{"id":1512,"date":"2021-07-16T13:32:44","date_gmt":"2021-07-16T17:32:44","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/back-matter\/__unknown__\/"},"modified":"2025-04-25T18:16:38","modified_gmt":"2025-04-25T22:16:38","slug":"formula-sheet","status":"publish","type":"back-matter","link":"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/formula-sheet\/","title":{"raw":"Appendix A: Formula Sheet","rendered":"Appendix A: Formula Sheet"},"content":{"raw":"
\r\n

Important Notations<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Measures<\/td>\r\nPopulation<\/td>\r\nSample<\/td>\r\n<\/tr>\r\n<\/thead>\r\n
Sample Size<\/td>\r\n[latex]N[\/latex]<\/td>\r\n[latex]n[\/latex]<\/td>\r\n<\/tr>\r\n
Mean<\/td>\r\n[latex]\\mu[\/latex]<\/td>\r\n[latex]\\bar{\\mu}[\/latex]<\/td>\r\n<\/tr>\r\n
Standard Deviation<\/td>\r\n[latex]\\sigma[\/latex]<\/td>\r\n[latex]s[\/latex]<\/td>\r\n<\/tr>\r\n
Proportion<\/td>\r\n[latex]p[\/latex]<\/td>\r\n[latex]\\hat{p}[\/latex]<\/td>\r\n<\/tr>\r\n
Slope<\/td>\r\n[latex]\\beta_1[\/latex]<\/td>\r\n[latex]b_1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Descriptive Measures<\/strong><\/p>\r\n\r\n

    \r\n \t
  • Five-number summary: minimum, [latex]Q_1[\/latex], [latex]Q_2[\/latex], [latex]Q_3[\/latex], and maximum<\/li>\r\n \t
  • Outliers: [latex]\\text{lowerlimit}=Q_1-1.5 \\times IQR;[\/latex] \u00a0[latex]\\text{upperlimit}=Q_3+1.5 \\times IQR;[\/latex] [latex]IQR=Q_3-Q_1 [\/latex]<\/li>\r\n \t
  • Sample mean: [latex]\\frac{x_1 + x_2 + \\dots + x_n}{n} = \\frac{\\sum x_i}{n}[\/latex]<\/li>\r\n \t
  • Sample standard deviation:\u00a0[latex]s = \\sqrt{ \\frac{\\sum (x_i - \\bar{x})^2 }{n-1} } = \\sqrt{ \\frac{ \\left( \\sum x_i^2 \\right) - \\frac{ \\left( \\sum x_i \\right)^2 }{n} }{n-1} }[\/latex]<\/li>\r\n<\/ul>\r\n

    Probability Concepts<\/strong><\/p>\r\n\r\n

      \r\n \t
    • Equal-likely outcome model: Probability of event<\/strong> E\r\n<\/em><\/strong>\r\n

      [latex]P(E)= \\frac{\\text{\\# of sample points in event E}}{\\text{\\# of sample points in sample space S} } = \\frac{\\text{\\# of ways event E can occur}}{\\text{\\# of possible outcomes}}= \\frac{f}{N}[\/latex]<\/p>\r\n<\/li>\r\n \t

    • Complement rule: [latex]P(not \\: E)=1-P(E)[\/latex]<\/li>\r\n \t
    • Special addition rule: [latex]P(A \\: or \\: B)=P(A)+P(B)[\/latex]\u00a0if events [latex]A[\/latex]\u00a0and [latex]B[\/latex]\u00a0are mutually exclusive. More generally, if events [latex]A,B,C, \\dots[\/latex]\u00a0are mutually exclusive, then [latex]P(A \\: or \\: B \\: or \\: C \\: or \\: \\dots)=P(A)+P(B)+P(C) + \\dots [\/latex]<\/li>\r\n \t
    • General addition rule: [latex]P(A \\:or \\: B)=P(A)+P(B)-P(A \\: \\& \\: B)[\/latex]<\/li>\r\n \t
    • Conditional probability of [latex]A[\/latex]\u00a0given <\/span>[latex]B[\/latex]: [latex]P(A|B)=\\frac{P(A \\: \\& \\: B)}{P(B)}[\/latex]\u00a0for [latex]P(B) \\: \\gt \\: 0[\/latex]<\/li>\r\n \t
    • General multiplication rule: [latex]P(A \\: \\& \\:B)=P(B)P(A|B)=P(A)P(B|A)[\/latex]<\/li>\r\n \t
    • Special multiplication rule: [latex]P(A \\: \\& \\: B)=P(A)P(B)[\/latex] if events [latex]A[\/latex] and [latex]B[\/latex] are independent. More generally, if events [latex]A,B,C, \\dots [\/latex] are independent, [latex]P(A \\: \\& \\: B \\: \\& \\: C \\: \\& \\: \\dots) =P(A) \\times P(B) \\times P(C) \\times \\dots [\/latex]<\/li>\r\n \t
    • Two events [latex]A[\/latex]\u00a0and [latex]B[\/latex]\u00a0are <\/span>independent<\/strong> if <\/span>ANY<\/strong> of the following is true:\r\n<\/span>\r\n

      [latex]P(A|B)=P(A)[\/latex]\u00a0OR [latex]P(B|A)=P(B)[\/latex]\u00a0OR [latex]P(A \\: \\& \\: B)=P(A) \\times P(B)[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n

        \r\n \t
      • Permutation: [latex]nP_r = \\frac{n!}{(n-r)!}[\/latex]<\/li>\r\n \t
      • Combination: [latex]nC_r = \\frac{n!}{r!(n-r)!}[\/latex]<\/li>\r\n<\/ul>\r\n

        Discrete Random Variables<\/strong><\/p>\r\n\r\n

          \r\n \t
        • The mean (expected value) of a discrete random variable [latex]X: \\mu= \\sum xP(X=x)[\/latex]<\/li>\r\n \t
        • Standard deviation of [latex]X: \\sigma= \\sqrt{\\sum (x- \\mu)^2 P(X=x)} = \\sqrt{\\sum x^2 P(X=x)-\\mu^2}[\/latex]<\/li>\r\n<\/ul>\r\n

