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# T Test Calculator

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T test check the two sets of samples are different from each other and also is used when the two normal distributions are unknown. T test is a statistical hypothesis test where it checks if there is a null hypothesis followed by a t distribution. The T test calculator will calculate the t test value by performing hypothesis test of the two given samples.

## Steps to Find the T Test Value

Step 1: Note the given two sets of samples.

Step 2: First find the mean and then find the standard deviation of each set of sample.

Step 3: Use the T test formula given below,
t = $\frac{\bar{x_{1}}-\bar{x_{2}}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$

where,
$\bar{x_{1}}$ = mean of first the sample set.
$\bar{x_{2}}$ = mean of second the sample set.
$s_{1}^{2}$ = standard deviation of first sample set.
$s_{2}^{2}$ = standard deviation of second sample set.
$n_{1}$ = number of elements in the first sample set.
$n_{2}$ = number of elements in the second sample set.

## T Test Solved Problems

Below are given some of the problems based on T test.

### Solved Examples

Question 1: Determine the T test for the two sample sets given below:
Sample 1 : 5,3,3,9,5,7,5,3
Sample 2 : 6,4,1,2,8,1,4,6

Solution:

Step 1 : Given two sets of sample :
Sample 1 : 5,3,3,9,5,7,5,3
Sample 2 : 6,4,1,2,8,1,4,6

Step 2 : Let us consider first sample set as $x_1$ and second sample set as $x_2$.
$Mean \ \bar{x_{1}}$ = $\frac{5+3+3+9+5+7+5+3}{8}$ = $\frac{40}{8}$ = 5.
$Mean \ \bar{x_{2}}$ = $\frac{6+4+1+2+8+1+4+6}{8}$ = $\frac{32}{8}$ = 4.

 $x_1$ $(x_{1} - \bar{x_{1}})^{2}$ $x_2$ $(x_{2} - \bar{x_{2}})^{2}$ 5 0 6 4 3 4 4 0 3 4 1 9 9 16 2 4 5 0 8 16 7 4 1 9 5 0 4 0 3 4 6 4

$\sum (x_{1} - \bar{x_{1}})^{2}$ = 0+4+4+16+0+4+0+4 = 32
$\sum (x_{2} - \bar{x_{2}})^{2}$ = 4+0+9+4+16+9+0+4 = 46

Standard deviation for first sample,
$(s_{1})^{2}$ = $\frac{\sum (x_{1} - \bar{x_{1}})^{2}}{n_{1}}$

$(s_{1})^{2}$ = $\frac{32}{8}$ = 4

Standard deviation for second sample,
$(s_{2})^{2}$ = $\frac{\sum (x_{2} - \bar{x_{2}})^{2}}{n_{2}}$

$(s_{2})^{2}$ = $\frac{46}{8}$ = 5.75

t = $\frac{\bar{x_{1}}-\bar{x_{2}}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$

t = $\frac{5-4}{\sqrt{\frac{4}{8}+\frac{5.75}{8}}}$

t = $\frac{1}{\sqrt{1.218}}$

t = $\frac{1}{1.1039}$

t = 0.905

Therefore, the T test value = 0.905

Question 2: Determine the T test for the two sample sets given below:
Sample 1 : 10,9,7,14,5,8,11,3,7,12
Sample 2 : 8,5,5,12,7,6,14,4,7,8
Solution:

Step 1 : Given two sets of sample :
Sample 1 : 10,9,7,14,5,8,11,3,7,12
Sample 2 : 8,5,5,12,7,6,14,4,7,8

Step 2 : Let us consider first sample set as $x_1$ and second sample set as $x_2$.
$Mean \ \bar{x_{1}}$ = $\frac{10+9+7+14+5+8+11+3+7+12}{10}$ = $\frac{86}{10}$ = 8.6
$Mean \ \bar{x_{2}}$ = $\frac{8+5+5+12+7+6+14+4+7+8}{10}$ = $\frac{76}{10}$ = 7.6

 $x_1$ $(x_{1} - \bar{x_{1}})^{2}$ $x_2$ $(x_{2} - \bar{x_{2}})^{2}$ 10 1.96 8 0.16 9 0.16 5 6.76 7 2.56 5 6.76 14 29.16 12 19.36 5 12.96 7 0.36 8 0.36 6 2.56 11 5.76 14 40.96 3 31.36 4 12.96 7 2.56 7 0.36 12 11.56 8 0.16

$\sum (x_{1} - \bar{x_{1}})^{2}$ = 1.96 + 0.16 + 2.56 + 29.16 + 0.36 + 5.76 + 31.36 + 2.56 + 11.56 = 98.4
$\sum (x_{2} - \bar{x_{2}})^{2}$ = 0.16 + 6.76 + 6.76 + 19.36 + 0.36 + 2.56 + 40.96 + 12.96 + 0.36 + 0.16 = 90.4

Standard deviation for first sample,
$(s_{1})^{2}$ = $\frac{\sum (x_{1} - \bar{x_{1}})^{2}}{n_{1}}$

$(s_{1})^{2}$ = $\frac{98.4}{10}$ = 9.84

Standard deviation for second sample,
$(s_{2})^{2}$ = $\frac{\sum (x_{2} - \bar{x_{2}})^{2}}{n_{2}}$

$(s_{2})^{2}$ = $\frac{90.4}{10}$ = 9.04

t = $\frac{\bar{x_{1}}-\bar{x_{2}}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$

t = $\frac{8.6-7.6}{\sqrt{\frac{9.84}{10}+\frac{9.04}{10}}}$

t = $\frac{1}{\sqrt{1.888}}$

t = $\frac{1}{1.3740}$

t = 0.7277

Therefore, the T test value = 0.7277