T-Tests and Chi2

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T-Tests and Chi2

• Does your sample data reflect the population from
  which it is drawn from?

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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.
• Salkinds text skips a discussion of single
  group/sample T-Tests.

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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.

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Problem with Z Tests

• But because we do not often know the population
  variance, s2, 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.

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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
  from µ ?

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Students 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, Students
  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.

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Students 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 mean0) is used to create
  confidence intervals, like critical values.

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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.

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The t formula
For a .05 and N30 , t 2.045
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95 CI using t-test

• Mean 20
• Sy 5
• N 20

20 2.086 (5/19) 20.55 upper 19.45 lower
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T-Tests

• Independent Samples

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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.

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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 Two-Group
  Design.

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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.
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Two Sample Difference of Means T-Test
Pooled variance of the two groups
Sp2
common standard deviation of two groups
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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.

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An example

• Test on GRE verbal test scores by gender
• Females mean 50.9, variance 47.553, n6
• Males mean41.5, variance 49.544, n10

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Now what do we do with this obtained value?
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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.

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Steps of Testing and Significance

1. Comparison of the obtained and critical values.
2. If obtained value is more extreme than critical
   value, you may reject the null hypothesis. In
   other words, you have significant results.
3. If point seven above is not true, obtained is
   lower than critical, then null is not rejected.

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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
two-tailed test, depending on if a hypothesis is
directional (increase/decrease) in nature.
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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.

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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.

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Lets try another sample

• D\POLS 5300 FA04\Comparing Means examples.xls
• Type in the data in SPSS

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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.

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SD sum differences between groups, plus it is
squared. n number of paired groups
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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.

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Lets 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

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H0 µ scr1 µscr2 whereas research
hypothesis H1
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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.

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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?

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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.

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Cross-Tabs and Chi2

• One often encounters chi2 with cross-tabulations,
  which are usually used descriptively but can be
  used to test hypotheses.

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