Question

What are the similarities and differences between the two t tests for the difference between two population means? Under what condition should we use the t test for the matched samples?

Answer #1

The two-sample *t*-test is used when the data of two
samples are statistically independent, while the paired
*t*-test is used when data is in the form of matched
pairs.

To use the two-sample *t*-test, we need to assume that
the data from both samples are normally distributed and they have
the same variances. For paired *t*-test, we only require
that the difference of each pair is normally distributed. An
important parameter in the *t*-distribution is the degrees
of freedom.

For two independent samples with equal sample size n,
*df* = 2(*n*-1) for the two-sample *t*-test.
However, if we have *n* matched pairs, the actual sample
size is *n* (pairs) although we may have data from
2*n* different subjects. The paired *t*-test is, in
fact, a one-sample *t*-test, which makes its *df* =
n-1.

What are the similarities and differences between the two t
tests for the difference between two population means? Under what
condition should we use the t test for the matched samples?

Make comparisons between means and proportions regarding
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Recall the method used to obtain a confidence interval for the
difference between two population means for matched samples.
(a) The following data are from matched samples taken from two
populations. Compute the difference value for each element. (Use
Population 1 − Population 2.)
Element
Population
Difference
1
2
1
11
8
2
7
8
3
9
6
4
12
7
5
13
10
6
15
15
7
15
14
(b) Compute d.
(c) Compute the standard
deviation sd. (Round
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T-tests are used when you want to examine differences but you do
not know everything about the population. There are three types of
t-tests that you may choose to do: one-sample t-test, independent
sample t-test, or dependent sample t-test. You can calculate these
by hand, in SPSS, or in Excel. The instructions below can be used
for SPSS and your textbook offers instructions for using Excel.
Single-sample t-tests
These tests are used when you want to determine the
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11 . Choosing the appropriate test statistic
You are interested in the difference between two population means.
Both populations are normally distributed, and the population
variances σ212 and σ222 are known. You use an independent samples
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appropriate test statistic?
F = √[2/n1 + 2/n2]
F = s1/s2
z = (x̄1 – x̄2) / √[σ2/n1 + σ2/n2]
z = (p̂1 – p̂2) / √[p̂(1 – p̂)(1/n1 + 1/n2)]
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When we compute an independent-samples t-test, we are
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of the independent variable.
True or False
When we compute a paired-samples t-test, we are looking
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two independent groups, each measured on a different level of the
independent variable.
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What is the difference between a t test for independent
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Give an example of when you would use each?

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