To find the critical value we need to know 3 things, degrees of freedom, significance or confidence level, whether 1- or 2-tail test.
First, what is the claim: σ (population standard deviation) > 1 inch, and the counterclaim is σ≤1.
The null hypothesis, H0: σ=1 (based on the counterclaim which includes σ=1 as well as σ<1); the alternative hypothesis, H1: σ>1 (claim). This is a right tail test (1-tail) because the right tail of the distribution deals with standard deviations above the mean, that is, greater than.
The critical value is found from a t-table because we only know the sample standard deviation, s. We have dof=19, ɑ=0.01, 1-tail test, and the critical value from the table is 2.54. Now we need our test statistic:
(s-σ)/(s/√n)=(1.2-1)/(1.2/√20)=0.745. Since 0.745<2.54, we fail to reject H0, which means we have insufficient evidence to support the claim or counterclaim.