          Binomial Distribution<\/strong><\/p>\r\nAmong [latex]n[\/latex]\u00a0independent Bernoulli trials with probability of success [latex]p[\/latex], let [latex]X[\/latex]\u00a0be the number of successes. The probability of observing [latex]x[\/latex]\u00a0successes is:\r\n

          [latex]P(X=x)={n\u00a0\\choose x} p^x (1-p)^{n-x}=nC_x p^x(1-p)^{n-x}, x=0,1, \\dots , n[\/latex]<\/p>\r\nThe mean and standard deviation of a Binomial distribution are [latex]\\mu = np[\/latex], [latex]\\sigma = \\sqrt{np(1-p)}[\/latex], respectively.\r\n

          Normal Distribution<\/strong><\/p>\r\n\r\n

            \r\n \t
          • If random variable [latex]X \\sim N(\\mu, \\sigma)[\/latex], then the standardized variable [latex]Z=\\frac{X-\\mu}{\\sigma} \\sim N(0,1)[\/latex].<\/li>\r\n \t
          • Given an [latex]x[\/latex]\u00a0value, its\u00a0z-score is [latex]z=\\frac{x-\\mu}{\\sigma}[\/latex]<\/li>\r\n \t
          • Given the\u00a0z-score, find the [latex]x[\/latex]\u00a0value: [latex]x=\\mu+z \\times \\sigma [\/latex].<\/li>\r\n \t
          • If sample mean [latex]\\bar{X} \\sim N(\\mu_{\\bar{X}}=\\mu, \\sigma_{\\bar{X}}= \\frac{\\sigma}{\\sqrt{n}})[\/latex], the standardized variable [latex]Z=\\frac{\\bar{X}-\\mu_{\\bar{X}}}{\\sigma_{\\bar{X}}}= \\frac{\\bar{X} - \\mu}{\\sigma \/ \\sqrt{n}} \\sim N(0,1)[\/latex].<\/li>\r\n<\/ul>\r\n

            Sampling Distributions<\/strong><\/p>\r\n\r\n

              \r\n \t
            • Mean and standard deviation of the sample mean [latex]\\bar{X}: \\mu_{\\scriptsize \\bar{X}}= \\mu, \\sigma_{\\scriptsize \\bar{X}} = \\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/li>\r\n \t
            • Mean and standard deviation of a sample proportion [latex]\\hat{p}: \u00a0\\mu_{\\scriptsize \\hat{p}} = p; \\sigma_{\\scriptsize \\hat{p}} =\\sqrt{ \\frac{p(1-p)}{n}}[\/latex]<\/li>\r\n \t
            • Mean and standard deviation of [latex]\\bar{X}_1 - \\bar{X}_2: \\mu_{\\scriptsize \\bar{X}_1 - \\bar{X}_2} = \\mu_1 - \\mu_2; \\sigma_{\\scriptsize \\bar{X}_1 - \\bar{X}_2} = \\sqrt{ \\frac{\\sigma_1^2}{n_1} + \\frac{\\sigma_2^2}{n_2}}[\/latex]<\/li>\r\n \t
            • Mean and standard deviation of [latex]\\hat{p}_1\u00a0 - \\hat{p}_2: \\mu_{\\scriptsize \\hat{p}_1\u00a0 - \\hat{p}_2} = p_1 - p_2; \\sigma_{\\scriptsize \\hat{p}_1\u00a0 - \\hat{p}_2} = \\sqrt{ \\frac{p_1(1-p_1)}{n_1} + \\frac{p_2(1-p_2)}{n_2}}[\/latex]<\/li>\r\n<\/ul>\r\n

