T-Test

2MATHWORLD
Presented by GROUP 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

T-TEST | 2MATHWORLD | Presented by GROUP 2 5
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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

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

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

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

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t table

T-Test Assumptions
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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.

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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?

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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.

All other factors being equal, large differences between means are
less likely to occur by chance than small differences.
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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.

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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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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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
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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.

TYPESOFT-TEST
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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

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 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

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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

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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

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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.

TYPESOFT-TEST
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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.
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

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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

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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.
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

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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

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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
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

TYPESOFT-TEST
Independent
sampleS
t-test
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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.
 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

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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

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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
Independent Samples T-Test | Data Requirements

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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

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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

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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

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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

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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

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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
𝒏 𝟏= Sample size (i.e., number of observations)
of first sample
𝒏 𝟐= Sample size (i.e., number of observations) of
second sample

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where
UNEQUAL VARIANCES
𝒕 =
𝒙 𝟏 − 𝒙 𝟐
𝑺 𝑷
𝑺 𝟏
𝟐
𝒏 𝟏
+
𝑺 𝟐
𝟐
𝒏 𝟐
𝒏 𝟏= Sample size (i.e., number of observations) of
first sample
𝒏 𝟐 = Sample size (i.e., number of observations) 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

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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
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

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t table

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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

T-Test

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