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Examples - Inference on the variance of a population

A consumer group is investigating whether there is to little soda in cans from a certain soda factory. It is important that the filling process in the factory is stable, that is the variance is not higher than 10 ml$ ^2$ , so that there is not many bottles with too much or too little soda. To investigate this an experiment was performed where a random sample of 30 bottles was taken and the standard deviation calculated to, $ s = 3.5$ . It can be assumed that the soda level follows a normal distribution. Test the hypothesis that the variance is higher than 10 ml$ ^2$ . Use $ \alpha = 0.05$ .

  1. We would like to test a hypothesis regarding the variance of a normal distribution.

  2. $ \alpha = 0.05.$
  3. The hypotheses are:
    $\displaystyle H_0$ $\displaystyle :$ $\displaystyle \sigma^2 = 10$  
    $\displaystyle H_1$ $\displaystyle :$ $\displaystyle \sigma^2 > 10.$  

  4. The test statistic is:

    $\displaystyle \chi^2 = \frac{(n-1)S^2}{\sigma_0^2}.
$

    $ n - 1 = 30 - 1 = 29$ , $ \sigma_0$ = 10.

    $\displaystyle \chi^2 = \frac{29 \cdot 12.25}{10} = 35.53.
$

  5. $ \chi^2_{0.95,(29)}$ = 42.56. We reject the null hypothesis if $ \chi^2 > 42.56$ . We see that $ \chi^2 < 42.56$ .
  6. We cannot reject the null hypothesis so we cannot conclude that the variance is larger than 10 ml$ ^2$ .


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