              Confidence Intervals and Hypothesis Tests<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
              Parameter<\/td>\r\nEstimate<\/td>\r\nTest Statistic<\/td>\r\n\r\n
              [latex](1 - \\alpha) \\times 100 \\%[\/latex]\u00a0Confidence Interval<\/span><\/div><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
              [latex]\\mu[\/latex]<\/td>\r\n[latex]\\bar{x}[\/latex]<\/td>\r\n[latex]t_o = \\frac{\\bar{x} - \\mu_0}{\\left( \\frac{s}{\\sqrt{n}} \\right)}[\/latex]<\/td>\r\n\r\n
              [latex]\\bar{x} \\pm t_{\\alpha \/ 2} \\frac{s}{\\sqrt{n}}[\/latex]\u00a0with\u00a0[latex]df=n-1[\/latex]<\/div><\/td>\r\n<\/tr>\r\n
              [latex]p[\/latex]<\/td>\r\n[latex]\\hat{p}[\/latex]<\/td>\r\n[latex]z_o = \\frac{\\hat{p} - p_0}{\\sqrt{\\frac{p_0(1 - p_0)}{n}}}[\/latex]<\/td>\r\n[latex]\\hat{p} \\pm z_{\\alpha \/ 2} \\sqrt{\\frac{\\hat{p}(1 - \\hat{p})}{n}} , \\hat{p} = \\frac{x}{n}[\/latex]<\/td>\r\n<\/tr>\r\n
              [latex]\\mu_1 - \\mu_2[\/latex]<\/td>\r\n[latex]\\bar{x}_1 - \\bar{x}_2[\/latex]<\/td>\r\n[latex]t_o = \\frac{(\\bar{x}_1 - \\bar{x}_2) - \\Delta_0}{\\sqrt{\\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}}}[\/latex]<\/td>\r\n[latex](\\bar{x}_1 - \\bar{x}_2) \\pm t_{\\alpha \/ 2} \\sqrt{\\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}}[\/latex]\u00a0with\r\n
              [latex]df = \\frac{\\left( \\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}\u00a0\\right)^2}{ \\left( \\frac{1}{n_1 - 1} \\right) \\left( \\frac{s_1^2}{n_1} \\right)^2 + \\left( \\frac{1}{n_2 - 1} \\right) \\left( \\frac{s_2^2}{n_2} \\right)^2}[\/latex]<\/div><\/td>\r\n<\/tr>\r\n
              [latex]\\mu_1 - \\mu_2[\/latex]<\/td>\r\n[latex]\\bar{x}_1 - \\bar{x}_2[\/latex]<\/td>\r\n[latex]t_o = \\frac{(\\bar{x}_1 - \\bar{x}_2) - \\Delta_0}{ s_p \\sqrt{\\frac{1}{n_1} + \\frac{1}{n_2}}}[\/latex]<\/td>\r\n[latex](\\bar{x}_1 - \\bar{x}_2) \\pm t_{\\alpha \/ 2} s_p \\sqrt{\\frac{1}{n_1} + \\frac{1}{n_2}}[\/latex]\u00a0with\u00a0[latex]df=n_1+n_2-2[\/latex]<\/td>\r\n<\/tr>\r\n
              [latex]\\mu_1 - \\mu_2[\/latex]<\/td>\r\n[latex]\\bar{d}[\/latex]<\/td>\r\n[latex]t_o = \\frac{\\bar{d} - \\delta_0}{\\left( \\frac{s_d}{\\sqrt{n}} \\right)}[\/latex]<\/td>\r\n\r\n
              [latex]\\bar{d} \\pm t_{\\alpha \/ 2} \\frac{s_d}{\\sqrt{n}}[\/latex]\u00a0with [latex]df=n-1[\/latex], [latex]n\u00a0=[\/latex] # of pairs\r\n[latex]\\bar{d} = \\frac{\\sum d_i}{n}, s_d = \\sqrt{\\frac{\\left( \\sum d_i^2 \\right) - \\frac{\\left( \\sum d_i \\right)^2}{n} }{n-1}}[\/latex]<\/div><\/td>\r\n<\/tr>\r\n
              [latex]p_1-p_2[\/latex]<\/td>\r\n[latex]\\hat{p}_1 - \\hat{p}_2[\/latex]<\/td>\r\n[latex]z_o = \\frac{\\hat{p}_1 - \\hat{p}_2}{\\sqrt{\\hat{p}_p (1 - \\hat{p}_p)} \\sqrt{ \\frac{1}{n_1} + \\frac{1}{n_2} }}[\/latex]<\/td>\r\n[latex](\\hat{p}_1 - \\hat{p}_2) \\pm z_{\\alpha \/ 2} \\sqrt{\\frac{\\hat{p}_1 (1 - \\hat{p}_1)}{n_1} + \\frac{\\hat{p}_2 (1 - \\hat{p}_2)}{n_2}}[\/latex]\r\n[latex]\\hat{p}_p = \\frac{x_1 + x_2}{n_1 + n_2}, \\hat{p}_1 = \\frac{x_1}{n_1}, \\hat{p}_2 = \\frac{x_2}{n_2}[\/latex]<\/td>\r\n<\/tr>\r\n
              [latex]\\beta_1[\/latex]<\/td>\r\n[latex]b_1[\/latex]<\/td>\r\n[latex]t_o = \\frac{b_1}{\\left( \\frac{s_e}{\\sqrt{S_{xx}}} \\right)}[\/latex]<\/td>\r\n\r\n
              [latex]b_1 \\pm t_{\\alpha \/ 2} \\frac{s_e}{\\sqrt{S_{xx}}}[\/latex]\u00a0with\u00a0[latex]df=n-2[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

              Margin of Error and Sample Size Calculation<\/strong><\/p>\r\n\r\n

                \r\n \t
              • Margin of error for the estimate of [latex]\\mu: E = z_{\\alpha \/ 2} \\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/li>\r\n \t
              • Sample size calculation for [latex]\\mu : = \\left( \\frac{\\sigma \\times z_{\\alpha \/ 2}}{E} \\right)^2[\/latex]\u00a0round up to the nearest integer<\/li>\r\n \t
              • Margin of error for the estimate of [latex]p: E = z_{\\alpha \/ 2} \\sqrt{ \\frac{\\hat{p}(1 - \\hat{p})}{n} }[\/latex]<\/li>\r\n \t
              • Sample size calculation for [latex]p[\/latex]\u00a0without guessing [latex]\\hat{p}: n \\leq 0.05 (1 - 0.05) \\left( \\frac{z_{\\alpha \/ 2}}{E} \\right)^2 = 0.25 \\left( \\frac{z_{\\alpha \/ 2}}{E} \\right)^2[\/latex]<\/li>\r\n \t
              • Sample size calculation for [latex]p[\/latex] with guessing [latex]\\hat{p}: n = p_g (1 - p_g) \\left( \\frac{z_{\\alpha \/ 2}}{E} \\right)^2[\/latex] round up<\/li>\r\n<\/ul>\r\n

                Chi-Square Test<\/strong><\/p>\r\n\r\n

                  \r\n \t
                • Chi-square goodness-of-fit test for one categorical\/discrete variable:\r\n
                    \r\n \t
                  • Expected frequency: [latex]E=np[\/latex]<\/li>\r\n \t
                  • Test statistic: [latex]\\chi_o^2 = \\sum_{\\text{all cells}} \\frac{(O - E)^2}{E}[\/latex]\u00a0with [latex]df=k-1[\/latex], where [latex]k[\/latex]\u00a0is number of possible values of the variable<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
                  • Chi-square independence (or homogeneity) test of two variables:\r\n
                      \r\n \t
                    • Expected frequency: [latex]E=\\frac{\\text{(rth row total)} \\times \\text{(cth column total)}}{n}[\/latex]<\/li>\r\n \t
                    • Test statistic: [latex]\\chi_o^2 = \\sum_{\\text{all cells}} \\frac{(O - E)^2}{E}[\/latex]\u00a0with [latex]df=(r-1) \\times (c-1)[\/latex]\u00a0where [latex]r[\/latex]\u00a0is the number of rows and [latex]c[\/latex]\u00a0is number of columns of the cells.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n

