The document discusses the t-test, which is a statistical method used to determine if there is a significant difference between the means of two groups. It can be used to compare the means of two independent groups, related groups, or a group's mean to a hypothesized population mean. There are assumptions that must be met for a t-test, including independent observations, normal distribution of data, and homogeneity of variances. The t-test calculates a t-score or t-value which is compared to a critical value to determine if the null hypothesis can be rejected.
Parametric vs Nonparametric Tests: When to use whichGönenç Dalgıç
There are several statistical tests which can be categorized as parametric and nonparametric. This presentation will help the readers to identify which type of tests can be appropriate regarding particular data features.
Analysis of variance (ANOVA) everything you need to knowStat Analytica
Most of the students may struggle with the analysis of variance (ANOVA). Here in this presentation you can clear all your doubts in analysis of variance with suitable examples.
This presentation educates you about T-Test, Key takeways, Assumptions for Performing a t-test, Types of t-tests, One sample t-test, Independent two-sample t-test and Paired sample t-test.
For more topics Stay tuned with Learnbay
T test, Student’s t Test, Key Takeaways, Uses of t-test / Application , Type of t-test, Type of t-test Cont.., One-tailed or two-tailed t-test, Which t-test to Use, t-test Formula, The t-score, Understanding P-values, Degrees of Freedom, How is the t-distribution table used, Example, Example Cont.., Different t-test Formulae, Different t-test Formulae Cont.., Reference.
Parametric vs Nonparametric Tests: When to use whichGönenç Dalgıç
There are several statistical tests which can be categorized as parametric and nonparametric. This presentation will help the readers to identify which type of tests can be appropriate regarding particular data features.
Analysis of variance (ANOVA) everything you need to knowStat Analytica
Most of the students may struggle with the analysis of variance (ANOVA). Here in this presentation you can clear all your doubts in analysis of variance with suitable examples.
This presentation educates you about T-Test, Key takeways, Assumptions for Performing a t-test, Types of t-tests, One sample t-test, Independent two-sample t-test and Paired sample t-test.
For more topics Stay tuned with Learnbay
T test, Student’s t Test, Key Takeaways, Uses of t-test / Application , Type of t-test, Type of t-test Cont.., One-tailed or two-tailed t-test, Which t-test to Use, t-test Formula, The t-score, Understanding P-values, Degrees of Freedom, How is the t-distribution table used, Example, Example Cont.., Different t-test Formulae, Different t-test Formulae Cont.., Reference.
This presentation describes the concept of One Sample t-test, Independent Sample t-test and Paired Sample t-test. This presentation also deals about the procedure to do the t-test through SPSS.
11 T(EA) FOR TWO TESTS BETWEEN THE MEANS OF DIFFERENT GROUPS11 .docxnovabroom
11 T(EA) FOR TWO TESTS BETWEEN THE MEANS OF DIFFERENT GROUPS
11: MEDIA LIBRARY
Premium Videos
Core Concepts in Stats Video
· Testing the Difference Between Two Sample Means
Lightboard Lecture Video
· Independent t Tests
Time to Practice Video
· Chapter 11: Problem 5
Difficulty Scale
(A little longer than the previous chapter but basically the same kind of procedures and very similar questions. Not too hard, but you have to pay attention.)
WHAT YOU WILL LEARN IN THIS CHAPTER
· Using the t test for independent means when appropriate
· Computing the observed t value
· Interpreting the t value and understanding what it means
· Computing the effect size for a t test for independent means
INTRODUCTION TO THE T TEST FOR INDEPENDENT SAMPLES
Even though eating disorders are recognized for their seriousness, little research has been done that compares the prevalence and intensity of symptoms across different cultures. John P. Sjostedt, John F. Schumaker, and S. S. Nathawat undertook this comparison with groups of 297 Australian and 249 Indian university students. Each student was measured on the Eating Attitudes Test and the Goldfarb Fear of Fat Scale. High scores on both measures indicate the presence of an eating disorder. The groups’ scores were compared with one another. On a comparison of means between the Indian and the Australian participants, Indian students scored higher on both of the tests, and this was due mainly to the scores of women. The results for the Eating Attitudes Test were t(544) = −4.19, p < .0001, and the results for the Goldfarb Fear of Fat Scale were t(544) = −7.64, p < .0001.
