STUDENT T TEST

Student's t test

Values of 't'

Note 1: A difference between two means is significant (at the given probability level) if the calculated t value is greater than the value given in this table. A probability of p= 0.05 (95% probability of making a correct statement) is usually acceptable for biological work, but p = 0.1 can be used for a "one-tailed" t-test.

Note 2: When comparing two means, the number of degrees of freedom is (n₁ + n₂)-2, where n₁ is the number of replicates of treatment 1, and n₂ is the number of replicates of treatment 2.

Note 3: This table does not show all degrees of freedom. If you want a value between, say 30 and 40, then use the value for 30 df.

Degrees of Freedom	Probability, p
	0.1	0.05	0.01	0.001
1	6.31	12.71	63.66	636.62
2	2.92	4.30	9.93	31.60
3	2.35	3.18	5.84	12.92
4	2.13	2.78	4.60	8.61
5	2.02	2.57	4.03	6.87
6	1.94	2.45	3.71	5.96
7	1.89	2.37	3.50	5.41
8	1.86	2.31	3.36	5.04
9	1.83	2.26	3.25	4.78
10	1.81	2.23	3.17	4.59
11	1.80	2.20	3.11	4.44
12	1.78	2.18	3.06	4.32
13	1.77	2.16	3.01	4.22
14	1.76	2.14	2.98	4.14
15	1.75	2.13	2.95	4.07
16	1.75	2.12	2.92	4.02
17	1.74	2.11	2.90	3.97
18	1.73	2.10	2.88	3.92
19	1.73	2.09	2.86	3.88
20	1.72	2.09	2.85	3.85
21	1.72	2.08	2.83	3.82
22	1.72	2.07	2.82	3.79
23	1.71	2.07	2.82	3.77
24	1.71	2.06	2.80	3.75
25	1.71	2.06	2.79	3.73
26	1.71	2.06	2.78	3.71
27	1.70	2.05	2.77	3.69
28	1.70	2.05	2.76	3.67
29	1.70	2.05	2.76	3.66
30	1.70	2.04	2.75	3.65
40	1.68	2.02	2.70	3.55
60	1.67	2.00	2.66	3.46
120	1.66	1.98	2.62	3.37
�	1.65	1.96	2.58	3.29

1. Reference

http://archive.bio.ed.ac.uk/jdeacon/statistics/table1.html

t-test example

Problem: Sam Sleepresearcher hypothesizes that people who are allowed to sleep for only four hours will score significantly lower than people who are allowed to sleep for eight hours on a cognitive skills test. He brings sixteen participants into his sleep lab and randomly assigns them to one of two groups. In one group he has participants sleep for eight hours and in the other group he has them sleep for four. The next morning he administers the SCAT (Sam's Cognitive Ability Test) to all participants. (Scores on the SCAT range from 1-9 with high scores representing better performance).

SCAT scores
8 hours sleep group (X)	5	7	5	3	5	3	3	9
4 hours sleep group (Y)	8	1	4	6	6	4	1	2

x	(x-Mx)2	y	(y - My)2
5	0	8	16
7	4	1	9
5	0	4	0
3	4	6	4
5	0	6	4
3	4	4	0
3	4	1	9
9	16	2	4
Sx=40	S(x-Mx)2=32	Sy=32	S(y-My)2=46
Mx=5		My=4

*(according to the t sig/probability table with df = 14, t must be at least 2.145 to reach p < .05, so this difference is not statistically significant)

Interpretation: Sam's hypothesis was not confirmed. He did not find a significant difference between those who slept for four hours versus those who slept for eight hours on cognitive test performance.

Reference

http://web.mst.edu/~psyworld/texample.htm

Rosenthal and Jacobson (1968) informed classroom teachers that some of their students showed

unusual potential for intellectual gains. Eight months later the students identified to teachers as

having potentional for unusual intellectual gains showed significiantly greater gains performance

on a test said to measure IQ than did children who were not so identified. Below are the data for

the students in the first grade:

Table 1: Scores for First Graders

Experimental        Comparison

35                                  2

40                                 27

12                                 38

15                                 31

21                                   1

14                                 19

46                                    1

10                                 34

28                                   3

48                                   1

16                                  2

30                                  3

32                                  2

48                                  1

31                                  2

22                                 1

12                                 3

39                               29

19                               37

25                               2

Mean = 27.15 11.95

SD = 12.51 14.62

Formula for T-test for indepentdent groups

 Substituting our values:

Our obtained, or calculated t value is 3.54. Our degrees of freedom equals the total group size (40) minus 2, or 38. Entering a t table with 38 degrees of freedom, we see that for alpha = .05 the tabled value is 2.03 and for alpha = .01, the tabled value is 2.72.

Our calculated value is larger than the tabled value at alpha = .01, so we reject the null hypothesis and accept the alternative hypothesis, namely, that the difference in gain scores is likely the result of the experimental treatment and not the result of chance variation.

Reference

http://www.indiana.edu/~educy520/sec6342/week_10/ttest_exp.pdf