Student's T test distributions & its Applications

SIT DOLOR AMET
Student’s t
Distribution
& Its
Applications
Savitri Dasigi
Paul Paulson
Vidit Jain
Prof.Kavya S
Presented by: Mentor:

t - distribution
The t-distribution (also called Student’s t distribution) is
a family of distributions that look almost identical to the
normal distribution curve, only a bit shorter and fatter.
The t distribution is used instead of the normal
distribution when you have small samples (for more on
this, see: t-score vs. z-score). The larger the sample size,
the more the t distribution looks like the normal
distribution.

History
• Just past the turn of the 19th century, a major development in the science was
fermenting at Guinness Brewery.
• 1908 - William Sealy Gosset faces a problem at the Guinness brewery
where he worked.
• There didn't exist a better way to test samples at the time for accuracy of
representing the population and measure accuracy of their estimates.
• t-distribution thus arises - a sampling distribution of normal samples with
an unknown population variance.
• It thus became a primary way to understand likely error of an estimate and
becomes source of statistical significance tests.

PDF of the
t-distribution
𝒗 + 𝟏
𝟐
𝝅𝒗
𝒗
𝟐
𝟏 +
𝒕𝟐
𝒗
.−
𝒗+𝟏
𝟐 ; −∞ < 𝒕 < ∞
𝟎 ; 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
𝒇 𝒕 =
( (

• Mean = 0, for v>1 ,else undefined
• Median = 0
• Mode = 0
𝑣
𝑣 − 2
, 𝑓𝑜𝑟 𝑣 > 2
∞ 𝑓𝑜𝑟 1 < 𝑣 < 2
• MGF =Undefined
• Variance =
• Skewness=
0 𝑓𝑜𝑟 𝑣 > 3
Undefined ,otherwise
6
𝑣 − 4
𝑓𝑜𝑟 𝑣 > 4
∞ 𝑓𝑜𝑟 2 < 𝑣 ≤ 4
• Kurtosis =
Characteristics of
t-distribution

t -test
A t-test 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.
The t-test is one of many tests used for the
purpose of hypothesis testing in statistics.

Applications of t-test
Testing
significance
between
hypothetical
mean of sample
and population
mean.
01
Testing
significance
between mean of
2 different
samples.
02
Testing
significance
between mean of
sample post &
prior to stimuli
operation.
03
Testing significance
b/w hypothetical
rank correlation
coefficient and
population rank
correlation
coefficient
04
Testing line of
regression
analysis
05
Testing
significance of
partial and
multiple
correlation
coefficient .
06

The One Sample t Test
determines whether the
sample mean is
statistically different from
a known or hypothesized
population mean.
ONE
SAMPLE
TEST
A two-sample t-test is
used to test the difference
(d0) between two
population means. A
common application is to
determine whether the
means are equal.
INDEPEN
DENT
TWO
SAMPLE
TEST
The Paired Samples t Test
compares two means that
are from the same
individual, object, or
related units.
PAIRED
SAMPLE
TEST
Types of t test

Assumptions
•The scale of measurement applied to the data collected must follow a
continuous or ordinal scale.
•The data to be tested should be a simple random sample, (i.e.) the data
should be collected from a representative, randomly selected portion of
the total population.
•The population data must follow the normal distribution
•Homogeneity of variance is assumed and population variance is
unknown.

Test Procedure for Two Sample t-test
1.Defining the Hypothesis
• 𝑯𝟎 : 𝝁𝟏 = 𝝁𝟐 (Independent population means are same)
• 𝑯𝟏 : 𝝁𝟏 > 𝝁𝟐 ( right tailed)
Or
• 𝑯𝟏 : 𝝁𝟏 < 𝝁𝟐 ( left tailed)
0r
• 𝑯𝟏 : 𝝁𝟏 ≠ 𝝁𝟐 (2 tailed)

The test statistic is a t statistic (t) defined by the following equation:
2. Test statistic.
Where, 𝑥 =
𝑥
𝑛1
𝑦 =
𝑦
𝑛2
𝑠2
=
1
𝑛1+𝑛2−2
( (𝑥𝑖 − 𝑥 )2
+ (𝑦𝑖 − 𝑦 )2
𝑛1 𝑎𝑛𝑑 𝑛2 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒𝑠
𝑡 =
𝒙−𝒚
𝒔
𝟏
𝒏𝟏
+
𝟏
𝒏𝟐
~students t distribution with (n1 +
n2 − 2 )degrees of freedom

