Independent Samples t-test

1. Testing the difference between two means

Other than testing a simple population mean, hypothesis testing can also be used to test the difference between two population means. This concept is likely to be introduce during the university statistics class. However, the one-sample test is taught in JC H2 mathematics classes.

Among the formula to the for the mean difference, one of the formula is:

2. Formula for independent t-test

t = (Mean1 – Mean2)/Pooled Variance * Square root of the sum between the inverse of sample size 1 and sample size 2

3. Decision rules fo independent sample t-test

For such test, one can find out the reject region based on the critical value obtained from students t-distribution at certain significance level & degree of freedom. For the degree of freedom, df, it is equal to the sum of the two sample sizes and deducted by 2.

Exercise

Calcuate the test statistics for the following datasets.

Question 1

Y3	X3
9	5
5	8
8	5
10	4
10	7
8	6
3	4
9	5
1	3
8	7
7	5
6	8
10	6
6	1
2	8
7	5
9	6
4	3
4	5
4	3
3	3
3	7
5	1
8	10
3	9
2	8
9	9
8	6
4	3
9	9
9	7
9	4
7	4

Question 2

Y4	X4
8	6
8	2
4	9
3	4
2	8
1	8
8	3
10	8
1	1
8	8
10	8
10	9
4	1
7	4
1	5
10	6
2	3
6	8
10	1
5	2
5	8
5	5
1	9
7	4
5	8
8	9
5	6
10	10
8	10
10	7
2	4
5	6
1	7

Question 3

Y1	X1
10	4
1	6
3	1
8	3
9	7
7	7
9	4
5	2
1	10
3	6
9	2
6	3
5	5
3	6
2	8
10	4
1	6
4	2
9	6
4	8
6	7
7	1
6	2
6	9
7	6
7	6
8	8
5	1
3	3
1	2
1	1
4	7
3	4

Question 4

Y2	X2
5	1
7	10
8	5
8	5
9	3
10	8
4	9
3	3
1	1
1	9
4	4
10	5
3	9
4	7
1	5
4	3
7	4
1	5
9	2
4	10
8	5
10	1
8	7
10	10
7	4
2	10
3	1
4	8
6	6
4	7
5	6
1	7
6	7

Question 5

Y5	X5
8	10
6	3
10	2
6	3
9	2
8	6
3	6
10	1
2	4
2	2
5	9
7	3
9	3
1	10
4	6
4	2
6	2
6	1
4	6
8	7
2	7
6	7
6	5
2	10
9	3
5	10
6	5
9	1
3	9
7	8
7	2
1	3
8	3

Question 6

Y6	X6
3	5
6	6
8	9
1	3
2	2
7	7
8	7
3	9
7	5
2	9
1	2
6	8
8	5
7	3
1	10
9	1
6	9
7	5
7	6
8	10
7	2
9	3
2	5
8	7
5	1
4	7
8	5
10	10
10	5
4	1
2	8
1	6
2	5

Answer

0.231
-0.187
0.438
-0.056
-0.375
0.393

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Y3	X3
9	5
5	8
8	5
10	4
10	7
8	6
3	4
9	5
1	3
8	7
7	5
6	8
10	6
6	1
2	8
7	5
9	6
4	3
4	5
4	3
3	3
3	7
5	1
8	10
3	9
2	8
9	9
8	6
4	3
9	9
9	7
9	4
7	4

Y4	X4
8	6
8	2
4	9
3	4
2	8
1	8
8	3
10	8
1	1
8	8
10	8
10	9
4	1
7	4
1	5
10	6
2	3
6	8
10	1
5	2
5	8
5	5
1	9
7	4
5	8
8	9
5	6
10	10
8	10
10	7
2	4
5	6
1	7

Y1	X1
10	4
1	6
3	1
8	3
9	7
7	7
9	4
5	2
1	10
3	6
9	2
6	3
5	5
3	6
2	8
10	4
1	6
4	2
9	6
4	8
6	7
7	1
6	2
6	9
7	6
7	6
8	8
5	1
3	3
1	2
1	1
4	7
3	4

Y2	X2
5	1
7	10
8	5
8	5
9	3
10	8
4	9
3	3
1	1
1	9
4	4
10	5
3	9
4	7
1	5
4	3
7	4
1	5
9	2
4	10
8	5
10	1
8	7
10	10
7	4
2	10
3	1
4	8
6	6
4	7
5	6
1	7
6	7

Y5	X5
8	10
6	3
10	2
6	3
9	2
8	6
3	6
10	1
2	4
2	2
5	9
7	3
9	3
1	10
4	6
4	2
6	2
6	1
4	6
8	7
2	7
6	7
6	5
2	10
9	3
5	10
6	5
9	1
3	9
7	8
7	2
1	3
8	3

