15 APPENDIX B: MATHEMATICAL PHRASES, SYMBOLS,

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

Chapter (1st used)

Symbol

Spoken

Meaning

Sampling and Data

The square root of

same

Sampling and Data

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

Q3 – Q1 = IQR

Descriptive Statistics

x–

x-bar

sample mean

Descriptive Statistics

µ

mu

population mean

Descriptive Statistics

s

s

sample standard deviation

Descriptive Statistics

s2

s squared

sample variance

Descriptive Statistics

σ

sigma

population standard deviation

Descriptive Statistics

σ 2

sigma squared

population variance

Descriptive Statistics

Σ

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

prob. of A occurring given B has occurred

Probability Topics

P(A ∪ B)

prob. of A or B

prob. of A or B or both occurring

Probability Topics

P(A ∩ 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. density function

same

Discrete Random Variables

X

X

the random variable X

Discrete Random Variables

X ~

the distribution of X

same

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

Chapter (1st used)

Symbol

Spoken

Meaning

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

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

X–

X-bar

the random variable X-bar

The Central Limit Theorem

µ –x

mean of X-bars

the average of X-bars

The Central Limit Theorem

o –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 α

2

student t with α/2 area in right tail

same

Confidence Intervals

p′

p-prime

sample proportion of success

Confidence Intervals

q′

q-prime

sample proportion of failure

Hypothesis Testing

H0

H-naught, H-sub 0

null hypothesis

Hypothesis Testing

Ha

H-a, H-sub a

alternate hypothesis

Hypothesis Testing

H1

H-1, H-sub 1

alternate hypothesis

Hypothesis Testing

α

alpha

probability of Type I error

Hypothesis Testing

β

beta

probability of Type II error

Hypothesis Testing

X–1 – X2

X1-bar minus X2-bar

difference in sample means

Hypothesis Testing

µ 1 − µ 2

mu-1 minus mu-2

difference in population means

Hypothesis Testing

P′1 − P′2

P1-prime minus P2-prime

difference in sample proportions

Chapter (1st used)

Symbol

Spoken

Meaning

Hypothesis Testing

p1 − p2

p1 minus p2

difference in population proportions

Chi-Square Distribution

Χ 2

Ky-square

Chi-square

Chi-Square Distribution

O

Observed

Observed frequency

Chi-Square Distribution

E

Expected

Expected frequency

Linear Regression and Correlation

y = a + bx

y equals a plus b-x

equation of a straight line

Linear Regression and Correlation

y^

y-hat

estimated value of y

Linear Regression and Correlation

r

sample correlation coefficient

same

Linear Regression and Correlation

ε

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

Symbols You Must Know

Population

Sample

N

Size

n

µ

Mean

x

σ 2

Variance

s2

σ

Standard Deviation

s

p

Proportion

p′

Single Data Set Formulae

Population

Sample

N

µ = E(x) = 1 ∑ (xi)

Ni = 1

Arithmetic Mean

n

x– = 1n ∑ (xi)

i = 1

Geometric Mean

⎛ n⎞1n

~x = ⎜∏ Xi⎟

⎝i = 1 ⎠

Q3 = 3(n + 1) , Q1 = (n + 1)

44

Inter-Quartile Range

IQR = Q3 − Q1

Q3 = 3(n + 1) , Q1 = (n + 1)

44

N

σ 2 = 1 ∑ (xi − µ)2 Ni = 1

Variance

n

s2 = 1n ∑ ⎛xi − x_ ⎞2

⎝⎠

i = 1

Single Data Set Formulae

Population

Sample

N

µ = E(x) = 1 ∑ ⎛mi * f ⎞

Ni = 1 ⎝i⎠

Arithmetic Mean

n

x– = 1n ∑ ⎛mi * f ⎞

⎝i⎠

i = 1

Geometric Mean

⎛ n⎞1n

~x = ⎜∏ Xi⎟

⎝i = 1 ⎠

N

σ 2 = 1 ∑ (mi − µ)2 * fi Ni = 1

Variance

n

s2 = 1n ∑ ⎛mi − x_ ⎞2 * fi

⎝⎠

i = 1

CV = σ * 100

µ

Coefficient of Variation

CV = _s * 100

x

Basic Probability Rules

P(A ∩ B) = P(A|B) * P(B)

Multiplication Rule

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Addition Rule

P(A ∩ B) = P(A) * P(B) or P(A|B) = P(A)

Independence Test

Hypergeometric Distribution Formulae

nCx = ⎛nx⎞ = x !( n ! x)!