Now just what does all this mean? Read on.
Why was the t test for independent means used? Sjostedt and his colleagues were interested in finding out whether there was a difference in the average scores of one (or more) variable(s) between the two groups. The t test is called independent because the two groups were not related in any way. Each participant in the study was tested only once. The researchers applied a t test for independent means, arriving at the conclusion that for each of the outcome variables, the differences between the two groups were significant at or beyond the .0001 level. Such a small chance of a Type I error means that there is very little probability that the difference in scores between the two groups was due to chance and not something like group membership, in this case representing nationality, culture, or ethnicity.
Want to know more? Go online or to the library and find …
Sjostedt, J. P., Schumaker, J. F., & Nathawat, S. S. (1998). Eating disorders among Indian and Australian university students. Journal of Social Psychology, 138(3), 351–357.
LIGHTBOARD LECTURE VIDEO
Independent t Tests
THE PATH TO WISDOM AND KNOWLEDGE
Here’s how you can use Figure 11.1, the flowchart introduced in Chapter 9, to select the appropriate test statistic, the t test for independent means. Follow along the highlighted sequence of steps in Figure 1.
11 T(EA) FOR TWO TESTS BETWEEN THE MEANS OF DIFFERENT GROUPS11 .docxhyacinthshackley2629
11 T(EA) FOR TWO TESTS BETWEEN THE MEANS OF DIFFERENT GROUPS
11: MEDIA LIBRARY
Premium Videos
Core Concepts in Stats Video
· Testing the Difference Between Two Sample Means
Lightboard Lecture Video
· Independent t Tests
Time to Practice Video
· Chapter 11: Problem 5
Difficulty Scale
(A little longer than the previous chapter but basically the same kind of procedures and very similar questions. Not too hard, but you have to pay attention.)
WHAT YOU WILL LEARN IN THIS CHAPTER
· Using the t test for independent means when appropriate
· Computing the observed t value
· Interpreting the t value and understanding what it means
· Computing the effect size for a t test for independent means
INTRODUCTION TO THE T TEST FOR INDEPENDENT SAMPLES
Even though eating disorders are recognized for their seriousness, little research has been done that compares the prevalence and intensity of symptoms across different cultures. John P. Sjostedt, John F. Schumaker, and S. S. Nathawat undertook this comparison with groups of 297 Australian and 249 Indian university students. Each student was measured on the Eating Attitudes Test and the Goldfarb Fear of Fat Scale. High scores on both measures indicate the presence of an eating disorder. The groups’ scores were compared with one another. On a comparison of means between the Indian and the Australian participants, Indian students scored higher on both of the tests, and this was due mainly to the scores of women. The results for the Eating Attitudes Test were t(544) = −4.19, p < .0001, and the results for the Goldfarb Fear of Fat Scale were t(544) = −7.64, p < .0001.
Now just what does all this mean? Read on.
Why was the t test for independent means used? Sjostedt and his colleagues were interested in finding out whether there was a difference in the average scores of one (or more) variable(s) between the two groups. The t test is called independent because the two groups were not related in any way. Each participant in the study was tested only once. The researchers applied a t test for independent means, arriving at the conclusion that for each of the outcome variables, the differences between the two groups were significant at or beyond the .0001 level. Such a small chance of a Type I error means that there is very little probability that the difference in scores between the two groups was due to chance and not something like group membership, in this case representing nationality, culture, or ethnicity.
Want to know more? Go online or to the library and find …
Sjostedt, J. P., Schumaker, J. F., & Nathawat, S. S. (1998). Eating disorders among Indian and Australian university students. Journal of Social Psychology, 138(3), 351–357.
LIGHTBOARD LECTURE VIDEO
Independent t Tests
THE PATH TO WISDOM AND KNOWLEDGE
Here’s how you can use Figure 11.1, the flowchart introduced in Chapter 9, to select the appropriate test statistic, the t test for independent means. Follow along the highlighted sequence of steps in Figure 1.
