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Transcript
T-Tests and Chi2
Does your sample data reflect the
population from which it is drawn
from?
Single Group Z and T-Tests
• The basic goal of these simple tests is to
show that the distribution of the given data
under examination are not produced by
chance and that there is some systematic
pattern therein.
• Main point is to show the mean of a
sample is reflective of the population.
• Salkind’s text skips a discussion of single
group/sample T-Tests.
Review of Z-Tests
• Recall that a Z-score can measure the location of
a given value on a normal distribution, which can
be expressed as a probability.
• A Z-Test uses the normal distribution to obtain a
test statistic based on some data that can be
compared with a sampling distribution of chance,
which is an abstract construction drawn from the
data.
• This is a parameter estimation, which is an
inference of a sample based on a population of
data.
Problem with Z Tests
• But because we do not often know the
population variance, σ2, we estimate a
single “point estimate” or value
(sample mean).
• However, this sample mean may vary
greatly from the real population mean,
μ. This error is called “sampling error.”
Problem with Z Tests
• A confidence interval is set up to estimate μ.
This is a range of values that is likely to
include the value of the population mean (at
the center of the interval). The larger the
sample, the more the sample mean should
equal the population mean, but there may be
some error within the confidence interval.
How far is the X from μ ?
Student’s T-Test
• Problem: We may not know the mean and
variance of some populations, which
means we cannot do a Z-Test. In this
case, we use a T-test, Student’s T to be
specific, for use with a single group or
sample of data.
• Again, this is when we are not looking
at different groups but a sample of data
as an entirety. We will next examine
differences in groups.
Student’s T-Test
• One uses this test when the population
variance is unknown, as is usually the
case in the social sciences.
• The standard error of the sampling
distribution of the sample mean is
estimated.
• A t distribution (not normal curve, more
platykurtic but mean=0) is used to create
confidence intervals, like critical values.
T Distribution
• Very similar to the Z distribution by
assuming normality.
• Normality is obtained after about 100 data
observations.
• Basic rule of parameter estimation: the
higher the obs (N) of sample the more
reflective of overall population.
The t formula
y  y
t
Sy
N 1
CI  Y  t / 2 (S y / N  1)
For α =.05 and N=30 , t =2.045
95% CI using t-test
• Mean= 20
• Sy = 5
• N= 20
20± 2.086 (5/19) =
20.55 upper
19.45 lower
T-Tests
Independent Samples
T-Tests of Independence
• Used to test whether there is a significant
difference between the means of two
samples.
• We are testing for independence, meaning
the two samples are related or not.
• This is a one-time test, not over time with
multiple observations.
T-Test of Independence
• Useful in experiments where people are
assigned to two groups, when there should
be no differences, and then introduce
Independent variables (treatment) to see if
groups have real differences, which would
be attributable to introduced X variable.
This implies the samples are from different
populations (with different μ).
• This is the Completely Randomized TwoGroup Design.
For example, we can take a random set of
independent voters who have not made up their
minds about who to vote for in the 2004 election.
But we have another suspicion:
H1: watching campaign commercials increases
consumption of Twinkies (snackie cakes), or μ1≠ μ2
Null is μ1= μ2
After one group watches the commercials, but not
the other, we measure Twinkie in-take. We find that
indeed the group exposed to political commercials
indeed ate more Twinkies. We thus conclude that
political advertising leads to obesity.
Two Sample Difference of Means T-Test
t
X1  X 2
2


2


(
n

1
)
s

(
n

1
)
s
n

n
1
1
2
2
1
2





n1  n2  2
n1n2  



(n1  1) s1  (n2  1) s2
n1  n2  2
2
Sp2 =
 n1  n2 
 nn 
 1 2 
2
Pooled variance of the two groups
= common standard deviation of two groups
Two Sample Difference of Means T-Test
• The nominator of the equation captures
difference in means, while the
denominator captures the variation within
and between each group.
• Important point: of interest is the difference
between the sample means, not sample
and population means. However, rejecting
the null means that the two groups under
analysis have different population means.
An example
• Test on GRE verbal test scores by gender:
Females: mean = 50.9, variance = 47.553, n=6
Males: mean=41.5, variance= 49.544, n=10
t
50.9  41.5
 (6  1)47.553  (10  1)49.544  6  10  




6  10  2
 6(10)  

t
t

9.4
48.826(.26667)

