Statistical hypothesis testing, one-sided t-test

Question

These are the values:

18.2    17.6                                                
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18.9    18.6                                                
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20  19.7    19.8    19.6                                        
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21.4    20.6    21                                          
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21.9    22.1    22.2    22.2    21.6                                    
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23.1    22.6    23.2    23  22.9    23.2    23.2    23  22.5    22.5                
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23.8    24.1    23.8    23.6    24.3    23.7    24  24.2    24.4    23.8    23.5            
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24.7    24.9    25  25.4    25.1    25.3    25.3    25.2    25.2    24.7    24.6    24.5    24.5    
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25.6    26  25.6    25.6    25.6    25.7    25.6    26.3    26.3    26.2    26  25.9    26.2    26
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27  27.2    27  27.2    26.7    27  26.6    27.3    26.8    26.6    26.9    27  26.5    
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28.2    27.7    28.1    27.9    27.6    27.7    28.1    28.2    27.5                    
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28.6    29.1    28.7    29.1                                        
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30.3    29.6    29.7    29.5                                        
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30.9    31  31  30.5                                        
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32.3    31.6
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These are the groups:

 - 17.5 -   18.5
 - 18.5 -   19.5
 - 19.5 -   20.5
 - 20.5 -   21.5
 - 21.5 -   22.5
 - 22.5 -   23.5
 - 23.5 -   24.5
 - 24.5 -   25.5
 - 25.5 -   26.5
 - 26.5 -   27.5
 - 27.5 -   28.5
 - 28.5 -   29.5
 - 29.5 -   30.5
 - 30.5 -   31.5
 - 31.5 -   32.5

$$ \bar{x}= \frac{1}{100} \sum_{i=1}^{100} y_i= \frac{1}{100} \sum_{g=1}^{15} n_g \, y_g = 25.37 $$

$$ \hat{\sigma} = \sqrt{\frac{1}{99} \sum_{g=1}^{15} n_g (y_g - \bar{x})^2 } = 3.1031 $$

Now the EU said that the standard is:
a) at least 25 MPa
b) at least 24 MPa.

Decide if the concrete complies the standard with a reliability of 99%.
a) The hypothesis H1 is that the mean is at least 25 MPa.
b) The hypothesis H2 is that the mean is at least 24 MPa.

The results are
a) We cannot deny H1; we don't know if the concrete complies the standard 25 MPa.
b) The standard 24 MPa is met with a reliability of 99%.]

I know how to calculate the t-score: it's the difference between the measured mean (25.37) and the requested mean (25) divided by standard deviation(3.103) and the whole fraction is multiplied by $\sqrt{n}$ (n=100)...

I get:
a) t = 1.192
b) t = 4.41

In the table for the Student's t-distribution for df=99 and $\alpha=.01$, I found t* = 2.365

a) 1.192 < 2.365, so I cannot decide if H1 should be denied or not.
b) 4.41 > 2.365, so I can say that H2 cannot be deined and the standard of 24 MPa is met with a reliability of 99%.

So, my question is : When do I deny a hypothesis and when can I be sure that the standard is met?

Answer 1

You reject a null hypothesis whenever the observed statistic exceeds the critical value, in your case 2.365. Remember the old analogy to court cases... "Innocent until proven guilty." In statistics, it's essentially H0 until proven H1. So the correct interpretation of your problem is that you don't have enough evidence to conclude that it meets the 25 MPa standard at 99% confidence, but that you can conclude that it meets the 24 MPa standrad at 99% confidence.

As for being sure that the standard is met, if certainty is your goal, you will end up thoroughly disappointed by statistics.

Answer 2

You can never prove the null hypothesis because the Neyman Pearson approach fixes the significance level at some level and then picks the same size large enough to have sufficent power to reject the null hypothesis at a particular alternative of interest. If you want to "prove" the null hypothesis in this framework you need to reverse the null and alternative hypotheses as in equivalence testing (two-sided) or non-inferiority (one-sided).