A t-test is a type of inferential statistic which is used to determine if there is a significant difference between the means of two groups which may be
related in certain features. The T-test is used as a hypothesis testing tool, which allows testing of an assumption applicable to a population
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2. It is a type of inferential statistic used to determine if there is a significant
difference between the means of two groups, which may be related in certain
features
4. T-TEST | 2MATHWORLD | Presented by GROUP 2
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Background
The t-test is used to test hypotheses about means when the
population variance is unknown (the usual case). Closely
related to z, the unit normal.
Developed by Gossett for the quality control of beer.
Three Types of T-Test: One sample t-test, Paired samples t-
test, and Independent samples t-test
5. T-TEST | 2MATHWORLD | Presented by GROUP 2 7
The t score is a ratio between the difference between two groups and the
difference within the groups. The larger the t score, the more difference there is
between groups. The smaller the t score, the more similarity there is between
groups. A t score of 3 means that the groups are three times as different from each
other as they are within each other. When you run a t test, the bigger the t-value,
the more likely it is that the results are repeatable.
A large t-score tells you that the groups are different.
A small t-score tells you that the groups are similar.
The t Score
6. T-TEST | 2MATHWORLD | Presented by GROUP 2 8
We use t when the population variance is
unknown (the usual case) and sample size is
small (N<100, the usual case). If you use a stat
package for testing hypotheses about means, you
will use t.
The t distribution is a short, fat relative of the
normal. The shape of t depends on its df. As N
becomes infinitely large, t becomes normal.
The t Distribution
7. T-TEST | 2MATHWORLD | Presented by GROUP 2 9
For the t distribution, degrees of freedom are
always a simple function of the sample size, e.g.,
(N-1).
One way of explaining df is that if we know
the total or mean, and all but one score, the last
(N-1) score is not free to vary. It is fixed by the
other scores. 4+3+2+X = 10. X=1
Degrees of Freedom
9. T-TEST | 2MATHWORLD | Presented by GROUP 2
T-Test Assumptions
1 1
1. The first assumption made regarding t-tests concerns the scale of measurement. The assumption for a t-
test is that the scale of measurement applied to the data collected follows a continuous or ordinal scale,
such as the scores for an IQ test.
2. The second assumption made is that of a simple random sample, that the data is collected from a
representative, randomly selected portion of the total population.
3. The third assumption is the data, when plotted, results in a normal distribution, bell-shaped
distribution curve.
4. The fourth assumption is a reasonably large sample size is used. Larger sample size means the
distribution of results should approach a normal bell-shaped curve.
5. The final assumption is the homogeneity of variance. Homogeneous, or equal, variance exists when the
standard deviations of samples are approximately equal.
10. T-TEST | 2MATHWORLD | Presented by GROUP 2
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Spelling Test Scores
Suppose we conducted a study to compare two strategies for teaching spelling.
Group A had a mean score of 19. The range of scores was 16 to 22, and the
standard deviation was 1.5.
Group B had a mean score of 20. The range of scores was 17 to 23, and
the standard deviation was 1.5.
How confident can we be that the difference we found between the means of
Group A occurred because of differences in our strategies than by chance?
11. T-TEST | 2MATHWORLD | Presented by GROUP 2
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A t test allows us to compare the means of two groups and determine how likely
the difference between the two means occurred by chance when there was no
difference in population from which the sample was drawn.
The calculations for a t test requires three pieces of information:
- the difference between the means (mean difference)
- the standard deviation for each group
- and the number of subjects in each group.
12. T-TEST | 2MATHWORLD | Presented by GROUP 2
All other factors being equal, large differences between means are
less likely to occur by chance than small differences.
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13. T-TEST | 2MATHWORLD | Presented by GROUP 2
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The size of the standard deviation also influences the outcome of a t test.
Given the same difference in means, groups with smaller standard
deviations are more likely to report a significant difference than groups with
larger standard deviations.
14. T-TEST | 2MATHWORLD | Presented by GROUP 2
From a practical standpoint, we can see that smaller standard deviations produce less
overlap between the groups than larger standard deviations. Less overlap would indicate
that the groups are more different from each other.
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15. T-TEST | 2MATHWORLD | Presented by GROUP 2
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The size of our sample is also important. The more subjects that are involved in a
study, the more confident we can be that the differences we find between our groups did
not occur by chance.