                      Regression Analysis<\/strong><\/p>\r\n\r\n

                        \r\n \t
                      • Sums of squares\r\n

                        [latex]S_{xy}= \\sum x_i y_i - \\frac{\\left( \\sum x_i \\right) \\left( \\sum y_i \\right) }{n}; S_{xx} = x_i^2 - \\frac{\\left( \\sum x_i \\right)^2}{n}; S_{yy} = \\sum y_i^2 - \\frac{\\left( \\sum y_i \\right)^2}{n}[\/latex]<\/p>\r\n<\/li>\r\n \t

                      • The least-squares straight line: [latex]\\hat{y} = b_0 + b_1 x[\/latex]\u00a0, where [latex]b_1 = \\frac{S_{xy}}{S_{xx}}[\/latex]\u00a0and [latex]b_0 = \\bar{y} - b_1 \\bar{x} = \\frac{\\sum y_i}{n} \u00a0- b_1 \\frac{\\sum x_i}{n}[\/latex]<\/li>\r\n \t
                      • Total sum of squares: [latex]SST = \\sum (y_i - \\bar{y})^2 = S_{yy}[\/latex]<\/li>\r\n \t
                      • Regression sum of squares: [latex]SSR = \\sum (\\hat{y} - \\bar{y})^2 = r^2 S_{yy} = \\frac{S_{xy}^2}{S_{xx}}[\/latex]<\/li>\r\n \t
                      • Error sum of squares: [latex]SSE = \\sum (y_i - \\hat{y}_i)^2 = \\sum e_i^2 = SST - SSR = S_{yy} - \\frac{S_{xy}^2}{S_{xx}}[\/latex]<\/li>\r\n \t
                      • Regression identify: [latex]SST=SSE+SSR[\/latex]<\/li>\r\n \t
                      • Residual: [latex]e_i \u00a0= y_i -\\hat{y}_i = y_i - (b_0 + b_1 x_i)[\/latex]<\/li>\r\n \t
                      • Correlation coefficient: [latex]r = \\frac{S_{xy}}{\\sqrt{S_{xx} \\times S_{yy}}}[\/latex]<\/li>\r\n \t
                      • Coefficient of determination: [latex]R^2=r^2= \\frac{S_{xy}^2}{S_{xx} \\times S_{yy}}=\\frac{SSR}{SST}[\/latex]<\/li>\r\n \t
                      • Standard error of the estimate: [latex]s_e = \\sqrt{ \\frac{\\sum (e_i - \\bar{e})^2}{n-2} } = \\sqrt{ \\frac{\\sum e_i^2}{n-2}} = \\sqrt{\\frac{SSE}{n-2}}[\/latex]<\/li>\r\n \t
                      • Test statistic for [latex]\\beta_1: t_o = \\frac{b_1}{\\left( \\frac{s_e}{\\sqrt{S_{xx}}} \\right)}[\/latex] with [latex]df=n-2[\/latex]<\/li>\r\n \t
                      • A [latex](1 - \\alpha) \\times 100 \\% [\/latex] confidnece interval for [latex]\\beta_1: b_1 \\pm t_{\\alpha \/ 2} \\frac{s_e}{\\sqrt{S_{xx}}}[\/latex] with [latex]df = n-2[\/latex]<\/li>\r\n \t
                      • A [latex](1 - \\alpha) \\times 100 \\% [\/latex] confidence interval for\u00a0the conditional mean [latex]\\mu_p[\/latex]\u00a0is\r\n

                        [latex]\\hat{\\mu}_p \\pm t_{\\alpha \/ 2} \\times SE(\\hat{\\mu}_p) = (b_0 + b_1 x_p) \\pm t_{\\alpha \/ 2} \\times s_e \\sqrt{\\frac{(x_p - \\bar{x})^2}{S_{xx}} + \\frac{1}{n}}[\/latex]\u00a0with [latex]df=n-2[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n

                        \r\n
                          \r\n \t
                        • A [latex](1 - \\alpha) \\times 100 \\% [\/latex] confidence interval for a single response [latex]y_p[\/latex]\u00a0is\r\n

                          [latex]\\hat{y}_p \\pm t_{\\alpha \/ 2} \\times SE(\\hat{y}_p) = (b_0 + b_1 x_p) \\pm t_{\\alpha \/ 2} \\times s_e \\sqrt{\\frac{(x_p - \\bar{x})^2}{S_{xx}} + \\frac{1}{n} + 1}[\/latex]\u00a0with [latex]df=n-2[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n

                          Analysis of Variance (One-Way ANOVA F Test)<\/strong><\/p>\r\nCompare [latex]k[\/latex]\u00a0population means: [latex]\\mu_1, \\mu_2 , \\dots , \\mu_k[\/latex]. Denote sample sizes as [latex]n_1,n_2, \\dots ,n_k[\/latex], sample means as [latex]\\bar{x}_1, \\bar{x}_2, \\dots , \\bar{x}_k[\/latex] and sample standard deviations as [latex]s_1,s_2, \\dots ,s_k[\/latex]. Let [latex]n=n_1 + n_2 + \\dots + n_k[\/latex] and [latex]\\bar{x} = \\frac{\\sum x_{ij}}{n}[\/latex] where [latex]x_{ij}[\/latex] is the jth observation of sample [latex]i[\/latex].\r\n

                            \r\n \t
                          • Test statistic:\u00a0[latex]F_o = \\frac{SSTR \/ (k-1)}{SSE \/ (n-k)} = \\frac{MSTR}{MSE}[\/latex]\u00a0with [latex]df_n=k-1[\/latex]\u00a0and [latex]df_d=n-k[\/latex]<\/li>\r\n \t
                          • Total sum of squares:\u00a0[latex]SST = \\sum (x_{ij} - \\bar{x})^2 = \\sum x_{ij}^2 - \\frac{\\left( \\sum x_{ij} \\right)^2}{n}[\/latex]<\/li>\r\n \t
                          • Treatment sum of squares:\u00a0[latex]SSTR = \\sum_{i=1}^k n_i (\\bar{x}_i - \\bar{x})^2[\/latex]<\/span><\/li>\r\n \t
                          • Error sum of squares:\u00a0[latex]SSE = \\sum (x_{ij} - \\bar{x}_i)^2 = \\sum_{i=1}^k (n_i -1)s_i^2 = SST - SSTR[\/latex]<\/span><\/li>\r\n \t
                          • ANOVA identity: [latex]SST=SSE+SSTR[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"
                            \n