9.4
13.02
9.4
t
 2.605
3.608
Now what do we do with this
obtained value?
Steps of Testing and Significance
1. Statement of null hypothesis: if there is
not one then how can you be wrong?
2. Set Alpha Level of Risk: .10, .05, .01
3. Selection of appropriate test statistic:
T-test, chi2, regression, etc.
4. Computation of statistical value: get
obtained value.
5. Compare obtained value to critical
value: done for you for most methods
in most statistical packages.
Steps of Testing and Significance
6. Comparison of the obtained and
critical values.
7. If obtained value is more extreme than
critical value, you may reject the null
hypothesis. In other words, you have
significant results.
8. If point seven above is not true,
obtained is lower than critical, then
null is not rejected.
The critical values
are set by moving
toward the tails of the
distribution. The
higher the
significance
threshold, the more
space under the tail.
Also, hypothesis testing can entail a one or twotailed test, depending on if a hypothesis is
directional (increase/decrease) in nature.
Steps of Testing and Significance
• The curve represents all of the possible
outcomes for a given hypothesis.
• In this manner we move from talking
about a distribution of data to a
distribution of potential values for a
sample of data.
GRE Verbal Example
Obtained Value: 2.605
Critical Value?
Degrees of Freedom: number of cases left after
subtracting 1 for each sample.
Is the null hypothesis supported?
Answer: Indeed, women have higher verbal
skills and this is statistically significant. This
means that the mean scores of each gender
as a population are different.
Let’s try another sample
• D:\POLS 5300 FA04\Comparing Means
examples.xls
• Type in the data in SPSS
Paired T-Tests
• We use Paired T-Tests, test of
dependence, to examine a single sample
subjects/units under two conditions,
such as pretest - posttest experiment.
• For example, we can examine whether a
group of students improves if they retake
the GRE exam. The T-test examines if
there is any significant difference between
the two studies. If so, then possibly
something like studying more made a
difference.
D
n D  ( D )
2
(n  1)
2
ΣD = sum differences
between groups, plus it is
squared.
n = number of paired
groups
Paired T-Tests
• Unlike a test for independence, this test
requires that the two groups/samples being
evaluated are dependent upon each other.
• For example, we can use a paired t-test to
examine two sets of scores across time as
long as they come from the same students.
• If you are doing more than two groups, use
ANOVA.
Let’s Go to SPSS
• Using the data from last time, we will now
analyze the Pre-test/Post-test data for
GRE exams.
• D:\POLS 5300 FA04\Comparing Means
examples.xls
Paired Samples Statistics
Pair
1
TESTSCR1
TESTSCR2
Mean
409.69
448.88
N
16
16
Std. Deviation
200.459
152.679
Std. Error
Mean
50.115
38.170
Paired Samples Correlations
N
Pair 1
TESTSCR1 & TESTSCR2
16
Correlation
.959
Sig.
.000
Paired Samples Test
Paired Differences
Pair 1
TESTSCR1 - TESTSCR2
H0: μ scr1 = μscr2
Mean
-39.19
Std. Deviation
69.155
Std. Error
Mean
17.289
95% Confidence
Interval of the
Difference
Lower
Upper
-76.04
-2.34
whereas research hypothesis H1:
t
-2.267
df
15
Sig. (2-tailed)
.039
H1 :  scr 2   scr1
Nonparametric Test of Chi2
• Used when too many assumptions are violated in
T-Tests:
– Sample size to small to reflect population
– Data are not continuous and thus appropriate for
parametric tests based on normal distributions.
• Chi2 is another way of showing that some pattern
in data is not created randomly by chance.
• Chi2 can be one or two dimensional.
Nonparametric Test of Chi2
• Again, the basic question is what you are
observing in some given data created by
chance or through some systematic
process?
(O  E )
 
E
O  observedfrequency
E  exp ectedfrequecy
2
2
Nonparametric Test of Chi2
• The null hypothesis we are testing here
is that the proportion of occurrences in
each category are equal to each other.
Our research hypothesis is that they are
not equal.
Given the sample size, how many cases
could we expect in each category
(n/#categories)? The obtained/critical
value estimation will provide a coefficient
and a Pr. that the results are random.
Cross-Tabs and Chi2
• One often encounters chi2 with crosstabulations, which are usually used
descriptively but can be used to test
hypotheses.
Party Affiliation * gender (past, present) Crosstabulation
Count
Party Affiliation
Dem
GOP
Total
gender (pas t, present)
Fem
Male
5
6
5
8
10
14
Total
11
13
24
Chi-Square Tests
Pears on Chi-Square
Continuity Correctiona
Likelihood Ratio
Fisher's Exact Test
Linear-by-Linear
Ass ociation
N of Valid Cas es
Value
.120 b
.000
.120
.115
df
1
1
1
1
Asymp. Sig.
(2-s ided)
.729
1.000
.729
Exact Sig.
(2-s ided)
Exact Sig.
(1-s ided)
1.000
.527
.735
24
a. Computed only for a 2x2 table
b. 1 cells (25.0%) have expected count less than 5. The minimum expected count is
4.58.