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Statistics Final Exam Formula Sheet: Key Concepts, Cheat Sheet of Statistics

Formula sheet with descriptive statistics, probability theory, population and binomial distributions, sampling distributions and the analysis of variances.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

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Grant MacEwan University
STAT 151 Formula Sheet Final Exam
Dr. Karen Buro
Descriptive Statistics
Sample Variance: s2=Pn
i=1(xi¯x)2
n1=Σn
i=1x2
in
i=1xi)2
n
n1
Sample Standard Deviation: s=Sample Variance = s2
Median: Order the data from smallest to largest. The median Mis either the unique
middle value or the mean of the two middle values.
Lower Quartile: Order the data from smallest to largest. The lower quartile Q1is the
median of the smaller half of the values.
Upper Quartile: Order the data from smallest to largest. The upper quartile Q3is the
median of the upper half of the values.
Interquartile Range (iqr) = Upper Quartile Lower Quartile =Q3Q1
Outliers: lower fence =Q11.5iqr and upper fence=Q3+ 1.5iqr
Probability Theory
Addition Rule: P(Aor B) = P(A) + P(B)P(A&B)
Complement Rule: P(A does not occur) = P(not A) = 1 P(A)
Multiplication Rule: P(A&B) = P(A|B)P(B)
Multiplication Rule for Independent Events:
If A and B are independent, then P(A&B) = P(A)P(B)
Conditional Probability of Agiven B, if P(B)>0 : P(A|B) = P(A&B)
P(B)
Population Distributions
The mean (expected value) of a discrete random variable: µ=Pxp(x).
The variance of a discrete random variable: σ2=X(xµ)2p(x)
The standard deviation of a discrete random variable: σ=σ2
Binomial Distribution
Probability to observe ksuccesses in ntrials: p(k) = P(x=k) = ³n
k´pk(1 p)nk
³n
k´=n!
k!(nk)!
Mean and standard deviation of a binomial distribution: µ=np and σ=qnp(1 p)
pf3
pf4
pf5

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Download Statistics Final Exam Formula Sheet: Key Concepts and more Cheat Sheet Statistics in PDF only on Docsity!

Grant MacEwan University

STAT 151 – Formula Sheet – Final Exam

Dr. Karen Buro

Descriptive Statistics

  • Sample Variance: s

2

∑n i=1(xi^ −^ x¯)

2

n − 1

Σni=1x^2 i −

(Σni=1xi)^2 n n − 1

  • Sample Standard Deviation: s =

Sample Variance =

s^2

  • Median: Order the data from smallest to largest. The median M is either the unique

middle value or the mean of the two middle values.

  • Lower Quartile: Order the data from smallest to largest. The lower quartile Q 1 is the

median of the smaller half of the values.

  • Upper Quartile: Order the data from smallest to largest. The upper quartile Q 3 is the

median of the upper half of the values.

  • Interquartile Range (iqr) = Upper Quartile – Lower Quartile =Q 3 − Q 1
  • Outliers: lower fence =Q 1 − 1. 5 iqr and upper fence=Q 3 + 1. 5 iqr

Probability Theory

  • Addition Rule: P (A or B) = P (A) + P (B) − P (A&B)
  • Complement Rule: P (A does not occur) = P (not A) = 1 − P (A)
  • Multiplication Rule: P (A&B) = P (A|B)P (B)
  • Multiplication Rule for Independent Events:

If A and B are independent, then P (A&B) = P (A)P (B)

  • Conditional Probability of A given B, if P (B) > 0 : P (A|B) =

P (A&B)

P (B)

Population Distributions

  • The mean (expected value) of a discrete random variable: μ =

∑ xp(x).

  • The variance of a discrete random variable: σ

2

∑ (x − μ)

2 p(x)

  • The standard deviation of a discrete random variable: σ =

σ^2

Binomial Distribution

  • Probability to observe k successes in n trials: p(k) = P (x = k) =

( n k

) pk(1 − p)n−k

( n k

)

n! k!(n−k)!