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16. T-TEST | 2MATHWORLD | Presented by GROUP 2
One Sample t test whether the mean of a single
population is equal to a target value
A correlated (or paired) t test is concerned with the
difference between the average scores of a single
sample of individuals who is assessed at two
different times (such as before treatment and after
treatment) or on two different measures. It can also
compare average scores of samples of individuals
who are paired in some way (such as siblings,
mothers and daughters, persons who are matched in
terms of a particular characteristics).
Types of t tests | Overview
1 8
17. T-TEST | 2MATHWORLD | Presented by GROUP 2 1 9
Types of t tests | Overview
An independent t test compares the averages of
two samples that are selected independently of
each other (the subjects in the two groups are
not the same people). There are two types of
independent t tests: equal variance and
unequal variance.
19. 21
The One Sample t Test determines whether the sample mean is statistically
different from a known or hypothesized population mean. The One Sample t Test
is a parametric test.
This test is also known as:
Single Sample t Test
The variable used in this test is known as:
Test variable
In a One Sample t Test, the test variable is compared against a "test value", which
is a known or hypothesized value of the mean in the population.
One Sample T-Test
20. 22
Statistical difference between a sample mean and a known or hypothesized value of the
mean in the population.
Statistical difference between the sample mean and the sample midpoint of the test
variable.
Statistical difference between the sample mean of the test variable and chance.
This approach involves first calculating the chance level on the test variable. The
chance level is then used as the test value against which the sample mean of the test
variable is compared.
Statistical difference between a change score and zero.
This approach involves creating a change score from two variables, and then
comparing the mean change score to zero, which will indicate whether any change
occurred between the two time points for the original measures. If the mean change
score is not significantly different from zero, no significant change occurred.
One Sample T-Test | Common Uses
21. 23
1. Test variable that is continuous (i.e., interval or ratio level)
2. Scores on the test variable are independent (i.e., independence of observations)
There is no relationship between scores on the test variable
Violation of this assumption will yield an inaccurate p value
3. Random sample of data from the population
4. Normal distribution (approximately) of the sample and population on the test variable
• Non-normal population distributions, especially those that are thick-tailed or
heavily skewed, considerably reduce the power of the test
• Among moderate or large samples, a violation of normality may still yield
accurate p values
5. Homogeneity of variances (i.e., variances approximately equal in both the sample and
population)
6. No outliers
One Sample T-Test | Data Requirements
22. 24
The null hypothesis (H0) and (two-tailed) alternative hypothesis (H1) of the
one sample T test can be expressed as:
H0: µ = 𝑥 ("the sample mean is equal to the [proposed] population mean")
H1: µ ≠ x ("the sample mean is not equal to the [proposed] population
mean")
where µ is a constant proposed for the population mean and x is the
sample mean.
One Sample T-Test | Hypotheses
23. 25
One Sample T-Test | Test Statistic
𝒕 =
𝒙 − 𝝁
𝒔 𝒙
𝑺 𝒙 =
𝑺
𝒏
The test statistic for a One Sample t Test is denoted t, which is calculated using the following
formula:
where
μ = Proposed constant for the population mean
𝑥 = Sample mean
n = Sample size (i.e., number of observations)
𝑠 = Sample standard deviation
𝑠 𝑥 = Estimated standard error of the mean (s/sqrt(n))
The calculated t value is then compared to the critical t value from the t distribution
table with degrees of freedom df = n - 1 and chosen confidence level. If the calculated t
value > critical t value, then we reject the null hypothesis.
25. 28
The Paired Samples t Test compares two means that are from the same individual, object, or
related units. The two means typically represent two different times (e.g., pre-test and post-
test with an intervention between the two time points) or two different but related
conditions or units (e.g., left and right ears, twins). The purpose of the test is to determine
whether there is statistical evidence that the mean difference between paired observations
on a particular outcome is significantly different from zero. The Paired Samples t Test is a
parametric test.
This test is also known as:
The variable used in this test is known as:
Dependent variable, or test variable (continuous), measured at two different times or for
two related conditions or units
Paired Samples T-Test
Dependent t Test Paired t Test Repeated Measures t Test
26. 29
The Paired Samples t Test is commonly used to test the following:
Statistical difference between two time points
Statistical difference between two conditions
Statistical difference between two measurements
Statistical difference between a matched pair
To compare unpaired means between two groups on a continuous outcome that is
normally distributed, choose the Independent Samples t Test.
