Example 1
Two doctors recommend treating colds with two different methods. 
The results (number of days of the treatment) are
x1={5 8 7 8 4 5 5 6 9 3 5 8 6 8 7 5 8 5 7 5 6 8 4 7 7 5 6};
x2={3 4 9 5 4 9 9 8 3 3 5 3 6 4 5 6 2 2 3 4 2 3};
Test equality of the methods. Normality cannot be assumed.

Results
 pv=0.005
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Example 2
At certain process we have measured the data
xi = {5 12 20 26 29 38 40 45};
yi = {9 7 12 12 27 35 44 76};
Perform the 3rd order polynomial regression and
the exponential regression. Using prediction errors
decide which type of regression is better.  

Results
 SE_p  = 5.98
 SE_e  = 0.32
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Example 3
We are going to test if the tire removal on left and right sides 
of the front wheels of cars is equal. The measured values are 'xL' a 'xP'.
Test at the level 0.05.
xL = {1.8 1.0 2.2 0.9 1.5};
xP = {1.5 1.1 2.0 1.1 1.4};

Results
pv  = 0.55
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Example 4
To learn the accuracy of a method for measuring the volume
of manganese in the steel, we performed independent measurements
of several variances. We would like to know the border for which it holds 
that only 5% of possibly measured variances will be greater than 
this border. 
The measured values are
x = {4.3 2.9 5.1 3.3 2.7 4.8 3.6};

Results
 The border is 4.5
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Example 5
A factory produces some products whose weight must
be constant. For the production it uses four machines.
A sample of products has been taken from all machines
to test equality of the product weights. The measured 
values are 
x1={39.4 34.8 35.6 35.1 35.8};
x2={34.4 34.2 35.1 31.1 32.5 33.8}; 
x3={30.2 35.1 34.2 36.3 30.8 35.6 35.2}; 
x4={39.1 34.3 38.6 34.5 36.4 36.1}; 
Test the equality. The data are not normally distributed.

Results
 pv  = 0.033
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Example 6
At a crossroads we have written down numbers of passing cars. 
The measured data are: 
d = 15 10 20 35 10 50    - length of monitoring and  
x = 71 56 98 121 44 271  - number of cars
At the level 0.05 test the hypothesis that the cars go uniformly.  

Results
 pv  = 0.0013
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Example 7
The accuracy of setting of certain machine can be verified 
according to the variance of the products. If the variance is 
greater then the level 90, it is necessary to perform new setting. 
A data sample has been meaasured with values 
mx   = {100 95 100 110 105 110 125 90 100 120 110};
On the level 0.05 test if it is necessary to set the machine. 
Test on the level al = 0.05;

Results
 pv  = 0.28
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Example 8
Tree inspectors are to evaluate functionality of five fast 
food stands. Each inspector evaluates each stand. The result
is the table Tab: rows correspond to inspectors, columns columns
to stands. Evaluation is 1,2,..,10. 10 is the best.
Test if the quality of the stans is equal.
Tab={{10 8 3 9  7}
     {8  7 5 9 10}
     {8  9 5 7  6}}

Results
 pv = 0.155
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Example 9
A harmful substance has leaked into the water tank. A neutralizing 
agent was used and the concentration of the pollutant was measured 
at time points 'xi' . The time instances 'xi' and 
measured concentrations 'yi' are
xi = {5 12 20 26 29 38 65 126};
yi = {19 17 18 17 17 15 14 7};
Compute the correlation coefficient of linear regression
and conclude about its suitability. If suitable, estimate when 
the concentration will be zero.

Results
 Correletion coefficient
  r  = -0.9832531
 Parameters
  p  = -0.0948295xi + 19.305033
 Zero concentration at x_zero
  x_zero  = 203.57626
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Example 10
We monitor three machines. Randomly, we measure their productions
per hour 'x1', 'x2' and 'x3. At the level 0.05, test the equality 
of their production if normality of data is assumed.
x1 = {53 55 49 58 52 61 56 55};
x2 = {49 56 52 45 51 56 44 51};
x3 = {52 53 52 54 55 53 53 52};

Results
  pv  = 0.0541908
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Example 11
At certain process we have measured the data
x1i = {15 12 11  9  9  8  5  3}';
x2i = {3   9  5 11 28 14 32 58}';
yi = {9 7 22 12 27 31 44 36}';
Perform multivariate linear regression and test its
suitability.

Results
  pv = 0.045:
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Example 12
A connection between color of eyes and hair has been investigated.
In a collected data sample we obtained the following frequencies

eyes \ hair   light   brown   dark
blue           {90      75     55}
gray           {96     136     88}
brown         {108     135    119}

At the level 0.05 test the hypothesis that the color of eyes 
and hair are independent.

Results
 pv  = 0.017
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Example 13
At the motorway with recommended speed 80 km/h we monitored
the speeds of passing cars and obtained data 
x = {78 86 65 92 83 92 85 66 42 82 
	 99 92 75 81 66 76 89 76 97 76
	 75 56 76 78 96 77 86 79 86 93};
Test H0: the ratio of drivers that exceed the recommended speed 
by more than 3 km/h is not greater then 0.2. Test at alpha = 0.05.

Results
  pv=0.033
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