1) *This test is sensitive to the assumption of normality, so hopefully the problem says the data follows a normal distribution.
Claim: σ < 15
H0: σ >= 15
Ha: σ < 15
Data: n = 12, s = 13.6, alpha = 0.05
Test Stat: X2 = (12 - 1)13.62 / 152 = 9.0425
To get a critical value, use a standard chi-square table and n - 1 degrees of freedom. Here we look up 0.05 and 11 degrees of freedom. This is a left-tailed test with 5% in the left tail, that leaves 95% to the right of the critical value. Look up 95% on the type of chi-square table in STATSprofessor. This gives us a chi-square critical value of 4.575. Thus, we reject if the test stat is less than 4.575. The test stat is not in the rejection region (not less than 4.575), so do not reject H0 and do not support Ha. The data does not support the claim.
The second one is the same overall type of problem, but there are two critical values: 2.5% in each tail (5%/2=2.5%), so we should look up the 23 degrees of freedom with 97.5% to the right and the critical value with 2.5% to the right. 11.689 and 38.076 are the two critical values. If the test stat is greater than 38.076, reject. If it is less than 11.689, reject.
Hope that helps!