One sample Χ2 hypothesis test for VAR and SD

Null hypothesis H0: already known, established, default, status quo, old, pre-existing, current practice, well-known, working assumption, nothing new, boring. The (generic) parameter φ equals some number a; there is no difference.
Alternative hypothesis HA: new, exciting, hoped/wished, changed, different, research, challenger, the conjecture. Either the parameter σ<a, or σ>a, or σ≠a; there is a difference, there is an effect.
Test if the sample (i.e. its statistic s and its size n) provides enough evidence to overthrow ("warrant rejection of") the null hypothesis. Is the sample statistic s extreme enough.
Either "reject" or "fail to reject" the null hypothesis; never "accept" it. Rejecting it ≡ "support" the alternative.
The alternative hypothesis is neither rejected nor accepted.
Nothing is ever "proven". (would need entire population to prove anything)

Χ2-test for standard deviation σ. Uses σ, s, and n. Test statistic is Χ2.
Population must be normal.
Sample must be SRS random.

The test statistic Χ2 is a measure of discrepancy between the sample statistic s and the H0 claimed value of the population parameter σ.

Given null hypothesis H0: parameter σ = a
Choose one:
HA: parameter σ < a "HA < H0" Left-tailed
HA: parameter σ > a "HA > H0" Right-tailed
HA: parameter σ ≠ a "HA ≠ H0" Two-tailed

σ:
s:
n:

Χ2:         df:   
   Critical value: α=0.05: α=0.01:
   If Left-tailed and Χ2≤CritValue then Reject H0 at that α level.
   If Right-tailed and Χ2≥CritValue then Reject H0 at that α level.

p-value (Chisqr_CDF(Χ2,df)): if p < α, reject H0

Chance that the test statistic would be as much or more if H0 were true.
"If the p is low, the null must go."
Typically the critical/rejection region ("level of significance", α) is chosen to be .05 or .01, so if p is less than it reject H0; if p is not less than the critical value don't reject H0 ("fail to reject").
Probability (area) in a tail (or two) of the test statistic's PDF curve.
If p is high (bigger than α), can't reject H0.
Selecting Two-tailed case doubles the p-value over the One-tailed cases.
Tip: if the p-value is like .9, check that you selected the appropriate "tail" above before failing to reject.

Exs. 

Quarter weights
σ=0.062     n=24  s=.0480164    left-tailed

Video Triola
σ=15     n=13   s=7.2      two-tailed
                                 left-tailed

dollar coin weights
σ=0.04      n=16  s=.02176

body temperatures
σ=2.08       n=106   s=0.62

birth weights   boy vs girl SD
σ=660.2    n=30   s=829.5  

MMs weights
σ= 0.04     n=20   s=0.0337
NB. Also possible to have:
H0: φ≤a and HA: φ>a
H0: φ≥a and HA: φ<a