Inferential Statistics

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Inferential Statistics Test of Significance
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Confidence Interval (CI)
Y mean Z Z score related with a 95 CI s
standard error
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Building a CI

• Assume the following

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CI
Why do we use 1.96?
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Source Knoke Bohrnstead (1991167)
Is there a sample that is different from the
mean?
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Significance Testing

• When we explain some phenomenon we move beyond
  description to inferential statistics and
  hypothesis testing.
• Tests of significance allow us to test
  hypotheses, and when we find a relationship
  between variables, reject the null hypothesis.

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Hypothesis testing

• Hypothesis testing means that we are testing our
  null hypothesis (Ho) against some competing or
  alternative hypothesis (H1)
• Normally we choose statements such as
• Ho µy 100
• H1 µy ?100
• Or
• H1 µy gt 100
• Or
• H1 µy lt 100

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Significance Testing

• Even with high powered statistical measures,
  there will be results that pop up that are
  affected by chance. If we were to keep running
  our models a thousand times, or fewer, we would
  likely see some results that do not stem from
  systematic processes.
• Thus, we need to determine at what level of
  significance we are willing to frame our results.
  We can never be 100 confident.
• Conventional levels of significance where we
  reject the null hypothesis are usually .05 or
  .01. The probability .10 is weakly significant.

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Significance Testing

• When you erroneously reject the null hypothesis
  when it is true, you make a Type I error. This
  means you are accepting a False Positive
  result.
• Think of this as a fiancé test. The chances of
  rejecting or saying no to mister or miss right

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Significance Testing

• A Type II error occurs when you accept the null
  hypothesis when it is not true.
• This is a False Negative, when you have say yes
  to Mr. or Miss wrong
• Type II errors in statistical testing result from
  too little data, omitted variable bias, and
  multicollinearity.

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Other distributions

• The normal distribution assumes
• We know the standard error of the population,
  however, often we dont know it.
• The t-distribution become the best alternative
  when we dont know the standard error but we know
  the standard deviation.
• As the sample gets bigger the t-distribution
  approaches the normal distribution
• There are other distribution such as chi square
  and the that we will discuss latter.

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T- Distribution Normal Distribution
The form of the t-distribution depends on the
sample size. As the sample gets Larger there is
not difference between the normal and the
t-distribution
Source Gujarati (199276)
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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 2.093 (5/v20) 22.34 upper 18.88 lower
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Why do we care about CI?

• We use CI interval for hypothesis testing
• For instance, we want to know if there is a
  difference of home values between El Paso and
  Boston
• We want to know whether or not taking class at
  Kaplan makes a difference in our GRE scores
• We want to know if there is a difference between
  the treatment and control groups.

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Mean Difference testing
Mean USA
Boston
Las Cruces
El Paso
Home Values
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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.
• Example The values of homes between El Paso and
  Boston

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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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T-Test of Independence

• For example, we can take a random sample of high
  school students and divided into two groups. One
  gets tutoring for the SAT and the other does not.
  
• Ho µ1? µ2
• H1 µ1 µ2
• After one group gets tutoring, but not the other,
  we compare the scores. We find that indeed the
  group exposed to tutoring outperformed the other
  group. We thus conclude that tutoring makes a
  difference.

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• Positive increments at a different rate

Treatment
Control
Post-test
Pre-test
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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,
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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GRE Verbal Example

• Obtained Value 2.605
• Critical Value?
• Degrees of Freedom number of cases left after
  subtracting 1 for each sample. (14)
• Ho µf µm
• H1 µf ?µm
• Is the null hypothesis (Ho) supported?
• Answer No, 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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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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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.
• This is appropriate for a pre-test post-test
  research design

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SD sum differences between groups, plus it is
squared. n number of paired groups
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Comparing Test Scores
Midterm Final
48 71.2
69 73.3
95 96
87 94.2
50 81.4
75 86.7
74 72.8
88 88
92 95
69 88
75 91.8
86 93.6
73 71.8
60 80.1
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What can we conclude?