1. Parametric test - t-test for the following data
3,11,17,28,34
5,8,13,19,28, Significance Level `alpha=0.05` and One-tailed test
Solution:
Step-1: Take the hypothesis
Null Hypothesis `H_0` : There is no significant differentiating between samples
Alternative Hypothesis `H_1` : There is significant differentiating between samples
Step-2: Calculate `S_1^2,S_2^2`
`bar x_1=18.6` and Variance `S_(1)^2=157.3` for `3,11,17,28,34`
| `x` | `dx = x - A = x - 19` | `dx^2` |
| 3 | -16 | 256 |
| 11 | -8 | 64 |
| 17 | -2 | 4 |
| 28 | 9 | 81 |
| 34 | 15 | 225 |
| --- | --- | --- |
| `sum x=93` | `sum (dx)=-2` | `sum (dx)^2=630` |
Mean `bar x = (sum x)/n`
`=(3 + 11 + 17 + 28 + 34)/5`
`=93/5`
`=18.6`
`bar x = 18.6` is not an integer, use assumed mean method
`A = 19`
Sample Variance `S^2 = (sum dx^2 - (sum dx)^2/n)/(n-1)`
`=(630 - (-2)^2/5)/4`
`=(630 - 0.8)/4`
`=629.2/4`
`=157.3`
`bar x_2=14.6` and Variance `S_(2)^2=84.3` for `5,8,13,19,28`
| `x` | `dx = x - A = x - 15` | `dx^2` |
| 5 | -10 | 100 |
| 8 | -7 | 49 |
| 13 | -2 | 4 |
| 19 | 4 | 16 |
| 28 | 13 | 169 |
| --- | --- | --- |
| `sum x=73` | `sum (dx)=-2` | `sum (dx)^2=338` |
Mean `bar x = (sum x)/n`
`=(5 + 8 + 13 + 19 + 28)/5`
`=73/5`
`=14.6`
`bar x = 14.6` is not an integer, use assumed mean method
`A = 15`
Sample Variance `S^2 = (sum dx^2 - (sum dx)^2/n)/(n-1)`
`=(338 - (-2)^2/5)/4`
`=(338 - 0.8)/4`
`=337.2/4`
`=84.3`
Step-3: Calculate `t`
`t=|x_1-x_2|/sqrt((S_1^2)/n_1 + (S_2^2)/n_2)`
`=|18.6-14.6|/sqrt(157.3/5 + 84.3/5)`
`=|4|/sqrt(31.46 + 16.86)`
`=|4|/sqrt(48.32)`
`=|4|/6.9513`
`=0.5754`
Step-4:
Degree of freedom `= n_1 + n_2 - 2 = 5+5-2=8`
Step-5:
`df=8,t_(0.05)=1.8595`
As calculated `t=0.5754 < 1.8595`
So, `H_0` is accepted, Hence there is no significant differentiating between samples
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
