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11.2: B - Mathematical Phrases, Symbols, and Formulas

  • Page ID
    79103

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    English Phrases Written Mathematically

    When the English says: Interpret this as:

    \(X\) is at least 4.
    The minimum of \(X\) is 4.
    \(X\) is no less than 4.
    \(X\) is greater than or equal to 4.

    \(X \geq 4\)
    \(X\) is at most 4.
    The maximum of \(X\) is 4.
    \(X\) is no more than 4.
    \(X\) is less than or equal to 4.
    \(X\) does not exceed 4.
    \(X \leq 4\)
    \(X\) is greater than 4.
    \(X\) is more than 4.
    \(X\) exceeds 4.
    \(X > 4\)
    \(X\) is less than 4. \(X < 4\)
    \(X\) is 4.
    \(X\) is equal to 4.
    \(X\) is the same as to 4.​​​​​
    \(X = 4\)
    \(X\) is not 4.
    \(X\) is not equal to 4.
    \(X\) is not the same as 4.
    \(X\) is different than 4.
    \(X \neq 4\)
     

    Symbols and Their Meanings

    Chapter (1st used) Symbol Spoken Meaning
    Sampling and Data \(\sqrt{ } \) The square root of same
    Descriptive Statistics \(Q_1\) quartile one the first quartile
    Descriptive Statistics \(Q_2\) quartile two the second quartile
    Descriptive Statistics \(Q_3\) quartile three the third quartile
    Descriptive Statistics \(IQR\) interquartile range \(Q_3 – Q_1 = IQR\)
    Descriptive Statistics \(\overline x\) \(x\)-bar sample mean
    Descriptive Statistics \(\mu\) mu population mean
    Descriptive Statistics \(s\) \(s\) sample standard deviation
    Descriptive Statistics \(s^2\) \(s\) squared sample variance
    Descriptive Statistics \(\sigma\) sigma population standard deviation
    Descriptive Statistics \(\sigma^2\) sigma squared population variance
    Descriptive Statistics \(\Sigma\) capital sigma sum
    Probability Topics \(\{ \}\) brackets set notation
    Probability Topics \(S\) \(S\) sample space
    Probability Topics \(A\) event \(A\) event \(A\)
    Probability Topics \(P(A)\) probability of \(A\) probability of \(A\) occurring
    Probability Topics \(P(A|B)\) probability of \(A\) given \(B\) probability of \(A\) occurring given \(B\) has occurred
    Probability Topics \(P(A\cup B)\) probability of \(A\) or \(B\) probability of \(A\) or \(B\) or both occurring
    Probability Topics \(P(A\cap B)\) probability of \(A\) and \(B\) probability of both \(A\) and \(B\) occurring (same time)
    Probability Topics \(A^{\prime}\) \(A\)-prime; complement of \(A\) complement of \(A\); not \(A\)
    Probability Topics \(P(A^{\prime})\) probability of the complement of \(A\) same
    Probability Topics \(G_1\) green on first pick same
    Probability Topics \(P(G_1)\) probability of green on first pick same
    The Normal Distribution \(N\) normal distribution same
    The Normal Distribution \(z\) \(z\)-score same
    The Normal Distribution \(Z\) standard normal distribution same
    The Central Limit Theorem \(\overline x\) \(x\)-bar the random variable \(x\)-bar
    The Central Limit Theorem \(\mu_{\overline{x}}\) mean of \(x\)-bars the average of \(x\)-bars
    The Central Limit Theorem \(\sigma_{\overline{x}}\) standard deviation of \(x\)-bars same
    Confidence Intervals \(CL\) confidence level same
    Confidence Intervals \(CI\) confidence interval same
    Confidence Intervals \(EBM\) error bound for a mean same
    Confidence Intervals \(EBP\) error bound for a proportion same
    Confidence Intervals \(t\) Student's \(t\)-distribution same
    Confidence Intervals \(df\) degrees of freedom same
    Confidence Intervals \(t_{\frac{\alpha}{2}}\) Student's \(t\) with \(\alpha\)/2 area in each tail same
    Confidence Intervals \(P^{\prime}\) \(P\)-prime sample proportion of success or interest
    Hypothesis Testing \(H_0\) \(H\)-naught, \(H\)-sub-0 null hypothesis
    Hypothesis Testing \(H_a\) \(H\)-a, \(H\)-sub a alternative (or research) hypothesis
    Hypothesis Testing \(H_1\) \(H\)-1, \(H\)-sub 1 alternative (or research) hypothesis
    Hypothesis Testing \(\alpha\) alpha probability of Type I error
    Hypothesis Testing \(\beta\) beta probability of Type II error
    Hypothesis Testing \(\overline{x}_1-\overline{x}_2\) \(x\)1-bar minus \(x\)2-bar difference in sample means
    Hypothesis Testing \(\mu_{1}-\mu_{2}\) mu-1 minus mu-2 difference in population means
    Hypothesis Testing \(P_{1}^{\prime}-P_{2}^{\prime}\) \(P\)1-prime minus \(P\)2-prime difference in sample proportions
    Hypothesis Testing \(P_{1}-P_{2}\) \(P\)1 minus \(P\)2 difference in population proportions
    Linear Regression and Correlation \(Y = a + bX\) \(Y\) equals \(a\) plus \(b\)-\(X\) equation of a straight line
    Linear Regression and Correlation \(\hat Y\) \(Y\)-hat estimated value of \(Y\)
    Linear Regression and Correlation \(r\) sample correlation coefficient same
    Linear Regression and Correlation \(\varepsilon\) error term for a regression line same
    Linear Regression and Correlation \(SSE\) Sum of Squared Errors same
    F-Distribution and ANOVA \(F\) \(F\)-ratio \(F\)-ratio
     

