Chapter 8 Formulas and Vocabulary

• Confidence interval (p. 430)
• Confidence level (p. 430)
• Margin of error $E$ for the $Z$ interval for $\mu$ (p. 433).
  

  $E=Z_{\alpha \text{/}2}\left(\sigma \text{/}\sqrt{n}\right)$

• point estimate (p. 428)
• point estimation (p. 428)
• Sample size for estimating the population mean (p. 440).
  

  $n={\left[{\displaystyle \frac{\left(Z_{\alpha \text{/}2}\right)\sigma }{E}}\right]}^2$

• $Z$ interval for $\mu$ (p. 431).
  

  $\begin{array}{l}\text{lower bound}=\overline{x}-Z_{\alpha \text{/}2}\left(\sigma \text{/}\sqrt{n}\right)\\ \text{upper bound}=\overline{x}-Z_{\alpha \text{/}2}\left(\sigma \text{/}\sqrt{n}\right)\end{array}$

• Degrees of freedom (p. 449)
• Margin of error $E$ for the $t$ interval for $\mu$ (p. 454).
  

  $E=t_{\alpha \text{/}2}\left(s\text{/}\sqrt{n}\right)$

• $t$ Distribution (p. 449)
• $t$ Distribution characteristics (p. 450)
• $t$ interval for $\mu$ (p. 451).
  

  $\begin{array}{l}\text{lower bound}=\overline{x}-t_{\alpha \text{/}2}\left(\sigma \text{/}\sqrt{n}\right)\\ \text{upper bound}=\overline{x}+t_{\alpha \text{/}2}\left(s\text{/}\sqrt{n}\right)\end{array}$

• Central limit theorem for proportions (p. 464)
• Margin of error $E$ for the $Z$ interval for $p$ (p. 466).
  

  $E=Z_{\alpha \text{/}2}\sqrt{{\displaystyle \frac{\hat{p}\cdot \hat{q}}{n}}}$

• Sample size for estimating a population proportion when $\hat{p}$ is known (p. 468).
  

  $n=\hat{p}\cdot \hat{q}{\left({\displaystyle \frac{Z_{\alpha \text{/}2}}{E}}\right)}^2$

• Sample size for estimating a population proportion when $\hat{p}$ is not known (p. 468).
  

  $n={\left[{\displaystyle \frac{\left(0.5\right)\left(Z_{\alpha \text{/}2}\right)}{E}}\right]}^2$

• $Z$ interval for $p$ (p. 464).
  

  $\begin{array}{l}\text{lower bound}=\hat{p}-Z_{\alpha \text{/}2}\sqrt{{\displaystyle \frac{\hat{p}\cdot \hat{q}}{n}}}\\ \text{upper bound}=\hat{p}-Z_{\alpha \text{/}2}\sqrt{{\displaystyle \frac{\hat{p}\cdot \hat{q}}{n}}}\end{array}$

• ${\mathbf{\chi }}^{\mathbf{2}}$ (chi square) distribution properties (p. 474)
• Confidence interval for the population standard deviation $\mathbf{\sigma }$ (p. 477).
  

  $\text{lower bound=}\sqrt{{\displaystyle \frac{(n-1)s^2}{{\chi }_{\alpha \text{/}2}^2}}}\text{upper bound}=\sqrt{{\displaystyle \frac{(n-1)s^2}{{\chi }_{1-\alpha \text{/}2}^2}}}$

• Confidence interval for the population variance ${\mathbf{\sigma }}^{\mathbf{2}}$ (p. 476).
  

  $\begin{array}{cc}\text{lower bound=}{\displaystyle \frac{(n-1)s^2}{{\chi }_{\alpha \text{/}2}^2}}& \text{upper bound}={\displaystyle \frac{(n-1)s^2}{{\chi }_{1-\alpha \text{/}2}^2}}\end{array}$