Student's T test distributions & its Applications

T test Z test

1. Sit Dolor Amet Student’s t D istribution & Its Applications Savitri Dasigi Paul Paulson Vidit Jain Prof.Kavya S Presented by: Mentor:

2. 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.

3. History Just past the turn of the 19 th 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.

4. PDF of the t -distribution   =   ( (

5. Mean = 0, for v>1 ,else undefined Median = 0 Mode = 0 MGF =Undefined   Variance = Skewness=   Undefined ,otherwise     Kurtosis = Characteristics of t-distribution

6. t -test

7. Applications of t-test

8. Types of t test

9. Assumptions T he 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.

10. Test Procedure for Two Sample t-test 1.Defining the Hypothesis (Independent population means are same) ( right tailed) Or ( left tailed) 0r (2 tailed)  

11. The test statistic is a t statistic (t) defined by the following equation : 2 . T est statistic. Where,     ~ )degrees of freedom  

12. 3.Critical region and conclusion If > ,then the critical value will be ( + we reject if > ( + or P ( T > t ) 𝛼 (ii) If ,then the critical value will be ( + we reject if < - ( + or P ( T < -t ) ≤ 𝛼  

13. iii) If ,then the critical value will be ( + and ( + we reject if | > ( + Or if is falling outside the region (- , ) or P ( T >| t| ) ≤ 𝛼  

14. Test Procedure of Paired t test 1.Defining the Hypothesis (Two related population means are same) ( right tailed) Or ( left tailed) 0r (2 tailed)  

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

16. 3.Critical region and conclusion If > ,then the critical value will be we reject if > (n – 1 ) or P ( T > t ) 𝛼 (ii) If ,then the critical value will be (n - 1 we reject if < - or P ( T < -t ) ≤ 𝛼  

17. iii) If ,then the critical value will be (n -1 and we reject if| > (n - 1 Or if t cal is falling outside the region (- , ) or P ( T > |t| ) ≤ 𝛼  

18. Introduction

19. Psychometric Test Psychometric tests are written, visual , or verbal evaluations administered to assess the cognitive and emotional functioning of children and adults. American Psychological Association (APA)

20. Big Five Personality Test

21. 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

22. Variables Used:

23. t- T est 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:

24. Hypothesis for Independent two sample t-test for Level of Education Let H o : There is no significant difference in the mean scores of UG and PG students (i.e.) H 1 : There is a significant difference in the mean scores of UG and PG students (i.e.) The above general hypothesis is applied for all five domains namely Extroversion, Agreeableness, Consciousness, Emotional Stability and Imagination and the analysis is done .  

25. Independent Two Sample t-test R Code:

26. 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:

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

28. Hypothesis for Independent t-test taking for Gender Let H o : There is no significant difference in the mean scores of Male and female students (i.e.) H 1 : There is a significant difference in the mean scores of Male and Female students (i.e.) The above general hypothesis is applied for all five domains namely Extroversion, Agreeableness, Consciousness, Emotional Stability and Imagination and the analysis is done .  

29. 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:

30. 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 Since p-value > 0.05 , we accept the null hypothesis and hence we conclude that there is no significant difference in the Consciousness between the male and female students. Emotional Stability Since p-value > 0.05 , we accept the null hypothesis and hence we conclude that there is no significant difference in the Emotional Stability between the male and female students. Imagination Since p-value > 0.05 , we accept the null hypothesis and hence we conclude that there is no significant difference in the Imagination between the male and female students.

31. Hypothesis for Paired t -test Analysis: Let H o : There is no significant difference between the mean scores of any two pairs of behavioural traits (i.e.) H 1 : There is a significant difference between the mean scores of any two pairs of behavioural traits (i.e.) 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  

32. 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 95 percent confidence interval: -5.3500924 0.2167591 sample estimates: mean of the differences -2.566667

33. 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

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

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

36. 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.

37. 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

38. Acknowledgemen t 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

39. Thank you