1. Chapter 10
Estimation and Hypothesis Testing II: Independent and Paired Sample T-Test

2. Learning Objectives:
Describe the difference between an independent and a dependent sample
Describe the difference between and uses of an independent sample t-test and a paired sample t-test
List the assumptions for the independent sample t-test
Test a hypothesis using two independent samples
Construct confidence intervals for the difference in means for the independent sample t-test
List the assumptions for the paired sample t-test
Test a hypothesis using two dependent samples
Construct confidence intervals for the difference in means for the paired sample t-test
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3. Independent Samples
Independent samples are considered to be independent of one another, if they consist of different individuals and the selection of the individuals in the first group does not influence the selection of the second group.
For example, you want to determine whether grade 9 students involved in afterschool extracurricular activities have different grade point averages than grade 9 students who are not involved in after-school extracurricular activities.
LO1
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4. Dependent Samples
Dependent samples are considered to be dependent on one another if they consist of the same individuals and/or the selection of the individuals in the first group determines the selection of the second group.
For example, you want to know if a certain diet is effective at reducing LDL cholesterol levels (the bad kind of cholesterol). You randomly select 20 individuals and measure their LDL cholesterol level. You then have the 20 individuals participate in the diet for three months. At the end of the three months, you measure the LDL cholesterol levels of the same 20 individuals. You now have two samples (or groups).
LO1
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5. Independent and Paired Sample t-test
Independent sample t-test:
When we have two samples that are independent from each other and we want to compare the means of the two samples, we use a independent sample t-test.
Paired sample t-test:
When we have two samples that are dependent on each other and we want to compare the means of the two samples, we use a paired sample t-test.
LO2
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6. Assumptions of the Independent Sample t-test
The variables must be measured at either the interval or ratio level of measurement.
The two groups from which you collect the data must be independent of one another.
The data must be normally distributed.
The variance in the population must be equal for both groups, which means they are not statistically significantly different. This is also called the assumption of homogeneity of variances.
LO3
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7. Independent Sample t-test for Two Means
The general formula for the independent sample t-test is:
Technically, the formula for the independent sample t-test is:
Since we are testing the null hypothesis where μ1= μ2, and in the null hypothesis μ1-μ2=0, we set the term (μ1-μ2) to zero. Since is the same as writing we often leave the term (μ1-μ2,), which equals 0, out of the equation.
LO4
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8. Independent Sample t-test for Two Means
The point estimator for the difference in means is still
However, as we are assuming that σ = σ , we label this common variance σ2, and estimate it by pooling the two sample variances. The pooled estimate of the variance is given by:
For an independent sample t-test, the degrees of freedom are calculated as df = (n1 + n2 – 2)
LO-1; LO-2
LO4
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9. Independent Sample t-test for Two Means
The pooled variance is the weighted average of the two sample variances. So that the group with more observations is more reliable, and gets a higher weighting.
The standard error for a difference in two means is given in (9.2). As we assume that the two population variances are equal, we estimate them both with the common pooled variance s2p. So the standard error becomes:
LO-1; LO-2
LO4
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10. Independent Sample t-test for Two Means
The confidence interval for (μ1 – μ2) will use the t- distribution for small sample sizes. The degrees of freedom for s21 is (n1 – 1) and s22 is (n2 – 1).
When we pool the information we have (n1 – 1) + (n2 – 1) = (n1 + n2 – 2) = f
Where f is the degrees of freedom.
The 95% confidence interval for (μ1 – μ2) will be the point estimator plus or minus the t-table value times the standard error.
LO-1; LO-2
LO5
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11. Independent Sample t-test for Two Means
Then, a 95% confidence interval for μ1 – μ2
Where the degrees of freedom for the t-distribution is f = (n1 + n2 – 2).
LO-1; LO-2
LO5
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12. Test of Hypotheses Recall the 5 steps of hypothesis testing:
Define the Null and Alternative Hypothesis.
Define the Sampling Distribution and Critical Values
Calculate the Test Statistic Using the Sample Data
Make the Decision Regarding the Hypothesis
Interpret the results
LO4
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13. Test of Hypotheses
If the data is collected using two independent samples then we use the corresponding t-test. Again the hypotheses are:
H0: µ1 = µ2
Ha: µ1 ≠ µ2
We can also perform one sided tests if we have prior knowledge of what we would be looking for.