3.Critical region and conclusion
(i) If 𝐻1: 𝜇1>𝜇2 ,then the critical value will be 𝑡𝛼(𝑛1+𝑛2 − 2)
we reject 𝐻0 if 𝑡𝑐𝑎𝑙 > 𝑡𝛼(𝑛1+𝑛2 − 2) or P ( T > t ) ≤ 𝛼
(ii) If 𝐻1: 𝜇1 < 𝜇2 ,then the critical value will be −𝑡𝛼(𝑛1+𝑛2 − 2)
we reject 𝐻0 if 𝑡𝑐𝑎𝑙 < - 𝑡𝛼(𝑛1+𝑛2 − 2) or P ( T < -t ) ≤ 𝛼

iii) If 𝐻1: 𝜇1 ≠ 𝜇2 ,then the critical value will be 𝑡𝛼/2(𝑛1+𝑛2 − 2) and −𝑡𝛼/2(𝑛1+𝑛2 − 2)
we reject 𝐻0 if |𝑡𝑐𝑎𝑙| > 𝑡𝛼/2 (𝑛1+𝑛2 − 2)
Or if 𝑡𝑐𝑎𝑙 is falling outside the region (-𝑡𝛼/2 , 𝑡𝛼/2 ) or P ( T >|t| ) ≤ 𝛼

Test Procedure of Paired t test
1.Defining the Hypothesis
• 𝑯𝟎 : 𝝁𝒙 = 𝝁𝒚 (Two related population means are same)
• 𝑯𝟏 : 𝝁𝒙 > 𝝁𝒚 ( right tailed)
Or
• 𝑯𝟏 : 𝝁𝒙 < 𝝁𝒚 ( left tailed)
0r
• 𝑯𝟏 : 𝝁𝒙 ≠ 𝝁𝒚 (2 tailed)

2. Test Statistic
The test statistic for the paired Samples t test, denoted t, is given by :
Where, d=𝑥𝑖 - 𝑦𝑖
𝑑 = 𝑖=1
𝑛
𝑑𝑖
𝑛
s=
1
𝑛−1 𝑖=1
𝑛
(𝑑𝑖 − 𝑑)2
t =
𝒅
𝒔
𝒏
~ students t distribution with (n-1) degrees of freedom

3.Critical region and conclusion
(i) If 𝐻1: 𝜇𝑥>𝜇𝑦 ,then the critical value will be 𝑡𝛼(𝑛 − 1)
we reject 𝐻0 if 𝑡𝑐𝑎𝑙 >𝑡𝛼(n – 1 ) or P ( T > t ) ≤ 𝛼
(ii) If 𝐻1: 𝜇𝑥 < 𝜇𝑦 ,then the critical value will be −𝑡𝛼(n - 1)
we reject 𝐻0 if 𝑡𝑐𝑎𝑙 < - 𝑡𝛼(𝑛 − 1) or P ( T < -t ) ≤ 𝛼

iii) If 𝐻1: 𝜇𝑥 ≠ 𝜇𝑦 ,then the critical value will be 𝑡𝛼/2(n -1 ) and −𝑡𝛼
2
(𝑛 − 1)
we reject 𝐻0 if| 𝑡𝑐𝑎𝑙| > 𝑡𝛼/2 (n - 1)
Or if tcal is falling outside the region (-𝑡𝛼/2 , 𝑡𝛼/2 ) or P ( T > |t| ) ≤ 𝛼

Introduction
And this
proved to be
the best
methodology
for
understating
and predicting
human
behavior
Since the
dawn of the
Sciences of
Psychology
and Statistics
,both had
worked hand
in hand
And with the
breakthrough in
the past many
cooperate
sectors use
psychometric
assessments
This Project
presents a basic
model ----------
big 5
psychometric test
paired with
sampling survey
and Statistical
tools like T-test.

Psychometric
Test
Psychometric tests a
or verbal evaluations
assess the cognitive a
functioning of childr
American Psycholog
(APA)

Big Five
Personality
Test
The big five personality are the most commonly used model of
personality in academic psychology.
The five personality variables which stand out in explaining a
personality are :
Extraversion,neuroticism,aggreableness,conscientiouness and
openness to experience.
This test uses the Big-Five Factor Markers from the International
Personality Item Pool developed by Goldberg(1992)

Our Sample Survey:
• OBJECTIVE: To compare five basic behavioural traits among the different group of
students at NMIMS, Bangalore
• Source of Questionnaire: openpsychometrics.com
• Method of Data Collection: Primary data through google forms
• Total Samples collected :30

