The document provides an overview of t-tests, including one-sample, paired, and independent t-tests, used for hypothesis testing to evaluate population means. It outlines assumptions, calculations for test statistics, and interpretation of results, including examples that demonstrate their application. The document is aimed at understanding how to apply these statistical methods effectively.

1. Psychological Statistics
INTRODUCTION
TO T-TESTS
GROUP 1

2. What is t Test?
A t-test (also known as Student's t-test) is a tool for evaluating
the means of one or two populations using hypothesis testing. A t-test
may be used to evaluate whether a single group differs from a known
value (a one-sample t-test), whether two groups differ from each other
(an independent two-sample t-test), or whether there is a significant
difference in paired measurements (a paired, or dependent samples t-
test).

3. The one sample t test,
also referred to as a single
sample t test, is a statistical
hypothesis test used to
determine whether the mean
calculated from sample data
collected from a single group is
different from a designated
value specified by the researcher.
3
The t Test for a Single Sample

4. 4
One Sample t Test Assumptions
For a valid test, we need data values that are:
• Independent (values are not related to one another).
• Continuous.
• Obtained via a simple random sample from the population.
Also, the population is assumed to be normally distributed.

5. Example:
In the population, the average IQ is 100. A
team of scientists wants to test a new medication to
see if it has either a positive or negative effect on
intelligence or no effect at all. A sample of 30
participants who have taken the medication has a
mean of 140 with a standard deviation of 20. Did the
medication affect intelligence?
Alpha = 0.05
5

6. • Define Null and Alternative Hypothesis
• State Alpha
• Calculate Degress of Freedom
• State Decision Rule
• Calculate Test Statistic
• State Results
• State Conclusion
6
One Sample t Test

7. • Define Null and Alternative Hypothesis
Ho ; µ = 100
H1 ; µ ≠ 100
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One Sample t Test

8. • Define Null and Alternative Hypothesis
• State Alpha
α = 0.05
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One Sample t Test

9. • Define Null and Alternative Hypothesis
• State Alpha
• Calculate Degress of Freedom
N – 1 = 30 – 1 = 29
9
One Sample t Test

10. • Define Null and Alternative Hypothesis
• State Alpha
• Calculate Degress of Freedom
• State Decision Rule
10
One Sample t Test

11. • Calculate Test Statistic
=140
11
One Sample t Test

12. • State Results
Decision rule: If t is less than -2.0452, reject the
null hypothesis.
t = 10.96
Reject: H0
12
One Sample t Test

13. • State Conclusion
Medication significantly affected
intelligence, t=10.96,p=
13
One Sample t Test

14. The t-test for dependent means (also called a paired samples t-
test or repeated measures t-test) is used when comparing two
sets of related measurements, such as:- The same group of
people tested twice (e.g., pre-test and post-test).- Two
conditions from a within-subjects design (e.g., participants
experience both conditions A and B).
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t-Test for Dependent Means

15. 15
Key Assumptions:
1. Normality: The differences between paired scores
should be approximately normally distributed.
2. Dependent Samples: The data consists of paired
measurements (e.g., before and after for the same
group).
3. 3. Continuous Data: The data should be interval or
ratio.
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Formula
Formula:
The formula for the t-statistic in a dependent means t-test is:

17. 17
Where:
- D is the mean difference between the paired scores.
- SD is the standard deviation of the differences.
- n is the number of paired scores.
t-Test for Dependent Means

18. 18
Steps to perform the t-test:
1. Calculate the differences between the paired measurements.
2. Compute the mean of the differences (D).
3. Find the standard deviation of the differences, (sD).
4. Plug the values into the formula and calculate the t-statistic.
5. Determine the degrees of freedom, (df = n - 1), where ( n ) is the
number of paired observations.
6. Compare the t-value to the critical value from the t-distribution table (or
use the p-value) to determine statistical significance.

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Example of paired scores for dependent t-test:
Imagine a teacher wants to know if a new teaching method
improves student performance. She gives a group of 5
students a math test before and after using the new
method. In this example:
Student Pre-Test Post-Test Difference
(Post-Pre)
1 70 75
2 80 85
3 60 70
4 90 95
5 85 80

20. Effect Size
Tells you how meaningful the relationship
between variables or the difference
between group is. It indicates the practical
of a research outcome.
Effect Size Formula:
20
𝑀1 − 𝑀2
𝜎

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The Distribution of Difference Between Means
difference between sample means from two
independent groups.

22. It’s the difference between the average (means) of two
groups.
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Example: Group 1 has a mean score of 75, and Group 2
has a mean score of 85. The difference is 85 - 75 = 10.
What is the Difference Between Means?

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Sample mean vs Population Mean
● Sample means are taken from subsets of the population.
● Example: Comparing the average scores of students from two
different schools.
• It’s the distribution of all possible differences between two sample
means.
Sampling Distribution of the Difference Between
Means

24. 24
Formula for Difference Between Means
● Formula
Difference =
Where:
● = Mean of sample 1

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Standard Error of the Difference
Formula:
SE
Where:
• , = variances of the two samples
• = sample sizes

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Hypothesis Testing Example
Null Hypothesis : No difference between the means
Alternative Hypothesis
Example:
Group 1: Mean = 100, SD = 10, n = 30
Group 2: Mean = 95, SD = 12, n = 30
Calculate the Difference between means and test for significance.

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• The t-test for independent means compares the difference
between two independent sample means to an expectation about
the difference in the population.
• The Independent Samples t Test can only compare the means for
two (and only two) groups.
• It cannot make comparisons among more than two groups.
• The t-test for independent means requires that there is no
overlap between the two groups in the research design.
HYPOTHESIS TESTING WITH THE T-TEST
FOR INDEPENDENT MEANS

28. The Independent Samples t Test is commonly used
to test the following:
•Statistical differences between the means of two
groups
•Statistical differences between the means of two
interventions
•Statistical differences between the means of two
change scores
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29. Assumptions of the t-Test for
Independent Means
1. Data are Numeric
2. Independence of Observation
3. Normality
4. Equal Variances

30. Thank You
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