For solving problems involving the chi-square distribution in my syllabus, we use this $\chi^2$ statistical table. However, I have a doubt in reading it.
Suppose there is an experiment where a die is thrown many times. We tabulate the times for which $1$ turns up, $2$ turns up and so on. Now we want to check whether the die is fair.
We can construct an expected values table, in which number of times each digit must show up will be $1/6$ times the total number of throws. Then we calculate $\chi^2=\sum\cfrac{(O_i-E_i)^2}{E_i}$ where $O_i$ and $E_i$ are the observed and the expected values at the $i^{th}$ trial respectively.
Then we consult the table. Degrees of freedom will be $\nu=6-1=5$. Now, my text says that the critical point of rejection is $p=0.05$. That means, if $\chi^2_{calc}<\chi^2_{table}(p=0.05)$, then my null hypothesis is false, and the die is fair.
But then, it also goes on to say that if $\chi^2_{calc}<\chi^2_{table}(p=0.95)$, then we would say that the die is more than $95\%$ fair. So, that must mean we are rejecting null hypothesis if we find that the die is just more than $5\%$ fair. Is that correct? I find it unusual that we would reject the null hypothesis only on $5\%$ confidence against it.