                            Important Notations<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n
                            Measures<\/td>\nPopulation<\/td>\nSample<\/td>\n<\/tr>\n<\/thead>\n
                            Sample Size<\/td>\n[latex]N[\/latex]<\/td>\n[latex]n[\/latex]<\/td>\n<\/tr>\n
                            Mean<\/td>\n[latex]\\mu[\/latex]<\/td>\n[latex]\\bar{\\mu}[\/latex]<\/td>\n<\/tr>\n
                            Standard Deviation<\/td>\n[latex]\\sigma[\/latex]<\/td>\n[latex]s[\/latex]<\/td>\n<\/tr>\n
                            Proportion<\/td>\n[latex]p[\/latex]<\/td>\n[latex]\\hat{p}[\/latex]<\/td>\n<\/tr>\n
                            Slope<\/td>\n[latex]\\beta_1[\/latex]<\/td>\n[latex]b_1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                            Descriptive Measures<\/strong><\/p>\n

                              \n
                            • Five-number summary: minimum, [latex]Q_1[\/latex], [latex]Q_2[\/latex], [latex]Q_3[\/latex], and maximum<\/li>\n
                            • Outliers: [latex]\\text{lowerlimit}=Q_1-1.5 \\times IQR;[\/latex] \u00a0[latex]\\text{upperlimit}=Q_3+1.5 \\times IQR;[\/latex] [latex]IQR=Q_3-Q_1[\/latex]<\/li>\n
                            • Sample mean: [latex]\\frac{x_1 + x_2 + \\dots + x_n}{n} = \\frac{\\sum x_i}{n}[\/latex]<\/li>\n
                            • Sample standard deviation:\u00a0[latex]s = \\sqrt{ \\frac{\\sum (x_i - \\bar{x})^2 }{n-1} } = \\sqrt{ \\frac{ \\left( \\sum x_i^2 \\right) - \\frac{ \\left( \\sum x_i \\right)^2 }{n} }{n-1} }[\/latex]<\/li>\n<\/ul>\n

                              Probability Concepts<\/strong><\/p>\n

                                \n
                              • Equal-likely outcome model: Probability of event<\/strong> E
                                \n<\/em><\/strong><\/p>\n

                                [latex]P(E)= \\frac{\\text{\\# of sample points in event E}}{\\text{\\# of sample points in sample space S} } = \\frac{\\text{\\# of ways event E can occur}}{\\text{\\# of possible outcomes}}= \\frac{f}{N}[\/latex]<\/p>\n<\/li>\n

                              • Complement rule: [latex]P(not \\: E)=1-P(E)[\/latex]<\/li>\n
                              • Special addition rule: [latex]P(A \\: or \\: B)=P(A)+P(B)[\/latex]\u00a0if events [latex]A[\/latex]\u00a0and [latex]B[\/latex]\u00a0are mutually exclusive. More generally, if events [latex]A,B,C, \\dots[\/latex]\u00a0are mutually exclusive, then [latex]P(A \\: or \\: B \\: or \\: C \\: or \\: \\dots)=P(A)+P(B)+P(C) + \\dots[\/latex]<\/li>\n
                              • General addition rule: [latex]P(A \\:or \\: B)=P(A)+P(B)-P(A \\: \\& \\: B)[\/latex]<\/li>\n
                              • Conditional probability of [latex]A[\/latex]\u00a0given <\/span>[latex]B[\/latex]: [latex]P(A|B)=\\frac{P(A \\: \\& \\: B)}{P(B)}[\/latex]\u00a0for [latex]P(B) \\: \\gt \\: 0[\/latex]<\/li>\n
                              • General multiplication rule: [latex]P(A \\: \\& \\:B)=P(B)P(A|B)=P(A)P(B|A)[\/latex]<\/li>\n
                              • Special multiplication rule: [latex]P(A \\: \\& \\: B)=P(A)P(B)[\/latex] if events [latex]A[\/latex] and [latex]B[\/latex] are independent. More generally, if events [latex]A,B,C, \\dots[\/latex] are independent, [latex]P(A \\: \\& \\: B \\: \\& \\: C \\: \\& \\: \\dots) =P(A) \\times P(B) \\times P(C) \\times \\dots[\/latex]<\/li>\n
                              • Two events [latex]A[\/latex]\u00a0and [latex]B[\/latex]\u00a0are <\/span>independent<\/strong> if <\/span>ANY<\/strong> of the following is true:
                                \n<\/span><\/p>\n

                                [latex]P(A|B)=P(A)[\/latex]\u00a0OR [latex]P(B|A)=P(B)[\/latex]\u00a0OR [latex]P(A \\: \\& \\: B)=P(A) \\times P(B)[\/latex]<\/p>\n<\/li>\n<\/ul>\n

                                  \n
                                • Permutation: [latex]nP_r = \\frac{n!}{(n-r)!}[\/latex]<\/li>\n
                                • Combination: [latex]nC_r = \\frac{n!}{r!(n-r)!}[\/latex]<\/li>\n<\/ul>\n

                                  Discrete Random Variables<\/strong><\/p>\n

                                    \n
                                  • The mean (expected value) of a discrete random variable [latex]X: \\mu= \\sum xP(X=x)[\/latex]<\/li>\n
                                  • Standard deviation of [latex]X: \\sigma= \\sqrt{\\sum (x- \\mu)^2 P(X=x)} = \\sqrt{\\sum x^2 P(X=x)-\\mu^2}[\/latex]<\/li>\n<\/ul>\n

                                    Binomial Distribution<\/strong><\/p>\n

                                    Among [latex]n[\/latex]\u00a0independent Bernoulli trials with probability of success [latex]p[\/latex], let [latex]X[\/latex]\u00a0be the number of successes. The probability of observing [latex]x[\/latex]\u00a0successes is:<\/p>\n

                                    [latex]P(X=x)={n\u00a0\\choose x} p^x (1-p)^{n-x}=nC_x p^x(1-p)^{n-x}, x=0,1, \\dots , n[\/latex]<\/p>\n