Variables Used:
Independent
Variables
Gender
Male
Female
Level of
education
UG
PG
Dependant
Variables
Extroversion
Agreeableness
Consciousness
Imagination
Emotional
Stability

t- Test Assumptions Verification : Normality of Data and Random Samples :
Shapiro Wilks Test for Normality:
R Code with Sample Output:
Q-Q plot (or quantile-quantile
plot) draws the correlation
between a given sample and the
normal distribution.
R Code with Sample Output:

Hypothesis for Independent two sample t-test for Level of Education
Let Ho : There is no significant difference in the mean scores of UG and PG students (i.e.) 𝜇1 = 𝜇2
H1 : There is a significant difference in the mean scores of UG and PG students (i.e.) 𝜇1 ≠ 𝜇2
The above general hypothesis is applied for all five domains namely Extroversion, Agreeableness,
Consciousness, Emotional Stability and Imagination and the analysis is done .

Independent Two Sample t-test R Code:

S.No Educational Level
Size(
N)
Mean S.D S.E. t-value df p-value
1 Extroversion
UG 15 19.33 5.53 1.43
-0.17505 28 0.8623
PG 15 19.47 4.88 1.26
2 Agreeableness
UG 15 19.34 4.97 1.28
-0.91814 28 0.3664
PG 15 20.73 3.20 0.83
3 Consciousness
UG 15 20.73 4.13 1.07
0.47768 28 0.6366
PG 15 20.00 4.28 1.10
4 Emotional Stability
UG 15 17.07 3.88 1.00
-1.6756 28 0.1049
PG 15 19.73 4.79 1.24
5 Imagination
UG 15 22.20 2.76 0.71
1.7172 28 0.09699
PG 15 19.73 4.83 1.25
Independent t-test Analysis with Education level as a Parameter:

Conclusions
• Extroversion:
Since p-value > 0.05 , we accept the null hypothesis and hence we conclude that there is no significant
difference in the Extroversion between the students of UG and PG.
• Agreeableness
difference in the Agreeableness between the students of UG and PG.
• Consciousness
difference in the Consciousness between the students of UG and PG.
• Emotional Stability
difference in the Emotional Stability between the students of UG and PG.
• Imagination
difference in the Imagination between the students of UG and PG.

Hypothesis for Independent t-test taking for Gender
Let Ho : There is no significant difference in the mean scores of Male and female students (i.e.) 𝜇1 = 𝜇2
H1 : There is a significant difference in the mean scores of Male and Female students (i.e.) 𝜇1 ≠ 𝜇2
The above general hypothesis is applied for all five domains namely Extroversion, Agreeableness,
Consciousness, Emotional Stability and Imagination and the analysis is done .

S.No Domain Size(N) Mean S.D S.E. T value D.o.F P-value
1 Extroversion
Male 15 17.33 6.24 1.69
-2.2421 28 0.03306
Female 15 21.266 5.13 0.94
2 Agreeableness
Male 15 21.60 5.28 1.36
2.1914 28 0.03691
Female 15 18.466 1.68 0.43
3 Consciousness
Male 15 19.33 3.96 1.02
-1.386 28 0.1768
Female 15 21.400 4.21 1.09
4
Emotional
Stability
Male 15 17.0 3.21 0.83
-1.7686 28
0.08767
Female 15 19.8 5.23 1.35
5 Imagination
Male 15 21.0 3.76 0.94
0.044145 28 0.9657
Female 15 21.933 4.48 1.16
Independent two sample t-test Analysis with Gender as a Parameter:

Conclusions
• Extroversion:
Since p-value < 0.05 , we reject the null hypothesis and hence we conclude that there is a significant difference
in the Extroversion between the male and female students (i.e.) female students are more extrovert than the
male students.
• Agreeableness
Since p-value < 0.05 , we reject the null hypothesis and hence we conclude that there is a significant
difference in the Agreeableness between the male and female students.
• Consciousness
difference in the Consciousness between the male and female students.
• Emotional Stability
difference in the Emotional Stability between the male and female students.
• Imagination
difference in the Imagination between the male and female students.

