Chapter 10 Formulas and Vocabulary

• $\mathbf{100}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{\alpha }\mathbf{)}\%$ Confidence Interval for ${\mathbf{\mu }}_{\mathbf{d}}$ (p. 582)
  

  ${\overline{x}}_d\pm t_{\alpha /2}(s_d/\sqrt{n})$

• Dependent Samples (p. 576)
• Independent Samples (p. 576)
• Matched-Pair Samples (p. 576)
• Test Statistic for the Paired Sample $t$ Test (p. 580)
  

  $t_{\text{data}}=\frac{{\overline{x}}_d}{s_d/\sqrt{n}}$

• $100(1-\alpha )\%$ Confidence Interval for ${\mathbf{\mu }}_{\mathbf{1}}\mathbf{-}{\mathbf{\mu }}_{\mathbf{2}}$ (p. 594)

  $({\overline{x}}_1-{\overline{x}}_2)\pm t_{\alpha /2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$

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• Margin of Error $E$ (p. 594). For a $100(1-\alpha )\%$ confidence interval for ${\mu }_1-{\mu }_2$,
  

  $E=t_{\alpha /2}\cdot \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$

• Sampling distribution of ${\overline{x}}_1-{\overline{x}}_2$ (p. 590)
• The pooled variance method for $t$ inference may be applied when the data analyst has reason to believe that the variances of the two populations are equal.
• When the population standard deviation ${\sigma }_1$ and ${\sigma }_2$ are known, the data analyst may prefer to use $Z$ inference for
• Pooled estimate for the common variance ${\sigma }_2$ (p. 596):
  

  $s_{\text{pooled}}^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$

• Test statistic ${\mathit{t}}_{\mathbf{\text{data}}}$ for ${\mathbf{\mu }}_{\mathbf{1}}\mathbf{-}{\mathbf{\mu }}_{\mathbf{2}}$ using pooled variance (p. 596):
  

  $t_{\text{data}}=\frac{({\overline{x}}_1-{\overline{x}}_2)}{\sqrt{s_{\text{pooled}}^2\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$

• Pooled variance $t$ confidence interval for μ (p. 598):
  

  ${\overline{x}}_1-{\overline{x}}_2\pm t_{\alpha /2}\sqrt{s_{\text{pooled}}^2\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}$

• $Z$ test statistic for ${\mu }_1-{\mu }_2$ when ${\sigma }_1$ and ${\sigma }_2$ are known (p. 598):
  

  $Z_{\text{data}}=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}}}$

• $Z$ confidence interval for ${\mu }_1-\mu 2$ when ${\sigma }_1$ and ${\sigma }_2$ are known (p. 599):
  

  ${\overline{x}}_1-{\overline{x}}_2\pm Z_{\alpha /2}\sqrt{\left(\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}\right)}$

• $100(1-\alpha )\%$ Confidence Interval for $p_1-p_2$ (p. 612).
  

  ${\hat{p}}_1-{\hat{p}}_2\pm Z_{\alpha /2}\sqrt{\frac{{\hat{p}}_1\cdot {\hat{p}}_2}{n_1}+\frac{{\hat{p}}_2\cdot {\hat{p}}_2}{n_2}}$

• Margin of error $E$ (p. 612). For a $100(1-\alpha )\%$ confidence interval for $p_1-p_2$,
  

  $E=Z_{\alpha /2}\cdot \sqrt{\frac{{\hat{p}}_1\cdot {\hat{p}}_2}{n_1}+\frac{{\hat{p}}_2\cdot {\hat{p}}_2}{n_2}}$

• Pooled estimate of $p$ (p. 608).
  

  ${\hat{p}}_{\text{pooled}}=\frac{x_1+x_2}{n_1+n_2}$

• Sampling distribution of ${\hat{p}}_1-{\hat{p}}_2$ (p. 607)
• Test statistic for the independent samples $Z$ test for $p_1-p_2$ (p. 608):
  

  $Z_{\text{data}}=\frac{({\hat{p}}_1-{\hat{p}}_2)}{\sqrt{{\hat{p}}_{\text{pooled}}\cdot (1-{\hat{p}}_{\text{pooled}})\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$