                                    The mean and standard deviation of a Binomial distribution are [latex]\\mu = np[\/latex], [latex]\\sigma = \\sqrt{np(1-p)}[\/latex], respectively.<\/p>\n

                                    Normal Distribution<\/strong><\/p>\n

                                      \n
                                    • If random variable [latex]X \\sim N(\\mu, \\sigma)[\/latex], then the standardized variable [latex]Z=\\frac{X-\\mu}{\\sigma} \\sim N(0,1)[\/latex].<\/li>\n
                                    • Given an [latex]x[\/latex]\u00a0value, its\u00a0z-score is [latex]z=\\frac{x-\\mu}{\\sigma}[\/latex]<\/li>\n
                                    • Given the\u00a0z-score, find the [latex]x[\/latex]\u00a0value: [latex]x=\\mu+z \\times \\sigma[\/latex].<\/li>\n
                                    • If sample mean [latex]\\bar{X} \\sim N(\\mu_{\\bar{X}}=\\mu, \\sigma_{\\bar{X}}= \\frac{\\sigma}{\\sqrt{n}})[\/latex], the standardized variable [latex]Z=\\frac{\\bar{X}-\\mu_{\\bar{X}}}{\\sigma_{\\bar{X}}}= \\frac{\\bar{X} - \\mu}{\\sigma \/ \\sqrt{n}} \\sim N(0,1)[\/latex].<\/li>\n<\/ul>\n

                                      Sampling Distributions<\/strong><\/p>\n

                                        \n
                                      • Mean and standard deviation of the sample mean [latex]\\bar{X}: \\mu_{\\scriptsize \\bar{X}}= \\mu, \\sigma_{\\scriptsize \\bar{X}} = \\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/li>\n
                                      • Mean and standard deviation of a sample proportion [latex]\\hat{p}: \u00a0\\mu_{\\scriptsize \\hat{p}} = p; \\sigma_{\\scriptsize \\hat{p}} =\\sqrt{ \\frac{p(1-p)}{n}}[\/latex]<\/li>\n
                                      • Mean and standard deviation of [latex]\\bar{X}_1 - \\bar{X}_2: \\mu_{\\scriptsize \\bar{X}_1 - \\bar{X}_2} = \\mu_1 - \\mu_2; \\sigma_{\\scriptsize \\bar{X}_1 - \\bar{X}_2} = \\sqrt{ \\frac{\\sigma_1^2}{n_1} + \\frac{\\sigma_2^2}{n_2}}[\/latex]<\/li>\n
                                      • Mean and standard deviation of [latex]\\hat{p}_1\u00a0 - \\hat{p}_2: \\mu_{\\scriptsize \\hat{p}_1\u00a0 - \\hat{p}_2} = p_1 - p_2; \\sigma_{\\scriptsize \\hat{p}_1\u00a0 - \\hat{p}_2} = \\sqrt{ \\frac{p_1(1-p_1)}{n_1} + \\frac{p_2(1-p_2)}{n_2}}[\/latex]<\/li>\n<\/ul>\n

                                        Confidence Intervals and Hypothesis Tests<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n\n
                                        Parameter<\/td>\nEstimate<\/td>\nTest Statistic<\/td>\n\n
                                        [latex](1 - \\alpha) \\times 100 \\%[\/latex]\u00a0Confidence Interval<\/span><\/div>\n<\/td>\n<\/tr>\n<\/thead>\n
                                        [latex]\\mu[\/latex]<\/td>\n[latex]\\bar{x}[\/latex]<\/td>\n[latex]t_o = \\frac{\\bar{x} - \\mu_0}{\\left( \\frac{s}{\\sqrt{n}} \\right)}[\/latex]<\/td>\n\n
                                        [latex]\\bar{x} \\pm t_{\\alpha \/ 2} \\frac{s}{\\sqrt{n}}[\/latex]\u00a0with\u00a0[latex]df=n-1[\/latex]<\/div>\n<\/td>\n<\/tr>\n
                                        [latex]p[\/latex]<\/td>\n[latex]\\hat{p}[\/latex]<\/td>\n[latex]z_o = \\frac{\\hat{p} - p_0}{\\sqrt{\\frac{p_0(1 - p_0)}{n}}}[\/latex]<\/td>\n[latex]\\hat{p} \\pm z_{\\alpha \/ 2} \\sqrt{\\frac{\\hat{p}(1 - \\hat{p})}{n}} , \\hat{p} = \\frac{x}{n}[\/latex]<\/td>\n<\/tr>\n
                                        [latex]\\mu_1 - \\mu_2[\/latex]<\/td>\n[latex]\\bar{x}_1 - \\bar{x}_2[\/latex]<\/td>\n[latex]t_o = \\frac{(\\bar{x}_1 - \\bar{x}_2) - \\Delta_0}{\\sqrt{\\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}}}[\/latex]<\/td>\n[latex](\\bar{x}_1 - \\bar{x}_2) \\pm t_{\\alpha \/ 2} \\sqrt{\\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}}[\/latex]\u00a0with<\/p>\n
                                        [latex]df = \\frac{\\left( \\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}\u00a0\\right)^2}{ \\left( \\frac{1}{n_1 - 1} \\right) \\left( \\frac{s_1^2}{n_1} \\right)^2 + \\left( \\frac{1}{n_2 - 1} \\right) \\left( \\frac{s_2^2}{n_2} \\right)^2}[\/latex]<\/div>\n<\/td>\n<\/tr>\n
                                        [latex]\\mu_1 - \\mu_2[\/latex]<\/td>\n[latex]\\bar{x}_1 - \\bar{x}_2[\/latex]<\/td>\n[latex]t_o = \\frac{(\\bar{x}_1 - \\bar{x}_2) - \\Delta_0}{ s_p \\sqrt{\\frac{1}{n_1} + \\frac{1}{n_2}}}[\/latex]<\/td>\n[latex](\\bar{x}_1 - \\bar{x}_2) \\pm t_{\\alpha \/ 2} s_p \\sqrt{\\frac{1}{n_1} + \\frac{1}{n_2}}[\/latex]\u00a0with\u00a0[latex]df=n_1+n_2-2[\/latex]<\/td>\n<\/tr>\n
                                        [latex]\\mu_1 - \\mu_2[\/latex]<\/td>\n[latex]\\bar{d}[\/latex]<\/td>\n[latex]t_o = \\frac{\\bar{d} - \\delta_0}{\\left( \\frac{s_d}{\\sqrt{n}} \\right)}[\/latex]<\/td>\n\n
                                        [latex]\\bar{d} \\pm t_{\\alpha \/ 2} \\frac{s_d}{\\sqrt{n}}[\/latex]\u00a0with [latex]df=n-1[\/latex], [latex]n\u00a0=[\/latex] # of pairs
                                        \n[latex]\\bar{d} = \\frac{\\sum d_i}{n}, s_d = \\sqrt{\\frac{\\left( \\sum d_i^2 \\right) - \\frac{\\left( \\sum d_i \\right)^2}{n} }{n-1}}[\/latex]<\/div>\n<\/td>\n<\/tr>\n
                                        [latex]p_1-p_2[\/latex]<\/td>\n[latex]\\hat{p}_1 - \\hat{p}_2[\/latex]<\/td>\n[latex]z_o = \\frac{\\hat{p}_1 - \\hat{p}_2}{\\sqrt{\\hat{p}_p (1 - \\hat{p}_p)} \\sqrt{ \\frac{1}{n_1} + \\frac{1}{n_2} }}[\/latex]<\/td>\n[latex](\\hat{p}_1 - \\hat{p}_2) \\pm z_{\\alpha \/ 2} \\sqrt{\\frac{\\hat{p}_1 (1 - \\hat{p}_1)}{n_1} + \\frac{\\hat{p}_2 (1 - \\hat{p}_2)}{n_2}}[\/latex]
                                        \n[latex]\\hat{p}_p = \\frac{x_1 + x_2}{n_1 + n_2}, \\hat{p}_1 = \\frac{x_1}{n_1}, \\hat{p}_2 = \\frac{x_2}{n_2}[\/latex]<\/td>\n<\/tr>\n
                                        [latex]\\beta_1[\/latex]<\/td>\n[latex]b_1[\/latex]<\/td>\n[latex]t_o = \\frac{b_1}{\\left( \\frac{s_e}{\\sqrt{S_{xx}}} \\right)}[\/latex]<\/td>\n\n
                                        [latex]b_1 \\pm t_{\\alpha \/ 2} \\frac{s_e}{\\sqrt{S_{xx}}}[\/latex]\u00a0with\u00a0[latex]df=n-2[\/latex]<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                        Margin of Error and Sample Size Calculation<\/strong><\/p>\n