    Formulas

    Symbols you must know
    Population   Sample
    \(N\) Size \(n\)
    \(\mu\) Mean \(\overline x\)
    \(\sigma^2\) Variance \(s^2\)
    \(\sigma\) Standard deviation \(s\)
    \(P\) Proportion \(P^{\prime}\)
    Single data set formulae
    Population   Sample
    \(Q_{3}=\frac{3(N+1)}{4}, Q_{1}=\frac{(N+1)}{4}\) Inter-quartile range
    \(I Q R=Q_{3}-Q_{1}\)
    \(Q_{3}=\frac{3(n+1)}{4}, Q_{1}=\frac{(n+1)}{4}\)
    \(\sigma^{2}=\frac{1}{N} \sum_{i=1}^{N}\left(x_{i}-\mu\right)^{2}\) Variance \(s^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}\)
    \(\sigma^{2}=\frac{1}{N} \sum_{i=1}^{N}\left(x_{i}-\mu\right)^{2} \cdot f_{i}\) Variance \(s^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2} \cdot f_{i}\)
    Basic probability rules
    \(P(A \cap B)=P(A | B) \cdot P(B)\) Multiplication rule
    \(P(A \cup B)=P(A)+P(B)-P(A \cap B)\) Addition rule
    \(P(A \cap B)=P(A) \cdot P(B) \text { or } P(A | B)=P(A)\) Independence test
     
    The following formulae require the use of the \(z\), \(t\), or \(F\) tables.
    \(z=\frac{x-\mu}{\sigma}\) z-transformation for normal distribution
    Test statistics Confidence intervals
    [bracketed symbols equal margin of error]
    (subscripts denote locations on respective distribution tables)
    \(z_{obs}=\frac{\overline{x}-\mu_{0}}{\frac{\sigma}{\sqrt{n}}}\) Interval for the population mean when sigma is known
    \(\overline{x} \pm\left[z_{(\alpha / 2)} \frac{\sigma}{\sqrt{n}}\right]\)
    \(z_{obs}=\frac{\overline{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}\) Interval for the population mean when sigma is unknown and \(n > 100\)
    \(\overline{x} \pm\left[z_{(\alpha / 2)} \frac{s}{\sqrt{n}}\right]\)
    \(t_{obs}=\frac{\overline{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}\) Interval for the population mean when sigma is unknown and \(n < 100\)
    \(\overline{x} \pm\left[t_{(n-1),(\alpha / 2)} \frac{s}{\sqrt{n}}\right]\)
    \(z_{obs}=\frac{P^{\prime}-P_0}{\sqrt{\frac{P_0 (1-P_0)}{n}}}\) Interval for the population proportion
    \(P^{\prime} \pm\left[z_{(\alpha / 2)} \sqrt{\frac{P^{\prime} \left(1-P^{\prime}\right)}{n}}\right]\)
    \(t_{obs}=\frac{\bar{x}_d-\mu_{d}}{\frac{s_{d}}{\sqrt{n}}}\) Interval for difference between two means with matched pairs
    \(\bar{x}_d \pm\left[t_{(n-1),(\alpha / 2)} \frac{s_{d}}{\sqrt{n}}\right]\) where \(s_d\) is the deviation of the differences
    \(z_{obs}=\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left({\mu_{1}}-{\mu_{2}}\right)}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\) Interval for difference between two independent means when \(n > 100\)
    \(\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[z_{(\alpha / 2)} \sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}\right]\)
    \(z_{obs}=\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left({\mu_{1}}-{\mu_{2}}\right)}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\) Interval for difference between two independent means when \(n < 100\)
    \(\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[t_{(n_1+n_2-2),(\alpha / 2)} \sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}\right]\)
      Interval for difference between two population proportions
    \(\left(P_{1}^{\prime}-P_{2}^{\prime}\right) \pm\left[z_{(\alpha / 2)} \sqrt{\frac{P_{1}^{\prime}\left(1-P_{1}^{\prime}\right)}{n_{1}}+\frac{P_{2}^{\prime}\left(1-P_{2}^{\prime}\right)}{n_{2}}}\right]\)

