Inferential Statistics & Test ofSignificance
Confidence Interval (CI)Confidence Interval (CI)
Y = mean
Z = Z score related with a 95% CI
σ = standard error
Building a CIBuilding a CI
•Assume the following
CI
Why do we use 1.96?
Is there a sample that is different from the mean?
Source; Knoke & Bohrnstead (1991:167)
Significance TestingSignificance Testing
•When we explain some phenomenon wemove beyond description to inferentialstatistics and hypothesis testing.
•Tests of significance allow us to testhypotheses, and when we find arelationship between variables, reject thenull hypothesis.
Hypothesis testing
•Hypothesis testing means that we are testingour null hypothesis (Ho) against somecompeting or alternative hypothesis (H1)
•Normally we choose statements such as
Ho : μy = 100
H1: μy ≠100
Or
H1: μy > 100
Or
H1: μy < 100
Significance TestingSignificance Testing
•Even with high powered statistical measures,there will be results that pop up that are affectedby chance. If we were to keep running ourmodels a thousand times, or fewer, we wouldlikely see some results that do not stem fromsystematic processes.
•Thus, we need to determine at what level ofsignificance we are willing to frame our results.We can never be 100% confident.
•Conventional levels of significance where wereject the null hypothesis are usually .05 or .01.The probability .10 is weakly significant.
Significance TestingSignificance Testing
•When you erroneously reject the nullhypothesis when it is true, you make aType I error. This means you areaccepting a “False Positive” result.
•Think of this as a fiancé test. Thechances of rejecting or saying no tomister or miss “right”
Significance TestingSignificance Testing
•A Type II error occurs when you accept thenull hypothesis when it is not true.
•This is a “False Negative”, when youhave say yes to Mr. or Miss “wrong”
•Type II errors in statistical testing resultfrom too little data, omitted variable bias,and multicollinearity.
Other distributions
•The normal distribution assumes:
1.We know the standard error of the population,however, often we don’t know it.
2.The t-distribution become the best alternativewhen we don’t know the standard error but weknow the standard deviation.
3.As the sample gets bigger the t-distributionapproaches the normal distribution
4.There are other distribution such as chi squareand the that we will discuss latter.
T- Distribution & Normal Distribution
Source: Gujarati (1992:76)
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
The t formula
For α =.05 and N=30 , t =2.045
95% CI using t-test
•Mean= 20
•Sy = 5
•N= 20
20± 2.093 (5/√20) =
22.34 upper
18.88 lower
Why do we care about CI?Why do we care about CI?
•We use CI interval for hypothesis testing
•For instance, we want to know if there is adifference of home values between ElPaso and Boston
•We want to know whether or not takingclass at Kaplan makes a difference in ourGRE scores
•We want to know if there is a differencebetween the treatment and control groups.
Mean Difference testing
Home Values
El Paso
Boston
Las Cruces
Mean USA
T-Tests of Independence
•Used to test whether there is a significantdifference between the means of twosamples.
•We are testing for independence, meaningthe two samples are related or not.
•This is a one-time test, not over time withmultiple observations.
•Example: The values of homes between ElPaso and Boston
T-Test of Independence
•Useful in experiments where people areassigned to two groups, when there shouldbe no differences, and then introduceIndependent variables (treatment) to see ifgroups have real differences, which wouldbe attributable to introduced X variable.This implies the samples are from differentpopulations (with different μ).
•This is the Completely Randomized Two-Group Design.
T-Test of Independence
•For example, we can take a random sample ofhigh school students and divided into twogroups. One gets tutoring for the SAT and theother 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 thegroup exposed to tutoring outperformed theother group. We thus conclude that tutoringmakes a difference.
•Positive increments at a different rate
Treatment
Control
Pre-test
Post-test
Two Sample Difference of Means T-Test
Sp2 =
Pooled variance of the two groups
= common standard deviation of two groups
Two Sample Difference of Means T-Test
•The nominator of the equation capturesdifference in means, while thedenominator captures the variation withinand between each group.
•Important point: of interest is the differencebetween the sample means, not sampleand population means. However, rejectingthe null means that the two groups underanalysis 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
Now what do we do with thisobtained value?
Steps of Testing and Significance
1.Statement of null hypothesis: if there isnot 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: getobtained value.
5.Compare obtained value to criticalvalue: done for you for most methodsin most statistical packages.
Steps of Testing and Significance
6.Comparison of the obtained andcritical values.
7.If obtained value is more extreme thancritical value, you may reject the nullhypothesis. In other words, you havesignificant results.
8.If point seven above is not true,obtained is lower than critical, thennull is not rejected.
GRE Verbal Example
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Paired T-Tests
•We use Paired T-Tests, test ofdependence, to examine a single samplesubjects/units under two conditions,such as pretest - posttest experiment.
•For example, we can examine whether agroup of students improves if they retakethe GRE exam. The T-test examines ifthere is any significant difference betweenthe two studies. If so, then possiblysomething like studying more made adifference.
Paired T-Tests
•Unlike a test for independence, this testrequires that the two groups/samples beingevaluated are dependent upon each other.
•For example, we can use a paired t-test toexamine two sets of scores across time aslong as they come from the same students.
•This is appropriate for a pre-test –post-testresearch design
ΣD = sum differencesbetween groups, plus it issquared.
n = number of pairedgroups
Comparing Test Scores
Midterm
Final
48
71.2
69
73.3
95
96
87
94.2
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What can we conclude?