sum of squares (SS) as noted here Slide 9 Hypothesis Testing with
the Independent-Measures t Statistic Book Example: Book Example:
For 10 students who watched Sesame Street (group 1) and 10 students
who did not watch the show (group 2), was there a difference in
their average high-school grades. For 10 students who watched
Sesame Street (group 1) and 10 students who did not watch the show
(group 2), was there a difference in their average high-school
grades. Key Information to Run t-tests for independent- measures
Key Information to Run t-tests for independent- measures n 1 = 10n
2 = 10 n 1 = 10n 2 = 10 M 1 = 93M 2 = 85 M 1 = 93M 2 = 85 SS 1 =
200SS 2 = 160 SS 1 = 200SS 2 = 160 Slide 10 Hypothesis Testing with
the Independent-Measures t Statistic Step 1: Step 1: State the
hypotheses and select the level. For the independent-measures test,
H 0 states that there is no difference between the two population
means. State the hypotheses and select the level. For the
independent-measures test, H 0 states that there is no difference
between the two population means. For a two-tailed test, the
hypotheses are as follows: For a two-tailed test, the hypotheses
are as follows: Example: H 0 : 1 - 2 = 0 Example: H 0 : 1 - 2 = 0 H
1 : 1 - 2 0 H 1 : 1 - 2 0 Select : = 0.01 (two tail) Select : =
0.01 (two tail) For a one-tail test, H 0 : 1 - 2 0 For a one-tail
test, H 0 : 1 - 2 0 H 1 : 1 - 2 > 0 H 1 : 1 - 2 > 0 Slide 11
Hypothesis Testing with the Independent-Measures t Statistic Step
2: Step 2: Locate the critical region. The critical values for the
t statistic are obtained using degrees of freedom that are
determined by adding together the df value for the first sample and
the df value for the second sample. For two samples with n = 10 in
each sample, df is calculated as follows: Locate the critical
region. The critical values for the t statistic are obtained using
degrees of freedom that are determined by adding together the df
value for the first sample and the df value for the second sample.
For two samples with n = 10 in each sample, df is calculated as
follows: Example: df = df 1 + df 2 Example: df = df 1 + df 2 df =
(n 1 - 1) + (n 2 -1) df = (n 1 - 1) + (n 2 -1) df = 9 + 9 = 18 df =
9 + 9 = 18 To find the critical t value for 18df at = 0.01, we look
at the t-distribution table: t = +/- 2.878 Slide 12 Hypothesis
Testing with the Independent-Measures t Statistic Step 3: Step 3:
Compute the test statistic. The t statistic for the
independent-measures design has the same structure as the single
sample t introduced in Chapter 9. However, in the
independent-measures situation, all components of the t formula are
doubled: there are two sample means, two population means, and two
sources of error contributing to the standard error in the
denominator. Three key parts to the t-statistic calculation:
Compute the test statistic. The t statistic for the
independent-measures design has the same structure as the single
sample t introduced in Chapter 9. However, in the
independent-measures situation, all components of the t formula are
doubled: there are two sample means, two population means, and two
sources of error contributing to the standard error in the
denominator. Three key parts to the t-statistic calculation: Part
A) Calculate the pooled variance Part A) Calculate the pooled
variance Part B) Use the pooled variance to calculate the estimated
standard error Part B) Use the pooled variance to calculate the
estimated standard error Part C) Compute t-statistic Part C)
Compute t-statistic Slide 13 Hypothesis Testing with the
Independent-Measures t Statistic Slide 14 Step 4: Step 4: Make a
decision. If the t statistic ratio indicates that the obtained
difference between sample means (numerator) is substantially
greater than the difference expected by chance (denominator), we
reject H 0 and conclude that there is a real mean difference
between the two populations or treatments. Make a decision. If the
t statistic ratio indicates that the obtained difference between
sample means (numerator) is substantially greater than the
difference expected by chance (denominator), we reject H 0 and
conclude that there is a real mean difference between the two
populations or treatments. T = 4.00>+2.878, thus reject H 0 and
conclude that there is a significant difference in high school
grades for those who watched Sesame Street as compared to those who
did not watch this show. T = 4.00>+2.878, thus reject H 0 and
conclude that there is a significant difference in high school
grades for those who watched Sesame Street as compared to those who
did not watch this show. Slide 15 Measuring Effect Size for the
Independent-Measures t Effect size for the independent-measures t
is measured in the same way that we measured effect size for the
single-sample t in Chapter 9. Effect size for the
independent-measures t is measured in the same way that we measured
effect size for the single-sample t in Chapter 9. Specifically, you
can compute an estimate of Cohens d or you can compute r 2 to
obtain a measure of the percentage of variance accounted for by the
treatment effect. Specifically, you can compute an estimate of
Cohens d or you can compute r 2 to obtain a measure of the
percentage of variance accounted for by the treatment effect. Slide
16 Measuring Effect Size for the Independent-Measures t Magnitude
of dEvaluation of Effect Size d = 0.2Small effect (mean difference
around 0.2 standard deviations) d = 0.5Medium effect (mean
difference around 0.5 standard deviations) d = 0.8Large effect
(mean difference around 0.8 standard deviations) Percent of
Variance Explained as Measured by r 2 Evaluation of Effect Size r 2
= 0.01 (0.01*100 = 1%)Small effect r 2 = 0.09 (0.09*100 = 9%)Medium
effect r 2 = 0.25 (0.25*100 = 25%)Large effect Slide 17 Removing
Treatment Effects NOTE: We added 4 points to those students who did
not watch Sesame Street NOTE: We subtracted 4 points to those
students who watched Sesame Street Slide 18 Confidence Intervals
and Hypothesis Tests NOTE: Because 0 is not in the 95% C Interval,
we can conclude that the value of 0 is not in the 95% Confidence
Interval. Alternatively, the value of 0 is rejected with 95%
confidence. This is the same as rejecting H 0 with = 0.05 Slide 19
Sample Variance and Sample Size Standard error is positively
related to sample variance (larger variance leads to larger sample
error). Standard error is positively related to sample variance
(larger variance leads to larger sample error). Standard error is
inversely related to sample size (large sample sizes lead to
smaller standard error). Standard error is inversely related to
sample size (large sample sizes lead to smaller standard error).