⎝ ⎠n −

Combinatorial Equation

⎛ A⎞⎛N − A⎞

P(x) = ⎝ x ⎠⎝ n − x ⎠

⎛N ⎞

⎝ n ⎠

Probability Equation

E⎛X⎞ = µ = np

⎝ ⎠

Mean

σ 2 = ⎛N − n⎞np(q)

⎝N − 1⎠

Variance

Binomial Distribution Formulae

P(x) = x !( n ! x)! px (q)n − x

n −

Probability Density Function

E⎛X⎞ = µ = np

⎝ ⎠

Arithmetic Mean

σ 2 = np⎛q⎞

⎝ ⎠

Variance

Geometric Distribution Formulae

P(X = x) = (1 − p) x − 1(p)

Probability when x

is the first success.

Probability when x is the number of failures before first success

P(X = x) = (1 − p) x(p)

µ = 1p

Mean

Mean

µ = 1 − p

p

σ 2 = ⎛1 − p⎞

⎝⎠

p2

Variance

Variance

σ 2 = ⎛1 − p⎞

⎝⎠

p2

Poisson Distribution Formulae

P(x) = e−µ µx

x !

Probability Equation

E(X) = µ

Mean

σ 2 = µ

Variance

Uniform Distribution Formulae

f (x) = b 1 a for a ≤ x ≤ b

−

PDF

E(X) = µ = a + b

2

Mean

σ 2 = (b − a)2

12

Variance

Exponential Distribution Formulae

P(X ≤ x) = 1 − e−mx

Cumulative Probability

E(X) = µ = m1 or m = 1µ

Mean and Decay Factor

σ 2 = 1 = µ 2

m2

Variance

The following page of formulae requires the use of the ” Z “, ” t “, ” χ 2 ” or ” F ” tables.

Z = x − µ

σ

Z-transformation for Normal Distribution

Z = x − np′

np′(q′)

Normal Approximation to the Binomial

Probability (ignores subscripts) Hypothesis Testing

Confidence Intervals

[bracketed symbols equal margin of error] (subscripts denote locations on respective distribution tables)