    Simple linear regression formulae for \(Y=a+b(X)\)

    \[r_{X Y}=\frac{\sum (X_{i}-\overline{X})*(Y_{i}-\overline{Y})}{\sqrt{\sum (X_{i}-\overline{X})^{2}*\sum (Y_{i}-\overline{Y})^{2}}}\nonumber\]

    \[r_{X Y}=\frac{\sum X_{i} Y_{i}-\frac{\left(\sum X_{i}\right)\left(\sum Y_{i}\right)}{n}}{\sqrt{\left[\sum X_{i}^{2}-\frac{\left(\sum X_{i}\right)^{2}}{n}\right]*\left[\sum Y_{i}^{2}-\frac{\left(\sum Y_{i}\right)^{2}}{n}\right]}}\nonumber\]

    Correlation coefficient

    \[b_{1}=\frac{\Sigma(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{\Sigma(X_{i}-\overline{X})^{2}}\nonumber\]

    \[b_{1}=\frac{\sum X_{i} Y_{i}-\frac{\left(\sum X_{i}\right)\left(\sum Y_{i}\right)}{n}}{\sum X_{i}^{2}-\frac{\left(\sum X_{i}\right)^{2}}{n}}\nonumber\]

    \[b_{1}=r_{X Y}\left(\frac{s_{Y}}{s_{X}}\right)\nonumber\]

    Coefficient \(b\) (or \(b_1\), slope)

    \[b_{0}=\overline{Y}-b_{1} \overline{X}\nonumber\]

    \(Y\)-intercept (\(a\), or \(b_0\))

    \(s_{e}^{2}=\frac{\Sigma\left(Y_{i}-\hat{Y}_{i}\right)^{2}}{n-k}=\frac{\sum_{i=1}^{n} e_{i}^{2}}{n-k}\)

    Estimate of the error variance

    \(s_{b}=\frac{s_{e}^{2}}{\sqrt{\left(X_{i}-\overline{X}\right)^{2}}}=\frac{s_{e}^{2}}{(n-1) s_{X}^{2}}\)

    Standard error for coefficient \(b\)

    \(t_{obs}=\frac{b-\beta_{0}}{s_b}\)

    Hypothesis test for coefficient \(\beta\)

    \(b \pm\left[t_{n-2, \alpha / 2} s_{b}\right]\)

    Interval for coefficient \(\beta\)

    \(\hat{Y} \pm\left[t_{\alpha / 2} * s_{e}\left(\sqrt{\frac{1}{n}+\frac{\left(X_{p}-\overline{X}\right)^{2}}{s_{X}}}\right)\right]\)

    Interval for expected value of \(Y\)

    \(\hat{Y} \pm\left[t_{\alpha / 2} * s_{e}\left(\sqrt{1+\frac{1}{n}+\frac{\left(X_{p}-\overline{X}\right)^{2}}{s_{X}}}\right)\right]\)

    Prediction interval for an individual \(Y\)

    ANOVA formulae

    \(SS_R=n_1\left(\bar{x}_{1}-\bar{x}\right)^2+\cdots+n_g\left(\bar{x}_{g}-\bar{x}\right)^2\)

    Sum of squares regression

    \(SS_E=\left(n_1-1\right)s_1^2+\cdots+\left(n_g-1\right)s_g^2\)

    Sum of squares error

    \(SS_T=SS_R + SS_E\)

    Sum of squares total

    \(R^{2}=\frac{SS_R}{SS_T}\)

    Coefficient of determination

     
    The following is the breakdown of a one-way ANOVA table for linear regression.
    Source of variation Sum of squares Degrees of freedom Mean squares \(F\)-ratio
    Regression

    \(n_1\left(\bar{x}_{1}-\bar{x}\right)^2+\cdots+n_g\left(\bar{x}_{g}-\bar{x}\right)^2\)

    \(1\) or \(g−1\) \(M S R=\frac{S S_R}{d f_{R}}\) \(F=\frac{M S_R}{M S_E}\)
    Error

    \(\left(n_1-1\right)s_1^2+\cdots+\left(n_g-1\right)s_g^2\)

    \(n-g\) \(M S E=\frac{S S_E}{d f_{E}}\)  
    Total \(SS_R + SS_E\) \(n−1\)    

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