# 11.2: B - Mathematical Phrases, Symbols, and Formulas

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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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