Z = x– – µ 0

cσ

n

Interval for the population mean when sigma is known

x– ± ⎡Z σ ⎤

⎣ (α / 2) n⎦

Z = x– – µ 0

cs

n

Interval for the population mean when sigma is unknown but n > 30 x– ± ⎡Z s ⎤

⎣ (α / 2) n⎦

t = x– – µ 0

cs

n

Interval for the population mean when sigma is unknown but n < 30 x– ± ⎡t s ⎤

⎣ (n − 1), (α / 2) n⎦

Zc = p′ – p0

p0 q0

n

Interval for the population proportion

p′ ± ⎡Z(α2) p′q′⎤

⎣/n ⎦

d– – δ0

tc =sd

Interval for difference between two means with matched pairs

d– ± ⎡t(n − 1), (α2) sd ⎤ where sd is the deviation of the differences

⎣/n⎦

Z = (x– – x– ) − δ

c1 2 22 0

σ1 + σ2 n1n2

Interval for difference betwee⎡n two means when⎤sigmas are known

(x– – x– ) ± ⎢Zσ 2 + σ 2⎥ 12⎣ (α / 2) n1n2 ⎦

12

⎛ x¯ – x¯ ⎞ – δ tc = ⎝ 12⎠0

⎛ (s1)2(s2)2⎞

⎜ n1 + n2 ⎟

⎝⎠

Interval for difference between two means with equal variances when sigmas

⎡are unknown⎤

( x¯ – x¯ ) ± ⎢t⎛(s1)2 + (s2)2⎞⎥ where

12⎣ d f , (α / 2) ⎜ n1n2 ⎟⎦

⎝⎠

⎛22⎞ 2

⎜ (s1) + (s2) ⎟

d f =⎝ n1n2 ⎠

⎛1 ⎞ ⎛ (s1)2⎞⎛1 ⎞ ⎛ (s2)2⎞

⎜ n1 − 1⎟ ⎜ n1 ⎟ + ⎜ n2 − 1⎟ ⎜ n2 ⎟

⎝⎠ ⎝⎠⎝⎠ ⎝⎠

Z = ⎛p′1 – p′ ⎞ − δ0

⎝2⎠

c p′ ⎛q′ ⎞p′ ⎛q′ ⎞ 1⎝ 1⎠ + 2⎝ 2⎠ n1n2

Interval for difference between two population proportions

⎡p′ ⎛q′ ⎞p′ ⎛q′ ⎞⎤

(p′1 – p′2) ± ⎢Z(α / 2)1⎝ 1⎠ +2⎝ 2⎠⎥

⎣n1n2⎦

χc2 = (n − 1)s2

σ 2

0

Tests for GOF, Independence, and Homogeneity

χc2 = Σ (O − E)2 where O = observed values and E = expected values

E

F = s2

cs1

2

2

Where s2 is the sample variance which is the larger of the two sample

1

variances

The Next 3 Formulae are for Determining Sample Size with Confidence Intervals

(note: E represents the margin of error)

Z⎛2a⎞ σ 2

n = ⎝2⎠

E 2

Use when sigma is known

E = x¯ − µ

Z⎛2a⎞(0.25)

n = ⎝2⎠ 2

E

Use when p′ is unknown

E = p′ − p

Z⎛2a⎞[p′(q′)]

n = ⎝2⎠ 2

E

Use when p′ is uknown

E = p′ − p

Simple Linear Regression Formulae for y = a + b⎛x⎞

⎝ ⎠

r =

Σ ⎡⎛x − x¯ ⎞⎛y − y¯ ⎞⎤

⎣⎝⎠⎝⎠⎦

Σ ⎛x − x¯ ⎞2 * Σ ⎛y − y¯ ⎞2

⎝⎠⎝⎠

= Sxy =

Sx Sy

SSR SST

Correlation Coefficient

Σ ⎡⎛x − x¯ ⎞⎛y − y¯ ⎞⎤S ⎛s ⎞ b =⎣⎝⎠⎝ 2 ⎠⎦ = S xy = ry, x⎜sy⎟ Σ ⎛x − x¯ ⎞Sx⎜ x⎟

⎝⎠⎝ ⎠

Coefficient b (slope)

a = y¯ − b⎛ x¯ ⎞

⎝ ⎠

y-intercept

⎛ ^ ⎞2n 2 s2 = Σ ⎝yi − y i⎠ = i =Σ 1 ei en − kn − k

Estimate of the Error Variance

S =s2e=s2e

b⎛x − x¯ ⎞2⎛n − 1⎞s2x

⎝ i⎠⎝⎠

Standard Error for Coefficient b

tc = b − β0

sb

Hypothesis Test for Coefficient β

b ± ⎡tS ⎤

⎣ n − 2, α / 2 b⎦

Interval for Coefficient β

⎡⎛⎛¯ ⎞2⎞⎤

y^ ± ⎢t* s ⎜ 1 + ⎝x p − x ⎠ ⎟⎥

⎢ α / 2e⎜ nsx⎟⎥

⎢⎜⎟⎥

⎣⎝⎠⎦

Interval for Expected value of y

⎡⎛⎛¯ ⎞2⎞⎤

y^ ± ⎢t* s ⎜ 1 + 1 + ⎝x p − x ⎠ ⎟⎥

⎢ α / 2e⎜nsx⎟⎥

⎢⎜⎟⎥

⎣⎝⎠⎦

Prediction Interval for an Individual y

ANOVA Formulae

SSR =n (y^ − y¯ )2

i =Σ 1i

Sum of Squares Regression

SSE =n (y^ − y¯ )2

i =Σ 1i i

Sum of Squares Error

SST =n (y − y¯ )2

i =Σ 1i

Sum of Squares Total

R2 = SSR

SST

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

SSR

1 or k − 1

MSR = SSR

d fR

F = MSR

MSE

Error

SSE

n − k

MSE = SSE

d fE

Total

SST

n − 1