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Introductory Statistics

F | Mathematical Phrases, Symbols, and Formulas

Introductory StatisticsF | Mathematical Phrases, Symbols, and Formulas
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  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Frequency, Frequency Tables, and Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. 1.5 Data Collection Experiment
    7. 1.6 Sampling Experiment
    8. Key Terms
    9. Chapter Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
    3. 2.2 Histograms, Frequency Polygons, and Time Series Graphs
    4. 2.3 Measures of the Location of the Data
    5. 2.4 Box Plots
    6. 2.5 Measures of the Center of the Data
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. 2.8 Descriptive Statistics
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables
    6. 3.5 Tree and Venn Diagrams
    7. 3.6 Probability Topics
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Bringing It Together: Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
    3. 4.2 Mean or Expected Value and Standard Deviation
    4. 4.3 Binomial Distribution
    5. 4.4 Geometric Distribution
    6. 4.5 Hypergeometric Distribution
    7. 4.6 Poisson Distribution
    8. 4.7 Discrete Distribution (Playing Card Experiment)
    9. 4.8 Discrete Distribution (Lucky Dice Experiment)
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. References
    16. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Continuous Probability Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution
    5. 5.4 Continuous Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Normal Distribution (Lap Times)
    5. 6.4 Normal Distribution (Pinkie Length)
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means (Averages)
    3. 7.2 The Central Limit Theorem for Sums
    4. 7.3 Using the Central Limit Theorem
    5. 7.4 Central Limit Theorem (Pocket Change)
    6. 7.5 Central Limit Theorem (Cookie Recipes)
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Single Population Mean using the Normal Distribution
    3. 8.2 A Single Population Mean using the Student t Distribution
    4. 8.3 A Population Proportion
    5. 8.4 Confidence Interval (Home Costs)
    6. 8.5 Confidence Interval (Place of Birth)
    7. 8.6 Confidence Interval (Women's Heights)
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Distribution Needed for Hypothesis Testing
    5. 9.4 Rare Events, the Sample, Decision and Conclusion
    6. 9.5 Additional Information and Full Hypothesis Test Examples
    7. 9.6 Hypothesis Testing of a Single Mean and Single Proportion
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Two Population Means with Unknown Standard Deviations
    3. 10.2 Two Population Means with Known Standard Deviations
    4. 10.3 Comparing Two Independent Population Proportions
    5. 10.4 Matched or Paired Samples
    6. 10.5 Hypothesis Testing for Two Means and Two Proportions
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Goodness-of-Fit Test
    4. 11.3 Test of Independence
    5. 11.4 Test for Homogeneity
    6. 11.5 Comparison of the Chi-Square Tests
    7. 11.6 Test of a Single Variance
    8. 11.7 Lab 1: Chi-Square Goodness-of-Fit
    9. 11.8 Lab 2: Chi-Square Test of Independence
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  13. 12 Linear Regression and Correlation
    1. Introduction
    2. 12.1 Linear Equations
    3. 12.2 Scatter Plots
    4. 12.3 The Regression Equation
    5. 12.4 Testing the Significance of the Correlation Coefficient
    6. 12.5 Prediction
    7. 12.6 Outliers
    8. 12.7 Regression (Distance from School)
    9. 12.8 Regression (Textbook Cost)
    10. 12.9 Regression (Fuel Efficiency)
    11. Key Terms
    12. Chapter Review
    13. Formula Review
    14. Practice
    15. Homework
    16. Bringing It Together: Homework
    17. References
    18. Solutions
  14. 13 F Distribution and One-Way ANOVA
    1. Introduction
    2. 13.1 One-Way ANOVA
    3. 13.2 The F Distribution and the F-Ratio
    4. 13.3 Facts About the F Distribution
    5. 13.4 Test of Two Variances
    6. 13.5 Lab: One-Way ANOVA
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  15. A | Review Exercises (Ch 3-13)
  16. B | Practice Tests (1-4) and Final Exams
  17. C | Data Sets
  18. D | Group and Partner Projects
  19. E | Solution Sheets
  20. F | Mathematical Phrases, Symbols, and Formulas
  21. G | Notes for the TI-83, 83+, 84, 84+ Calculators
  22. H | Tables
  23. Index