To compare unpaired means between more than two groups on a continuous outcome
that is normally distributed, choose ANOVA.
To compare paired means for continuous data that are not normally distributed, choose
the nonparametric Wilcoxon Signed-Ranks Test.
To compare paired means for ranked data, choose the nonparametric Wilcoxon Signed-
Ranks Test.
Paired Samples T-Test | Common Uses
27. 30
1. Dependent variable that is continuous (i.e., interval or ratio
level)
Note: The paired measurements must be recorded in two
separate variables.
2. Related samples/groups (i.e., dependent observations)
The subjects in each sample, or group, are the same. This
means that the subjects in the first group are also in the
second group.
3. Random sample of data from the population
4. Normal distribution (approximately) of the difference
between the paired values
5. No outliers in the difference between the two related groups
Paired Samples T-Test | Data Requirements
28. 31
The hypotheses can be expressed in two different ways that express the same idea
and are mathematically equivalent:
H0: µ1 = µ2 ("the paired population means are equal")
H1: µ1 ≠ µ2 ("the paired population means are not equal")
OR
H0: µ1 - µ2 = 0 ("the difference between the paired population means is equal to 0")
H1: µ1 - µ2 ≠ 0 ("the difference between the paired population means is not 0")
where
µ1 is the population mean of variable 1, and
µ2 is the population mean of variable 2.
Paired Samples T-Test | Hypotheses
29. 32
Paired Samples T-Test | Test Statistic
The test statistic for the Paired Samples t Test, denoted t, follows the same formula as the
one sample t test.
where
𝒙 𝐝𝐢𝐟𝐟 = Sample mean
n = Sample size (i.e., number of observations)
𝒔diff = Sample standard deviation
𝒔 𝒙 = Estimated standard error of the mean (s/sqrt(n))
The calculated t value is then compared to the critical t value with df = n - 1 from the t
distribution table for a chosen confidence level. If the calculated t value is greater than the
critical t value, then we reject the null hypothesis (and conclude that the means are
significantly different).
𝒕 =
𝒙 𝒅𝒊𝒇𝒇 − 𝟎
𝒔 𝒙
𝒔 𝒙 =
𝒔 𝒅𝒊𝒇𝒇
𝒏
30. T-TEST | 2MATHWORLD | Presented by GROUP 2
TYPESOFT-TEST
Independent
sampleS
t-test
3 4
31. 35
The Independent Samples t Test compares the means of two independent groups in order to
determine whether there is statistical evidence that the associated population means are
significantly different. The Independent Samples t Test is a parametric test.
This test is also known as:
Independent t Test
Independent Measures t Test
Independent Two-sample t Test
Student t Test
Two-Sample t Test
Uncorrelated Scores t Test
Unpaired t Test
Unrelated t Test
Independent Samples T-Test
The variables used in this test are known as:
Dependent variable, or test variable
Independent variable, or grouping variable
32. 3636
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
Independent Samples T-Test | Common Uses
33. 3737
Your data must meet the following requirements:
1. Dependent variable that is continuous (i.e., interval or ratio level)
2. Independent variable that is categorical (i.e., two or more groups)
3. Cases that have values on both the dependent and independent variables
4. Independent samples/groups (i.e., independence of observations)
There is no relationship between the subjects in each sample. This means that:
a. ▪ Subjects in the first group cannot also be in the second group
No subject in either group can influence subjects in the other group
No group can influence the other group
Violation of this assumption will yield an inaccurate p value
5. Random sample of data from the population
Independent Samples T-Test | Data Requirements
34. 3838
6. Normal distribution (approximately) of the dependent variable for each group
Non-normal population distributions, especially those that are thick-tailed or
heavily skewed, considerably reduce the power of the test
Among moderate or large samples, a violation of normality may still yield accurate
p values
7. Homogeneity of variances (i.e., variances approximately equal across groups)
When this assumption is violated and the sample sizes for each group differ, the p
value is not trustworthy. However, the Independent Samples t Test output also
includes an approximate t statistic that is not based on assuming equal population
variances; this alternative statistic, called the Welch t Test statistic1, may be used
when equal variances among populations cannot be assumed. The Welch t Test is
also known an Unequal Variance T Test or Separate Variances T Test.
