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`

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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`

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`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`

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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`

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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

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