Y6	X6
3	5
6	6
8	9
1	3
2	2
7	7
8	7
3	9
7	5
2	9
1	2
6	8
8	5
7	3
1	10
9	1
6	9
7	5
7	6
8	10
7	2
9	3
2	5
8	7
5	1
4	7
8	5
10	10
10	5
4	1
2	8
1	6
2	5

Y3	X3
9	5
5	8
8	5
10	4
10	7
8	6
3	4
9	5
1	3
8	7
7	5
6	8
10	6
6	1
2	8
7	5
9	6
4	3
4	5
4	3
3	3
3	7
5	1
8	10
3	9
2	8
9	9
8	6
4	3
9	9
9	7
9	4
7	4

Y4	X4
8	6
8	2
4	9
3	4
2	8
1	8
8	3
10	8
1	1
8	8
10	8
10	9
4	1
7	4
1	5
10	6
2	3
6	8
10	1
5	2
5	8
5	5
1	9
7	4
5	8
8	9
5	6
10	10
8	10
10	7
2	4
5	6
1	7

Y1	X1
10	4
1	6
3	1
8	3
9	7
7	7
9	4
5	2
1	10
3	6
9	2
6	3
5	5
3	6
2	8
10	4
1	6
4	2
9	6
4	8
6	7
7	1
6	2
6	9
7	6
7	6
8	8
5	1
3	3
1	2
1	1
4	7
3	4

Y2	X2
5	1
7	10
8	5
8	5
9	3
10	8
4	9
3	3
1	1
1	9
4	4
10	5
3	9
4	7
1	5
4	3
7	4
1	5
9	2
4	10
8	5
10	1
8	7
10	10
7	4
2	10
3	1
4	8
6	6
4	7
5	6
1	7
6	7

Y5	X5
8	10
6	3
10	2
6	3
9	2
8	6
3	6
10	1
2	4
2	2
5	9
7	3
9	3
1	10
4	6
4	2
6	2
6	1
4	6
8	7
2	7
6	7
6	5
2	10
9	3
5	10
6	5
9	1
3	9
7	8
7	2
1	3
8	3

Y6	X6
3	5
6	6
8	9
1	3
2	2
7	7
8	7
3	9
7	5
2	9
1	2
6	8
8	5
7	3
1	10
9	1
6	9
7	5
7	6
8	10
7	2
9	3
2	5
8	7
5	1
4	7
8	5
10	10
10	5
4	1
2	8
1	6
2	5

Y3	X3
9	5
5	8
8	5
10	4
10	7
8	6
3	4
9	5
1	3
8	7
7	5
6	8
10	6
6	1
2	8
7	5
9	6
4	3
4	5
4	3
3	3
3	7
5	1
8	10
3	9
2	8
9	9
8	6
4	3
9	9
9	7
9	4
7	4

Y4	X4
8	6
8	2
4	9
3	4
2	8
1	8
8	3
10	8
1	1
8	8
10	8
10	9
4	1
7	4
1	5
10	6
2	3
6	8
10	1
5	2
5	8
5	5
1	9
7	4
5	8
8	9
5	6
10	10
8	10
10	7
2	4
5	6
1	7

Y1	X1
10	4
1	6
3	1
8	3
9	7
7	7
9	4
5	2
1	10
3	6
9	2
6	3
5	5
3	6
2	8
10	4
1	6
4	2
9	6
4	8
6	7
7	1
6	2
6	9
7	6
7	6
8	8
5	1
3	3
1	2
1	1
4	7
3	4

Y2	X2
5	1
7	10
8	5
8	5
9	3
10	8
4	9
3	3
1	1
1	9
4	4
10	5
3	9
4	7
1	5
4	3
7	4
1	5
9	2
4	10
8	5
10	1
8	7
10	10
7	4
2	10
3	1
4	8
6	6
4	7
5	6
1	7
6	7

Y5	X5
8	10
6	3
10	2
6	3
9	2
8	6
3	6
10	1
2	4
2	2
5	9
7	3
9	3
1	10
4	6
4	2
6	2
6	1
4	6
8	7
2	7
6	7
6	5
2	10
9	3
5	10
6	5
9	1
3	9
7	8
7	2
1	3
8	3

Y6	X6
3	5
6	6
8	9
1	3
2	2
7	7
8	7
3	9
7	5
2	9
1	2
6	8
8	5
7	3
1	10
9	1
6	9
7	5
7	6
8	10
7	2
9	3
2	5
8	7
5	1
4	7
8	5
10	10
10	5
4	1
2	8
1	6
2	5