8. No outliers
Independent Samples T-Test | Data Requirements
35. 39
The null hypothesis (H0) and alternative hypothesis (H1) of the Independent
Samples t Test can be expressed in two different but equivalent ways:
H0: µ1 = µ2 ("the two population means are equal")
H1: µ1 ≠ µ2 ("the two population means are not equal")
OR
H0: µ1 - µ2 = 0 ("the difference between the two population means is equal to 0")
H1: µ1 - µ2 ≠ 0 ("the difference between the two population means is not 0")
where µ1 and µ2 are the population means for group 1 and group 2, respectively.
Notice that the second set of hypotheses can be derived from the first set by simply
subtracting µ2 from both sides of the equation.
Independent Samples T-Test | Hypotheses
36. 40
Independent Samples T-Test | Test Statistic
The test statistic for an Independent Samples t Test is denoted t. There are actually two
forms of the test statistic for this test, depending on whether or not equal variances are
assumed. Note that the null and alternative hypotheses are identical for both forms of the
test statistic.
TWO TYPES OF INDEPENDENT T-TEST
Equal Variances
Unequal Variances
37. 41
Independent Samples T-Test | Test Statistic
When the two independent samples are assumed to be drawn from populations with
identical population variances (i.e., σ12 = σ22) , the test statistic t is computed as
where
EQUAL VARIANCES ASSUMED
𝒕 =
𝒙 𝟏 − 𝒙 𝟐
𝒔 𝑷
𝟏
𝒏 𝟏
+
𝟏
𝒏 𝟐
𝑺𝒑 =
𝒏 𝟏 − 𝟏 𝑺 𝟏
𝟐
+ 𝒏 𝟐 − 𝟏 𝑺 𝟐
𝟐
𝒏 𝟏 + 𝒏 𝟐 − 𝟐
𝒙 𝟏 = Mean of first sample
𝒙 𝟐 = Mean of second sample
𝒏 𝟏= Sample size (i.e., number of observations)
of first sample
𝒏 𝟐= Sample size (i.e., number of observations)
of second sample
𝒔 𝟏 = Standard deviation of first sample
𝒔 𝟐= Standard deviation of second sample
𝒔 𝒑 = Pooled standard deviation
38. 42
Independent Samples T-Test | Test Statistic
The calculated t value is then compared to the critical t value from the t distribution
table with degrees of freedom df = n1 + n2 - 2 and chosen confidence level. If the calculated
t value is greater than the critical t value, then we reject the null hypothesis.
Note that this form of the independent samples T test statistic assumes equal variances.
Because we assume equal population variances, it is OK to "pool" the sample variances
( 𝒔 𝑷). However, if this assumption is violated, the pooled variance estimate may not be
accurate, which would affect the accuracy of our test statistic (and hence, the p-value).
EQUAL VARIANCES ASSUMED
39. 43
Independent Samples T-Test | Test Statistic
When the two independent samples are assumed to be drawn from populations with
unequal variances (i.e., σ12 ≠ σ22), the test statistic t is computed as:
where
UNEQUAL VARIANCES
𝒕 =
𝒙 𝟏 − 𝒙 𝟐
𝑺 𝑷
𝑺 𝟏
𝟐
𝒏 𝟏
+
𝑺 𝟐
𝟐
𝒏 𝟐
𝒙 𝟏 = Mean of first sample
𝒙 𝟐 = Mean of second sample
𝒏 𝟏= Sample size (i.e., number of observations)
of first sample
𝒏 𝟐= Sample size (i.e., number of observations) of
second sample
𝒔 𝟏 = Standard deviation of first sample
𝒔 𝟐= Standard deviation of second sample
40. 4444
where
UNEQUAL VARIANCES
𝒕 =
𝒙 𝟏 − 𝒙 𝟐
𝑺 𝑷
𝑺 𝟏
𝟐
𝒏 𝟏
+
𝑺 𝟐
𝟐
𝒏 𝟐
𝒙 𝟏 = Mean of first sample
𝒙 𝟐 = Mean of second sample
𝒏 𝟏= Sample size (i.e., number of observations) of
first sample
𝒏 𝟐 = Sample size (i.e., number of observations) of
second sample
𝒔 𝟏 = Standard deviation of first sample
𝒔 𝟐= Standard deviation of second sample
The calculated t value is then compared to the critical t value from the t distribution table with degrees of freedom
and chosen confidence level. If the calculated t value > critical t value, then we reject the null hypothesis.
