4.10 Reference tables

Table 4.4: Every one-sample test of a mean in this unit. Use z.test() when \(\sigma\) is known and t.test() when it is not; the argument below selects the direction. There are only two procedures, each run three ways.

\(\sigma\)	Direction	Hypotheses	Reject when	R argument
Known	Two-sided	\(H_0: \mu = \mu_0\) vs \(H_A: \mu \neq \mu_0\)	\(\lvert Z_{calc} \rvert > Z_{\alpha/2}\)	alternative = "two.sided"
Known	Left	\(H_0: \mu \geq \mu_0\) vs \(H_A: \mu < \mu_0\)	\(Z_{calc} < -Z_{\alpha}\)	alternative = "less"
Known	Right	\(H_0: \mu \leq \mu_0\) vs \(H_A: \mu > \mu_0\)	\(Z_{calc} > Z_{\alpha}\)	alternative = "greater"
Unknown	Two-sided	\(H_0: \mu = \mu_0\) vs \(H_A: \mu \neq \mu_0\)	\(\lvert t_{calc} \rvert > t_{\alpha/2,\,n-1}\)	alternative = "two.sided"
Unknown	Left	\(H_0: \mu \geq \mu_0\) vs \(H_A: \mu < \mu_0\)	\(t_{calc} < -t_{\alpha,\,n-1}\)	alternative = "less"
Unknown	Right	\(H_0: \mu \leq \mu_0\) vs \(H_A: \mu > \mu_0\)	\(t_{calc} > t_{\alpha,\,n-1}\)	alternative = "greater"

Read down the first column rather than across all six rows. The distribution changes when \(\sigma\) becomes unknown; the direction changes where the rejection region sits. Nothing else differs.

Table 4.5: Standard normal critical values. Every figure below is qnorm() evaluated inline, not copied from a printed table.

\(\alpha\)	Two-tailed, \(Z_{\alpha/2}\)	One-tailed, \(Z_{\alpha}\)
10%	1.645	1.282
5%	1.96	1.645
1%	2.576	2.326