  • Mean and standard deviation of a binomial distribution: μ = np and σ =

√ np(1 − p)

Sampling Distributions

  • Sampling Distribution of a Sample Mean, ¯x:

μ¯x = μ, σ¯x =

σ √ n

  • Sampling Distribution of the difference of two Sample Means, ¯x 1 − x¯ 2 :

μ¯x 1 −x¯ 2 = μ 1 − μ 2 , σx¯ 1 −¯x 2 =

√ σ^21

n 1

σ^22

n 2

  • Sampling Distribution of a Sample Proportion, ˆp:

μpˆ = p, σpˆ =

√ p(1 − p)

n

  • Sampling Distribution of the difference of two Sample Proportions, ˆp 1 − pˆ 2 :

μpˆ 1 −pˆ 2 = p 1 − p 2 , σpˆ 1 −pˆ 2 =

√ p 1 (1 − p 1 )

n 1

p 2 (1 − p 2 )

n 2

Estimation

Parameter Estimator SE(Estimator) Approximate Confidence Interval

μ x¯

σ √ n

x¯ ± tα/ 2

s √ n

p pˆ

√ p(1 − p)

n

pˆ ± zα/ 2

√ p ˆ(1 − pˆ)

n

μ 1 − μ 2 ¯x 1 − ¯x 2

√ σ^21

n 1

σ 22

n 2

(¯x 1 − x¯ 2 ) ± tα/ 2

√ s^21

n 1

s^22

n 2

μ 1 − μ 2 ¯xd

σd √ n

x¯d ± tα/ 2

sd √ n

p 1 − p 2 pˆ 1 − pˆ 2

√ p 1 (1 − p 1 )

n 1

p 2 (1 − p 2 )

n 2

(ˆp 1 − pˆ 2 ) ± zα/ 2

√ p ˆ 1 (1 − pˆ 1 )

n 1

pˆ 2 (1 − pˆ 2 )

n 2

Choosing the Sample Size (formulas)

  • Estimate a mean μ with a (1 − α) confidence interval within an amount of m.

n ≥

( z(1−α/2)σ

m

) 2

  • Estimate a proportion p with a (1 − α) confidence interval within an amount of m.

n ≥

( z(1−α/2)

m

) 2

p(1 − p)

Regression Analysis

  • Sum of Squares

SSxy =

∑ xiyi −

∑ xi)(

∑ yi)

n

, SSxx =

∑ x

2 i −^

∑ xi)^2

n

, SSyy =

∑ y

2 i −^

∑ yi)^2

n

  • Correlation Coefficient (r), Coefficient of Determination (R^2 )

r =

SSxy √ SSxxSSyy

, R

2 = r

2

  • Least Squares Regression line ˆy = b 0 + b 1 x, with

b 1 =

SSxy

SSxx

, and b 0 = ¯y − b 1 x¯

  • Estimation of σ

se =

√ SSE

n − 2

, with SSE =

∑ (ˆyi − yi)^2 = SSyy − b 1 SSxy

  • Confidence interval for β 1

b 1 ± t

∗ (^) √se SSxx

  • test statistic for a test about β 1

t 0 =

b 1

se/

SSxx

, df = n − 2

  • Confidence interval for the mean of y, E(y), at x = xp

y ˆ ± t

∗ se

√ 1

n

(xp − x¯)^2

SSxx

  • Prediction Interval for y at x = xp

y ˆ ± t

∗ se

n

(xp − x¯)^2

SSxx

The Analysis of Variance (ANOVA)

  • Total Sum of Squares

SST =

ij

(xij − x¯)

2

ij

x

2 ij −^ CM^ (df^ =^ n^ −^ 1)

with ¯x = sample mean of all measurements, G =

∑ ij xij and^ CM^ =^

G

2

n

  • Sum of Squares for groups

SST R =

i

ni(¯xi − x¯)

2

i

T (^) i^2

ni

− CM (df = k − 1)

with ¯xi = sample mean of observations in sample i,

Ti = Total of observations in sample i.

  • Sum of Squares for Error

SSE =

i

(ni − 1)s

2 i =^ SST^ −^ SSG^ (df^ =^ n^ −^ k)

with si is the standard deviation of observation from sample i.

  • ANOVA–Table

Source df SS M S=SS/df F

Treatments/Groups k − 1 SST R M ST R = SST R/(k − 1) M ST R/M SE

Error n − k SSE M SE = SSE/(n − k)

Total n − 1 SST