                                          \n
                                        • Margin of error for the estimate of [latex]\\mu: E = z_{\\alpha \/ 2} \\frac{\\sigma}{\\sqrt{n}}[\/latex]<\/li>\n
                                        • Sample size calculation for [latex]\\mu : = \\left( \\frac{\\sigma \\times z_{\\alpha \/ 2}}{E} \\right)^2[\/latex]\u00a0round up to the nearest integer<\/li>\n
                                        • Margin of error for the estimate of [latex]p: E = z_{\\alpha \/ 2} \\sqrt{ \\frac{\\hat{p}(1 - \\hat{p})}{n} }[\/latex]<\/li>\n
                                        • Sample size calculation for [latex]p[\/latex]\u00a0without guessing [latex]\\hat{p}: n \\leq 0.05 (1 - 0.05) \\left( \\frac{z_{\\alpha \/ 2}}{E} \\right)^2 = 0.25 \\left( \\frac{z_{\\alpha \/ 2}}{E} \\right)^2[\/latex]<\/li>\n
                                        • Sample size calculation for [latex]p[\/latex] with guessing [latex]\\hat{p}: n = p_g (1 - p_g) \\left( \\frac{z_{\\alpha \/ 2}}{E} \\right)^2[\/latex] round up<\/li>\n<\/ul>\n

                                          Chi-Square Test<\/strong><\/p>\n

                                            \n
                                          • Chi-square goodness-of-fit test for one categorical\/discrete variable:\n
                                              \n
                                            • Expected frequency: [latex]E=np[\/latex]<\/li>\n
                                            • Test statistic: [latex]\\chi_o^2 = \\sum_{\\text{all cells}} \\frac{(O - E)^2}{E}[\/latex]\u00a0with [latex]df=k-1[\/latex], where [latex]k[\/latex]\u00a0is number of possible values of the variable<\/li>\n<\/ul>\n<\/li>\n
                                            • Chi-square independence (or homogeneity) test of two variables:\n
                                                \n
                                              • Expected frequency: [latex]E=\\frac{\\text{(rth row total)} \\times \\text{(cth column total)}}{n}[\/latex]<\/li>\n
                                              • Test statistic: [latex]\\chi_o^2 = \\sum_{\\text{all cells}} \\frac{(O - E)^2}{E}[\/latex]\u00a0with [latex]df=(r-1) \\times (c-1)[\/latex]\u00a0where [latex]r[\/latex]\u00a0is the number of rows and [latex]c[\/latex]\u00a0is number of columns of the cells.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                Regression Analysis<\/strong><\/p>\n

                                                  \n
                                                • Sums of squares\n

                                                  [latex]S_{xy}= \\sum x_i y_i - \\frac{\\left( \\sum x_i \\right) \\left( \\sum y_i \\right) }{n}; S_{xx} = x_i^2 - \\frac{\\left( \\sum x_i \\right)^2}{n}; S_{yy} = \\sum y_i^2 - \\frac{\\left( \\sum y_i \\right)^2}{n}[\/latex]<\/p>\n<\/li>\n