LO-2; LO-5
LO4
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14. Test of Hypotheses We first calculate the pooled sample variance
The t-test for the difference in two means (independent sample case)
LO-2; LO-5
LO4
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15. Test of Hypotheses
We then reject the null hypothesis of equal means if the t-value is too large or too small. The degrees of freedom associated with the two sample t-test is f = n1 + n We look up the t- value with in the right tail and we get the value t0.025:f.
The null hypothesis of equal means is rejected if
|t| > t0.025:f
where t0.025:f is the table value with f=n1+n2-2 degrees of freedom.
LO-2; LO-5
LO4
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16. Independent Sample t-test for Two Means – Example
"There is no criminal type" according to Dr. Charles Goring, Deputy Medical Officer of H.M. Prison, London. He measured facial characteristics on 3000 convicts according to The New York times, November 2, 1913.
The article explains that there was no significant difference between facial measurements of convicts compared to the general population. A similar data set was taken in 1904.
LO-1; LO-2
LO4
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17. Independent Sample t-test for Two Means – Example
The following data represents left ear measurements on convicts at Parkhurst prison in There are two groups used here representing Ordinary Murderers and Other Criminals
Ordinary
murderers
Other Criminals 59 63 60 56 58 62 50 61 68 55
LO-1; LO-2
LO4
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18. Independent Sample t-test for Two Means – Example
For this data the summary statistics are:
Griffiths, G.B. (1904). Measurements of One Hundred and Thirty Criminals. Biometrika, 3,
Sexually Abused
Comparison Group
Sample Means
Sample Variances
Sample Sizes
n1 = 10
n2 = 10
Population Means
1
2
Population Variances
LO-1; LO-2
LO4
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19. Independent Sample t-test for Two Means – Example
LO-1; LO-2
LO4
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20. Independent Sample t-test for Two Means – Example
LO-2; LO-5
LO5
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21. Independent Sample t-test for Two Means – Example (alternative calculation)
LO5
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22. Assumptions for the Paired Sample t-test
The variables must be measured at either the interval or ratio level of measurement.
The two groups must be dependent on one another.
The differences between the two samples must be relatively normally distributed.
LO6
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23. Paired Sample t-test for Two Means
LO-6
LO7
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24. Paired Sample t-test for Two Means
LO-6
LO7
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25. Paired Sample t-test for Two Means
In the example we compared the means of two types of criminals. The subjects were not related in any way. We consider them to be independent samples.
The variance parameter measures the variability in ear measurements of people, and can be quite large as people are highly variable. When we compare two groups of people, we can reduce the variability in the data by using pairs of individuals.
So we see toothpaste commercials comparing two brands of toothpaste and measuring the number of cavities over a six month period.
LO-3
LO7
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26. Paired Sample t-test for Two Means
They could use two groups of subjects and randomly assign them to a brand of toothpaste. But they normally pick sets of twins and randomly assigned one of each twin to a brand of toothpaste.
In this example, the variance measures variability in twins rather than variability in people. We expect the variability to be much less.
We can choose twins, or pairs of subjects matched up by demographical measurements such as age, sex etc. The idea is to control for differences due to a number of other factors and to reduce the standard errors of our estimates.
LO-3
LO7
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27. Paired Sample t-test for Two Means
LO-3
LO7
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28. Paired Sample t-test for Two Means
LO-3
LO7
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29. Paired Sample t-test for Two Means
LO-3
LO8
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30. Paired Sample t-test for Two Means – Example
Consider the left and right ear measurements on the "other criminals" in the example. Note ∑d = 8 and ∑d2 = 84
Subject
Right ear (x1)
Left Ear (x2))
Difference
d=x1 - x2 1 63 2 57 56 3 62 4 58 59 -1 5 -4 6 50 7 8 61 9 55 10
LO-3
LO7
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31. Paired Sample t-test for Two Means – Example
LO-6
LO7,8
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32. Conclusion
The independent sample t-test and the paired sample t-test are useful methods for testing hypotheses with two sample means.
Hypotheses with two sample means and are considered independent use the independent sample t-test.
Testing two sample means from dependent sample use the paired sample t-test.
Our research often involves comparing more than two groups so we need to use what is called an analysis of variance (ANOVA).
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