8.6 Chapter 8 Formulas

Hypothesis Test for One Mean Use z-test when σ is given. Use t-test when s is given. If n < 30, population needs to be normal.	Type I Error- Reject H₀ when H₀ is true. Type II Error- Fail to reject H₀ when H₀ is false.
Z-Test: H₀: μ = μ₀ H₁: μ ≠ μ₀ $Z=\frac{\bar{x}-\mu_{0}}{\left(\frac{\sigma}{\sqrt{n}}\right)}$ TI-84: Z-Test	t-Test: H₀: μ = μ₀ H₁: μ ≠ μ₀ $t=\frac{\bar{x}-\mu_{0}}{\left(\frac{s}{\sqrt{n}}\right)}$ TI-84: T-Test
z-Critical Values Excel: Two-tail: $z_{\alpha / 2}$ = NORM.INV(1– $\alpha$ /2,0,1) Right-tail: $z_{1-\alpha}$ = NORM.INV(1– $\alpha$ ,0,1) Left-tail: $z_{\alpha}$ = NORM.INV( $\alpha$ ,0,1) TI-84: Two-tail: $z_{\alpha / 2}$ = invNorm(1– $\alpha$ /2,0,1) Right-tail: $z_{1-\alpha}$ = invNorm(1– $\alpha$ ,0,1) Left-tail: $z_{\alpha}$ = invNorm( $\alpha$ ,0,1)	t-Critical Values Excel: Two-tail: $t_{\alpha / 2}$ =T.INV(1– $\alpha$ /2,df) Right-tail: $t_{1-\alpha}$ = T.INV(1– $\alpha$ ,df) Left-tail: $t_{\alpha}$ = T.INV( $\alpha$ ,df) TI-84: Two-tail: $t_{\alpha / 2}$ = invT(1– $\alpha$ /2,df) Right-tail: $t_{1-\alpha}$ = invT(1– $\alpha$ ,df) Left-tail: $t_{\alpha}$ = invT( $\alpha$ ,df)
Hypothesis Test for One Proportion H₀: p = p₀ H₁: p ≠ p₀ $Z=\frac{\hat{p}-p_{0}}{\sqrt{\left(\frac{p_{0} q_{0}}{n}\right)}}$ TI-84: 1-PropZTest	Rejection Rules: P-value method: reject H₀ when the p-value ≤ $\alpha$ . Critical value method: reject H₀ when the test statistic is in the critical region (shaded tails).

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University, https://mostlyharmlessstat.wixsite.com/webpage), © Rachel Webb, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.