Solved – Statistic hypothesis testing – Standard deviation less than 0.4

Question

There was too much snow on the highways, so the mayor of the town sent snowplows to spread some chemicals on them. There is a standard of how much of one specific substance should be present in the compound that is used for spreading…
We measured how much of the substance was present in the compound in 30 different places of the town.
These are the results:

0.91 1.08 0.72 1.07 1.14 0.62 1.06 1.20 0.76 1.19   
0.96 0.73 0.83 0.55 0.79 1.34 0.60 1.19 1.35 1.13   
0.67 0.77 0.48 0.83 1.78 2.25 1.21 0.89 0.83 1.07

We expect that the values have normal distribution.
Verify with a reliability of 99% that the standard deviation is less than 0.4.
[Result: r = 24.546. Hypothesis H0 is not denied.]

I calculated

a) $\mu$ = 1.00…..and…..b) $\sigma$ = 0.367

Now I set …H0: $\sigma^{2} = \sigma^{2}_{o}$… versus…H1: $\sigma^{2} < \sigma^{2}_{o}$

I used this test:
$ \frac {(n-1) s_{n}^{2}} { \sigma_{0}^{2} } \leq \chi^{2} _{ \alpha } (n-1) $

Then, I calculated
$ \frac {(n-1) s_{n}^{2}} { \sigma_{0}^{2} } $ = 24.54 and
$\chi^{2} _{ \alpha } (n-1) $ = 49.58

Now, we see that the inequality holds good, so H0 should be denied!
However the result in the book says the opposite…

Answer 1

As Procrastinator pointed out the test statistic is not significantly large. Don't just look at the number and assume that it is large enough to reject! The chi square statistic has 29 degrees of freedom. It has a mean of 29 and a variance of 58. So the value of the test statistic being 24.54 is not large at all and with the estimate so close to 0.4, this is what we would expect.