English Phrases Written Mathematically

When the English says: Interpret this as:
X is at least 4. X ≥ 4
The minimum of X is 4. X ≥ 4
X is no less than 4. X ≥ 4
X is greater than or equal to 4. X ≥ 4
X is at most 4. X ≤ 4
The maximum of X is 4. X ≤ 4
X is no more than 4. X ≤ 4
X is less than or equal to 4. X ≤ 4
X does not exceed 4. X ≤ 4
X is greater than 4. X > 4
X is more than 4. X > 4
X exceeds 4. X > 4
X is less than 4. X < 4
There are fewer X than 4. X < 4
X is 4. X = 4
X is equal to 4. X = 4
X is the same as 4. X = 4
X is not 4. X ≠ 4
X is not equal to 4. X ≠ 4
X is not the same as 4. X ≠ 4
X is different than 4. X ≠ 4
Table F1

Formulas

Formula 1: Factorial

n ! = n ( n 1 ) ( n 2 ) . . . ( 1 ) n!=n(n1)(n2)...(1)

0 ! = 1 0!=1

Formula 2: Combinations

( n r )= n! (nr)!r! ( n r )= n! (nr)!r!

Formula 3: Binomial Distribution

X~B(n,p) X~B(n,p)

P(X=x)=( n x ) p x q nx P(X=x)=( n x ) p x q nx , for x = 0 , 1 , 2 , . . . , n x=0,1,2,...,n

Formula 4: Geometric Distribution

X~G(p) X~G(p)

P(X=x)= q x1 p P(X=x)= q x1 p , for x = 1 , 2 , 3 , . . . x=1,2,3,...

Formula 5: Hypergeometric Distribution

X~H(r,b,n) X~H(r,b,n)

P ( X = x ) = ( ( r x ) ( b n x ) ( r + b n ) ) P(X=x)= ( ( r x ) ( b n x ) ( r + b n ) )

Formula 6: Poisson Distribution

X~P(μ) X~P(μ)

P ( X = x ) = μ x e μ x ! P(X=x)= μ x e μ x !

Formula 7: Uniform Distribution

X~U(a,b) X~U(a,b)

f(X)= 1 ba f(X)= 1 ba , a < x < b a<x<b

Formula 8: Exponential Distribution

X~Exp(m) X~Exp(m)

f(x)=m e mx m>0,x0 f(x)=m e mx m>0,x0

Formula 9: Normal Distribution X~N(μ, σ 2 ) X~N(μ, σ 2 )

f ( x ) = 1 σ 2 π e ( x μ ) 2 2 σ 2 f(x)= 1 σ 2 π e ( x μ ) 2 2 σ 2 , <x < <x<

Formula 10: Gamma Function

Γ(z)= 0 x z1 e x dx Γ(z)= 0 x z1 e x dx z>0z>0

Γ ( 1 2 ) = π Γ( 1 2 )= π

Γ ( m + 1 ) = m ! Γ(m+1)=m! for mm, a nonnegative integer

otherwise: Γ(a+1)=aΓ(a) Γ(a+1)=aΓ(a)

Formula 11: Student's t-distribution

X~ t df X~ t df

f ( x ) = ( 1 + x 2 n ) ( n + 1 ) 2 Γ ( n + 1 2 ) Γ ( n 2 ) f(x)= ( 1 + x 2 n ) ( n + 1 ) 2 Γ ( n + 1 2 ) Γ ( n 2 )

X = Z Y n X= Z Y n

Z~N(0,1),Y~ Χ df 2 Z~N(0,1),Y~ Χ df 2 , nn = degrees of freedom

Formula 12: Chi-Square Distribution

X~ Χ df 2 X~ Χ df 2

f ( x ) = x n 2 2 e x 2 2 n 2 Γ ( n 2 ) f(x)= x n 2 2 e x 2 2 n 2 Γ ( n 2 ) , x>0x>0 , nn = positive integer and degrees of freedom

Formula 13: F Distribution

X~F df(n),df(d) X~F df(n),df(d)

df(n)= df(n)=degrees of freedom for the numerator

df(d)= df(d)=degrees of freedom for the denominator

f(x)= Γ( u+v 2 ) Γ( u 2 )Γ( v 2 ) ( u v ) u 2 x ( u 2 1) [1+( u v ) x 0.5(u+v) ] f(x)= Γ( u+v 2 ) Γ( u 2 )Γ( v 2 ) ( u v ) u 2 x ( u 2 1) [1+( u v ) x 0.5(u+v) ]