Hypothesis for Paired t -test Analysis:
Let Ho : There is no significant difference between the mean scores of any two pairs of behavioural traits
(i.e.) 𝜇1 = 𝜇2
H1: There is a significant difference between the mean scores of any two pairs of behavioural traits (i.e.)
𝜇1 ≠ 𝜇2
This general hypothesis is applied for all the permutations of the domains namely Extroversion &
Agreeableness, Extroversion & Consciousness, Extroversion & Emotional Stability, Extroversion &
Imagination ,Agreeableness & Consciousness, Agreeableness & Emotional Stability, Agreeableness &
Imagination, Consciousness & Emotional Stability, Consciousness & Imagination, Emotional Stability &
Imagination and the analysis is done

Paired t-test R Code:
 # R Code for Paired T Test & Sample output:
 > library("duplr")
 > library(“paireddata")
 > library(“duplr")
 t.test(data30$`Emotionional Stability`,data30$Imagination,
 + paired = TRUE, alternative = "two.sided")
 Paired t-test
 data: data30$`Emotionional Stability` and data30$Imagination
 t = -1.886, df = 29, p-value = 0.06935
 alternative hypothesis: true difference in means is not equalto
0
 95 percent confidence interval:
 -5.3500924 0.2167591
 sample estimates:
 mean of the differences
 -2.566667

Paired t-test Results
Pair of domains Mean Std.Deviation Std. Error t value df mean difference p value
Extroversion &
Agreeableness
19.30 5.13 0.94
-0.48392 29 -0.733 0.6321
20.03 4.16 0.76
Extroversion &
Consciousness
19.30 5.13 0.94
0.92621 29 -1.0667 0.362
20.37 4.15 0.76
Extroversion &
Emotional Stability
19.30 5.13 0.94
0.85646 29 0.9 0.3988
18.40 4.49 0.82
Extroversion &
Imagination
19.30 5.13 0.94
-1.2508 29 -1.6667 0.221
20.97 4.06 0.74
Agreeableness &
Consciousness
20.03 4.16 0.76
-0.2859 29 -0.333 0.777
20.37 4.15 0.76
Agreeableness &
Emotional Stability
20.03 4.16 0.76 1.3995
29
1.633 0.1723
18.40 4.49 0.82
Agreeableness &
Imagination
20.03 4.16 0.76 -0.89797
29
-0.933 0.3766
20.97 4.06 0.74
Consciousness &
Emotional Stability
20.37 4.15 0.76
1.6414 29 1.9667 0.115
18.40 4.49 0.82
Consciousness &
Imagination
20.37 4.15 0.76
-0.53282 29 -0.6 0.5982
20.97 4.06 0.74
Emotional Stability &
Imagination
18.40 4.49 0.82
-1.886 29 -2.5667 0.06935
20.97 4.06 0.74

Conclusions
• Extroversion & Agreeableness:
difference in the extroversion and agreeableness of the students .
• Extroversion & Consciousness
Since p-value >0.05 , we accept the null hypothesis and hence we conclude that there is no significant
difference in the Extroversion & Consciousness of the students .
• Extroversion & Emotional Stability
difference in Extroversion & Emotional Stability of the students .
• Extroversion & Imagination
difference in the Extroversion & Imagination of the students .
• Agreeableness & Consciousness
difference in the Agreeableness & Consciousness of the students.

Conclusions
• Agreeableness & Emotional Stability:
difference in the agreeableness and emotional stability of the students .
• Agreeableness & Imagination
difference in the Agreeableness & Imagination of the students .
• Consciousness & Emotional Stability
difference in the consciousness and emotional stability of the students.
• Consciousness & Imagination
difference in the Consciousness & Imagination of the students .
• Emotional Stability & Imagination
difference in the Emotional Stability & Imagination of the students .

Limitations
• Peer pressure, leading to not filling thesurvey
accurately
• There was no compulsion to fill in the survey. As a
result it leads to lack of data , and hence results
could be misleading .
• The employed psychometric test is of general
purpose and old ,newer industry-oriented
psychometric test could be employed.

Future Scope
The main scope of this project is to empower students and the
organizations ,which could be achieved in ways like:
• Tracking group of students who lack in previously discussed
psychometric domains and hence the organizations can arrange
workshops or other activities to boost the skills of the students.
• Assess the student's psychometric profile with the existing job
domains and hence counsel the student for a better career decision.
• Promote studies and research in psychological statistics to provide
the student with experience and exposure for this emerging field.
Future Scope

Acknowledgement
Lastly, we would take this opportunity to thank Our Mentor
Prof. Kavya S and Prof. Dr. Santosh D for their humble
guidance and time to time involvement throughout the project
as well as for taking their time to teach us about all topics
covered in project without which It would be possible to
compile the project .
We would also like to thank Our Dean Preethi Ma’am and
all our faculty professors for their suggestions and critical
assessment which helped us improve on our short comings in
our project and presentation.
We are highly grateful to All Students of NMIMS
Bengaluru to participated in survey and generously provided
us the data for our project

Student's T test distributions & its Applications

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