Independent Sample Tests lecture at online course on statistics

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Testing Statistical HypothesisIndependent Sample
t-Test
Heibatollah Baghi, and Mastee Badii
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Research Design
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Steps in Test of Hypothesis

• Determine the appropriate test
• Establish the level of significancea
• Determine whether to use a one tail or two tail
  test
• Calculate the test statistic
• Determine the degree of freedom
• Compare computed test statistic against a tabled
  value

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1. Determine the Appropriate Test

• If comparing a sample to a population, use one
  sample tests.
• If comparing two samples in order to draw
  inferences about group differences in the
  population use two sample t-test.
• Here the test statistic is based on a theoretical
  sampling distribution known as sampling
  distribution of the difference between two means.
• Mdiff
• The standard deviation of such a sampling
  distribution is referred to as the standard error
  of the difference.

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1. Determine the Appropriate Test

• Assumptions and Requirements for the two sample
  test (comparing groups means) are
• Independent variable consists of two levels of a
  nominal-level variable (when there are two and
  only two groups).
• Dependent variable approximates interval-scale
  characteristics or higher.
• Normal distribution or large enough sample size
  to assume normality due to the central limit
  theorem.
• Equal variance assumption of the homogeneity of
  variance
• ?1 2 ?12

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1. Determine the Appropriate Test

• If the two groups are independent of each other
  uses independent group t-test.
• If the two groups are not independent of each
  other use dependent group t-test also known as
  paired t-test.

This lecture focuses on independent sample
t-test which is a parametric test
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2. Establish Level of Significance

• a is a predetermined value
• The convention
• a .05
• a .01
• a .001

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3. Determine Whether to Use a One or Two Tailed
Test

• If testing for equality of means then two tailed
  test
• If testing whether one mean greater/smaller than
  the other then one tailed test

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4. Calculating Test Statistics

• For the independent groups t-test the formula is
• The numerator is the difference in means between
  the two samples, and the denominator is the
  estimated standard error of the difference.

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4. Calculating Test Statistics

• The estimated standard error of the difference is
  estimated on the basis of variances of the two
  samples (Pooled Variance t-test).
• Where
• S21 variance of Group 1
• S22 variance of Group 2
• n 1 number of cases in Group 1
• n 2 number of cases in Group 2

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5. Determine Degrees of Freedom

• Degrees of freedom, df, is value indicating the
  number of independent pieces of information a
  sample can provide for purposes of statistical
  inference.
• Df Sample size Number of parameters estimated
• Df is n1 n2 -2 for two sample test of means
  because the population variance is estimated from
  the sample

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6. Compare the Computed Test Statistic Against a
Tabled Value

• If tc gt ta Reject H0
• If p value lt a Reject H0

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Example of Independent Groups t-tests

• Suppose that we plan to conduct a study to
  alleviate the distress of preschool children who
  are about to undergo the finger-stick procedure
  for a hematocrit (Hct) determination.
• Note Hct of volume of a blood sample
  occupied by cells.

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Example of Independent Groups t-tests, Continued

• Twenty subjects will be used to examine the
  effectiveness of the special treatment.
• 10 subjects randomly assigned to treatment group.
• 10 assigned to a control group that receives no
  special preparation.

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1. Determine the Appropriate Test

• Testing hypothesis about two independent means
  (t-test)
• Dependent variable the childs pulse rate just
  prior to the finger-stick
• Independent variable or grouping variable
  treatment conditions (2 levels)

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1. Determine the Appropriate Test

• Two samples are independent.
• Two populations are normally distributed.
• The assumption of homogeneity of variance.
  (Examine Levenes Test)
• Ho ?1 2 ?12
• Ha ?1 2 ? ?12
• If sig. level or p-value is gt .05, the
  assumption is met.

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2. Establish Level of Significance

• The convention
• a .05
• a .01
• a .001
• In this example, assume a 0.05

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3. Determine Whether to Use a One or Two Tailed
Test

• H0 µ1 µ2
• Ha µ1 ? µ2
• Where
• µ1 population mean for the experimental group
• µ2 population mean for the control group

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4. Calculating Test Statistics
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Rearrange the Data
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4. Calculating Test Statistics (continued)
Group 1 (Experimental) Group
2 (Control) --------------------------------------
--------------------------------------------------
---------- X1
X2
------------ --------------
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4. Calculating Test Statistics (continued)
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6. Compare the Computed Test Statistic Against a
Tabled Value
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6. Compare the Computed Test Statistic Against a
Tabled Value

• If we had chosen a one tail test
• H0 µ1 µ2
• Ha µ1 lt µ2
• 1.73
• The null hypothesis can be rejected

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SPSS Output for Two Sample Independent t-test
Example
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Nature Magnitude of Relationship

• Going Beyond Test of Significance

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Point Biserial Correlation Measures Strength of
the relationship

• Point biserial correlation is similar to Pearson
  r and can be calculated using the same formula or
  using the following formula

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Measures of Practical Significance

• Point biserial correlation also provides
  information about the proportion of explained
  variation in the dependent variable.
• In our example 16 of the variation in the
  childrens pulse rates is explained by the group
  membership.

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Effect Size

• Effect size, gamma (?) is a measure of the
  strength of the relationship between two
  variables in the population and an index of how
  wrong the null hypothesis is.
• The higher the effect size the greater the power
  of the test.

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Effect Size

• To evaluate the magnitude of the difference
  between two means, a mean difference is divided
  by a pooled standard deviation.
• Since researches typically do not have the value
  of the population effect size, it is estimated
  from sample data.

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Most Statistical Tests Assume Randomness

• Perfect randomness is often impossible and so
  researchers try to minimize the different forms
  of bias in their selection of subjects
• Selection bias
• Attrition bias
• Non-response bias
• Cohort bias

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Take Home Lesson

• How to Compare Mean of Two Independent Samples