Note that this form of the independent samples T test statistic does not assume equal variances. This is why both the
denominator of the test statistic and the degrees of freedom of the critical value of t are different than the equal
variances form of the test statistic.
𝐝𝒇 =
𝒔 𝟏
𝟐
𝒏 𝟏
+
𝒔 𝟐
𝟐
𝒏 𝟐
𝟐
𝟏
𝒏 𝟏 − 𝟏
𝒔 𝟏
𝟐
𝒏 𝟏
𝟐
+
𝟏
𝒏 𝟐 − 𝟏
𝒔 𝟐
𝟐
𝒏 𝟐
𝟐
41. T-TEST | 2MATHWORLD | Presented by GROUP 2
4 5
A light bulb manufacturer guarantees that the mean life
of a certain type of light bulb is at least 750 hrs. A
random sample of 36 light bulbs has a mean life of 745
hours with a standard deviation of 60 hours. Do you
have enough evidence to reject the manufacturer’s
claim? Use a=0.05
Sample Problem
where
𝒙 𝐝𝐢𝐟𝐟 = Sample mean
n = Sample size (i.e., number of observations)
𝒔diff = Sample standard deviation
𝒔 𝒙 = Estimated standard error of the mean (s/sqrt(n))
𝒕 =
𝒙 𝒅𝒊𝒇𝒇 − 𝟎
𝒔 𝒙 𝒔 𝒙 =
𝒔 𝒅𝒊𝒇𝒇
𝒏
43. T-TEST | 2MATHWORLD | Presented by GROUP 2
4 7
GROUP 2
Guanzon, Eildriz Niño
Layug, Christine Kinsley
Garcia, Adrian Lash
Lapuz, Alexandra
Leal, Ivan
Mendoza, JC
Lumalanlan, Joseph Ian
Velasquez, Miguel
Magcalas, Neildrin
Marin, Sivel
Editor's Notes
The One Sample t Test can only compare a single sample mean to a specified constant. It can not compare sample means between two or more groups. If you wish to compare the means of multiple groups to each other, you will likely want to run an Independent Samples t Test (to compare the means of two groups) or a One-Way ANOVA (to compare the means of two or more groups).
The Paired Samples t Test can only compare the means for two (and only two) related (paired) units on a continuous outcome that is normally distributed. The Paired Samples t Test is not appropriate for analyses involving the following: 1) unpaired data; 2) comparisons between more than two units/groups; 3) a continuous outcome that is not normally distributed; and 4) an ordinal/ranked outcome.
Note: When testing assumptions related to normality and outliers, you must use a variable that represents the difference between the paired values - not the original variables themselves.
Note: When one or more of the assumptions for the Paired Samples t Test are not met, you may want to run the nonparametric Wilcoxon Signed-Ranks Test instead.
Note: 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. If you wish to compare the means across more than two groups, you will likely want to run an ANOVA.
Note: When one or more of the assumptions for the Independent Samples t Test are not met, you may want to run the nonparametric Mann-Whitney U Test instead.
Researchers often follow several rules of thumb:
Each group should have at least 6 subjects, ideally more. Inferences for the population will be more tenuous with too few subjects.
Roughly balanced design (i.e., same number of subjects in each group) are ideal. Extremely unbalanced designs increase the possibility that violating any of the requirements/assumptions will threaten the validity of the Independent Samples t Test.
Note: When one or more of the assumptions for the Independent Samples t Test are not met, you may want to run the nonparametric Mann-Whitney U Test instead.
Researchers often follow several rules of thumb:
Each group should have at least 6 subjects, ideally more. Inferences for the population will be more tenuous with too few subjects.
Roughly balanced design (i.e., same number of subjects in each group) are ideal. Extremely unbalanced designs increase the possibility that violating any of the requirements/assumptions will threaten the validity of the Independent Samples t Test.