Larger variance produces smaller t-statistic and reduces the
likelihood of a significant finding. Larger variance produces
smaller t-statistic and reduces the likelihood of a significant
finding. Larger variance also produces smaller measures of effect
size Larger variance also produces smaller measures of effect size
Larger samples produces larger values for t-statistic and increases
the likelihood of rejecting H 0. Larger samples produces larger
values for t-statistic and increases the likelihood of rejecting H
0. Sample size has no effect on Cohens d and only a small influence
on r 2 Sample size has no effect on Cohens d and only a small
influence on r 2 Slide 20 Homogeneity of Variance Assumption Most
hypothesis tests usually work reasonably well even if the set of
underlying assumptions are violated. Most hypothesis tests usually
work reasonably well even if the set of underlying assumptions are
violated. The one notable exception is the assumption of
homogeneity of variance for the independent- measures t test. The
one notable exception is the assumption of homogeneity of variance
for the independent- measures t test. Requires that the two
populations from which the samples are obtained have equal
variances Requires that the two populations from which the samples
are obtained have equal variances Necessary in order to justify
pooling the two sample variances and using the pooled variance in
the calculation of the t statistic Necessary in order to justify
pooling the two sample variances and using the pooled variance in
the calculation of the t statistic Slide 21 If the assumption is
violated, then the t statistic contains two questionable values:
(1) the value for the population mean difference which comes from
the null hypothesis, and (2) the value for the pooled variance. If
the assumption is violated, then the t statistic contains two
questionable values: (1) the value for the population mean
difference which comes from the null hypothesis, and (2) the value
for the pooled variance. The problem is that you cannot determine
which of these two values is responsible for a t statistic that
falls in the critical region. The problem is that you cannot
determine which of these two values is responsible for a t
statistic that falls in the critical region. In particular, you
cannot be certain that rejecting the null hypothesis is correct
when you obtain an extreme value for t. In particular, you cannot
be certain that rejecting the null hypothesis is correct when you
obtain an extreme value for t. Homogeneity of Variance Assumption
Slide 22 If the two sample variances appear to be substantially
different, you should use Hartleys F-max test to determine whether
or not the homogeneity assumption is satisfied. If the two sample
variances appear to be substantially different, you should use
Hartleys F-max test to determine whether or not the homogeneity
assumption is satisfied. If homogeneity of variance is violated,
Box 10.2 presents an alternative procedure for computing the t
statistic that does not involve pooling the two sample variances.
If homogeneity of variance is violated, Box 10.2 presents an
alternative procedure for computing the t statistic that does not
involve pooling the two sample variances. Homogeneity of Variance
Assumption Slide 23 NOTE: If the F-max test rejects the hypothesis
of equal variances, or if you suspect that the homogeneity of
variance assumption is not justified, you should not compute an
independent-measures t-statistic using pooled variance. In such
cases, use the alternative formula for the t-statistic on the next
slide. Slide 24 Alternative formula for t-statistic Alternative
formula for t-statistic Step 1: Calculate the standard error using
the two separate sample variances as in equation 10.1 Step 1:
Calculate the standard error using the two separate sample
variances as in equation 10.1 Step 2: The value of degrees of
freedom Step 2: The value of degrees of freedom for the t-statistic
is adjusted using the for the t-statistic is adjusted using the
following equation: following equation: Homogeneity of Variance
Assumption NOTE: The following alternative formula for the
t-statistic does not pool sample variances and does not require the
homogeneity of variance assumption. Decimal values for df should be
rounded down to the next lower integer. By lowering the df, you
push the boundaries of the critical region farther out. This makes
the test more demanding.