X = Y u W v X= Y u W v , YY, WW are chi-square

Symbols and Their Meanings

Chapter (1st used) Symbol Spoken Meaning
Sampling and Data                     The square root of same
Sampling and Data π π Pi 3.14159… (a specific number)
Descriptive Statistics Q1 Quartile one the first quartile
Descriptive Statistics Q2 Quartile two the second quartile
Descriptive Statistics Q3 Quartile three the third quartile
Descriptive Statistics IQR interquartile range Q3Q1 = IQR
Descriptive Statistics x ¯ x ¯ x-bar sample mean
Descriptive Statistics μμ mu population mean
Descriptive Statistics s sx sx s sample standard deviation
Descriptive Statistics s2s2 sx2 sx2 s squared sample variance
Descriptive Statistics σσ σxσx σx sigma population standard deviation
Descriptive Statistics σ2σ2 σx2 σx2 sigma squared population variance
Descriptive Statistics ΣΣ capital sigma sum
Probability Topics { } {} brackets set notation
Probability Topics SS S sample space
Probability Topics AA Event A event A
Probability Topics P ( A ) P ( A ) probability of A probability of A occurring
Probability Topics P(A|B) P(A|B) probability of A given B prob. of A occurring given B has occurred
Probability Topics P(A OR B) P(A OR B) prob. of A or B prob. of A or B or both occurring
Probability Topics P(A AND B) P(A AND B) prob. of A and B prob. of both A and B occurring (same time)
Probability Topics A A-prime, complement of A complement of A, not A
Probability Topics P(A') prob. of complement of A same
Probability Topics G1 green on first pick same
Probability Topics P(G1) prob. of green on first pick same
Discrete Random Variables PDF prob. distribution function same
Discrete Random Variables X X the random variable X
Discrete Random Variables X ~ the distribution of X same
Discrete Random Variables B binomial distribution same
Discrete Random Variables G geometric distribution same
Discrete Random Variables H hypergeometric dist. same
Discrete Random Variables P Poisson dist. same
Discrete Random Variables λλ Lambda average of Poisson distribution
Discrete Random Variables greater than or equal to same
Discrete Random Variables less than or equal to same
Discrete Random Variables = equal to same
Discrete Random Variables not equal to same
Continuous Random Variables f(x) f of x function of x
Continuous Random Variables pdf prob. density function same
Continuous Random Variables U uniform distribution same
Continuous Random Variables Exp exponential distribution same
Continuous Random Variables k k critical value
Continuous Random Variables f(x) = f of x equals same
Continuous Random Variables m m decay rate (for exp. dist.)
The Normal Distribution N normal distribution same
The Normal Distribution z z-score same
The Normal Distribution Z standard normal dist. same
The Central Limit Theorem CLT Central Limit Theorem same
The Central Limit Theorem X ¯ X ¯ X-bar the random variable X-bar
The Central Limit Theorem μ x μ x mean of X the average of X
The Central Limit Theorem μ x ¯ μ x ¯ mean of X-bar the average of X-bar
The Central Limit Theorem σ x σ x standard deviation of X same
The Central Limit Theorem σ x ¯ σ x ¯ standard deviation of X-bar same
The Central Limit Theorem ΣXΣX sum of X same
The Central Limit Theorem ΣxΣx sum of x 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 α 2 t α 2 student t with a/2 area in right tail same
Confidence Intervals p p ; p ^ p ^ p-prime; p-hat sample proportion of success
Confidence Intervals q q ; q ^ q ^ q-prime; q-hat sample proportion of failure
Hypothesis Testing H 0 H 0 H-naught, H-sub 0 null hypothesis
Hypothesis Testing H a H a H-a, H-sub a alternate hypothesis
Hypothesis Testing H 1 H 1 H-1, H-sub 1 alternate hypothesis
Hypothesis Testing αα alpha probability of Type I error
Hypothesis Testing β β beta probability of Type II error
Hypothesis Testing X1 ¯ X2 ¯ X1 ¯ X2 ¯ X1-bar minus X2-bar difference in sample means
Hypothesis Testing μ 1 μ 2 μ 1 μ 2 mu-1 minus mu-2 difference in population means
Hypothesis Testing P 1 P 2 P 1 P 2 P1-prime minus P2-prime difference in sample proportions
Hypothesis Testing p 1 p 2 p 1 p 2 p1 minus p2 difference in population proportions
Chi-Square Distribution Χ 2 Χ 2 Ky-square Chi-square
Chi-Square Distribution OO Observed Observed frequency
Chi-Square Distribution EE Expected Expected frequency
Linear Regression and Correlation y = a + bx y equals a plus b-x equation of a line
Linear Regression and Correlation y^y^ y-hat estimated value of y
Linear Regression and Correlation rr correlation coefficient same
Linear Regression and Correlation εε error same
Linear Regression and Correlation SSE Sum of Squared Errors same
Linear Regression and Correlation 1.9s 1.9 times s cut-off value for outliers
F-Distribution and ANOVA F F-ratio F-ratio
Table F2 Symbols and their Meanings
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