                                                • The least-squares straight line: [latex]\\hat{y} = b_0 + b_1 x[\/latex]\u00a0, where [latex]b_1 = \\frac{S_{xy}}{S_{xx}}[\/latex]\u00a0and [latex]b_0 = \\bar{y} - b_1 \\bar{x} = \\frac{\\sum y_i}{n} \u00a0- b_1 \\frac{\\sum x_i}{n}[\/latex]<\/li>\n
                                                • Total sum of squares: [latex]SST = \\sum (y_i - \\bar{y})^2 = S_{yy}[\/latex]<\/li>\n
                                                • Regression sum of squares: [latex]SSR = \\sum (\\hat{y} - \\bar{y})^2 = r^2 S_{yy} = \\frac{S_{xy}^2}{S_{xx}}[\/latex]<\/li>\n
                                                • Error sum of squares: [latex]SSE = \\sum (y_i - \\hat{y}_i)^2 = \\sum e_i^2 = SST - SSR = S_{yy} - \\frac{S_{xy}^2}{S_{xx}}[\/latex]<\/li>\n
                                                • Regression identify: [latex]SST=SSE+SSR[\/latex]<\/li>\n
                                                • Residual: [latex]e_i \u00a0= y_i -\\hat{y}_i = y_i - (b_0 + b_1 x_i)[\/latex]<\/li>\n
                                                • Correlation coefficient: [latex]r = \\frac{S_{xy}}{\\sqrt{S_{xx} \\times S_{yy}}}[\/latex]<\/li>\n
                                                • Coefficient of determination: [latex]R^2=r^2= \\frac{S_{xy}^2}{S_{xx} \\times S_{yy}}=\\frac{SSR}{SST}[\/latex]<\/li>\n
                                                • Standard error of the estimate: [latex]s_e = \\sqrt{ \\frac{\\sum (e_i - \\bar{e})^2}{n-2} } = \\sqrt{ \\frac{\\sum e_i^2}{n-2}} = \\sqrt{\\frac{SSE}{n-2}}[\/latex]<\/li>\n
                                                • Test statistic for [latex]\\beta_1: t_o = \\frac{b_1}{\\left( \\frac{s_e}{\\sqrt{S_{xx}}} \\right)}[\/latex] with [latex]df=n-2[\/latex]<\/li>\n
                                                • A [latex](1 - \\alpha) \\times 100 \\%[\/latex] confidnece interval for [latex]\\beta_1: b_1 \\pm t_{\\alpha \/ 2} \\frac{s_e}{\\sqrt{S_{xx}}}[\/latex] with [latex]df = n-2[\/latex]<\/li>\n
                                                • A [latex](1 - \\alpha) \\times 100 \\%[\/latex] confidence interval for\u00a0the conditional mean [latex]\\mu_p[\/latex]\u00a0is\n

                                                  [latex]\\hat{\\mu}_p \\pm t_{\\alpha \/ 2} \\times SE(\\hat{\\mu}_p) = (b_0 + b_1 x_p) \\pm t_{\\alpha \/ 2} \\times s_e \\sqrt{\\frac{(x_p - \\bar{x})^2}{S_{xx}} + \\frac{1}{n}}[\/latex]\u00a0with [latex]df=n-2[\/latex]<\/p>\n<\/li>\n<\/ul>\n

                                                  \n
                                                    \n
                                                  • A [latex](1 - \\alpha) \\times 100 \\%[\/latex] confidence interval for a single response [latex]y_p[\/latex]\u00a0is\n

                                                    [latex]\\hat{y}_p \\pm t_{\\alpha \/ 2} \\times SE(\\hat{y}_p) = (b_0 + b_1 x_p) \\pm t_{\\alpha \/ 2} \\times s_e \\sqrt{\\frac{(x_p - \\bar{x})^2}{S_{xx}} + \\frac{1}{n} + 1}[\/latex]\u00a0with [latex]df=n-2[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/div>\n

                                                    Analysis of Variance (One-Way ANOVA F Test)<\/strong><\/p>\n

                                                    Compare [latex]k[\/latex]\u00a0population means: [latex]\\mu_1, \\mu_2 , \\dots , \\mu_k[\/latex]. Denote sample sizes as [latex]n_1,n_2, \\dots ,n_k[\/latex], sample means as [latex]\\bar{x}_1, \\bar{x}_2, \\dots , \\bar{x}_k[\/latex] and sample standard deviations as [latex]s_1,s_2, \\dots ,s_k[\/latex]. Let [latex]n=n_1 + n_2 + \\dots + n_k[\/latex] and [latex]\\bar{x} = \\frac{\\sum x_{ij}}{n}[\/latex] where [latex]x_{ij}[\/latex] is the jth observation of sample [latex]i[\/latex].<\/p>\n

                                                      \n
                                                    • Test statistic:\u00a0[latex]F_o = \\frac{SSTR \/ (k-1)}{SSE \/ (n-k)} = \\frac{MSTR}{MSE}[\/latex]\u00a0with [latex]df_n=k-1[\/latex]\u00a0and [latex]df_d=n-k[\/latex]<\/li>\n
                                                    • Total sum of squares:\u00a0[latex]SST = \\sum (x_{ij} - \\bar{x})^2 = \\sum x_{ij}^2 - \\frac{\\left( \\sum x_{ij} \\right)^2}{n}[\/latex]<\/li>\n
                                                    • Treatment sum of squares:\u00a0[latex]SSTR = \\sum_{i=1}^k n_i (\\bar{x}_i - \\bar{x})^2[\/latex]<\/span><\/li>\n
                                                    • Error sum of squares:\u00a0[latex]SSE = \\sum (x_{ij} - \\bar{x}_i)^2 = \\sum_{i=1}^k (n_i -1)s_i^2 = SST - SSTR[\/latex]<\/span><\/li>\n
                                                    • ANOVA identity: [latex]SST=SSE+SSTR[\/latex]<\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":19,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[44],"contributor":[],"license":[],"class_list":["post-1512","back-matter","type-back-matter","status-publish","hentry","back-matter-type-resources"],"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/back-matter\/1512","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":28,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/back-matter\/1512\/revisions"}],"predecessor-version":[{"id":5427,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/back-matter\/1512\/revisions\/5427"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/back-matter\/1512\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=1512"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/back-matter-type?post=1512"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=1512"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=1512"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}