# Statistical hypothesis testing, one-sided t-test

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?

• So, it should look like this: $H_{0} : H_{1} \Longleftrightarrow \mu=\mu_{0} : \mu \geq \mu_{0}$ or like this: $H_{0} : H_{1} \Longleftrightarrow \mu<\mu_{0} : \mu \geq \mu_{0}$ – user1111261 Jun 9 '12 at 8:44
• I'm not sure that I'd write it either way, but if I had to choose one, it would be the first... I'd prefer $H_0:\mu \le \mu_0$ vs. $H_1: \mu > \mu_0$ – Gschneider Jun 9 '12 at 13:50