N=24 months. Sample standard deviation of monthly returns of 3.60 percent. You now want to test a claim that the standard deviation of monthly returns is less than 4 percent.
H0 >= 16.0, Ha <16.0 Chisq Distribution, 0.05 rejection point for df=23 at 13.091 (23 *3.60^2) / 4^2 = 18.63 for your calculated value of the test stat
First,I don’t really understand why the null is greater than or equal to 16. If the claim is that the stdev of monthly returns is less than 4%, why is the null not <16?
Second my intuition is that because your calculated value is larger, it falls out of the region under the curve you should reject the null hypothesis. The answer is the opposite. You do not reject it. Is this because of the way the null hypothesis is set up?
As a matter of protocol, the null hypothesis has to contain the equal sign, so _H_0: _σ_² < 16 isn’t a properly formed null hypothesis. The only way to create a properly formed null hypothesis that tests exactly what you want to test is to use _H_0: _σ_² ≥ 16, so that Ha: _σ_² < 16.
By the way, despite what financial types publish nonstop, if σ = 4%, then _σ_² = 0.0016, not 16. There’s nothing you or I can do about this reprehensible practice, but we should at least understand that wht they’re writing is garbage. (Note, however, that you arrive at the same value for the calculated χ² statistic.)
The critical values are 11.68 and 38.076 and the chi-square statistic we calculated is 18,63 which falls between the (11.68 - 38.07) critical values. We would reject it if it were outside this interval
S2000 Magician - thank you, the matter of protocol about the null hypothesis having the equal sign makes it a lot easier to understand. Also commiserate with your comment about practices around representation of percentages.
RogerF - the official answer states that the lower rejection point is 13.091 and we will reject the null if we find that chi squared is less than 13.091. May I ask how you computed the critical values you posted?