OPRE504 OPRE504 Business Statistics Section 1 Chapter 1 Descriptive Statistics • Descriptive statistics Inferential statistics • • Measurement scales Qualitative/quantitative • • Percentiles Quartiles • • Samples/populations Mean/median/mode • • Variance/standard deviation Skewness/kurtosis • • Visual analysis Recommended Problems: 1, 4, 5, 6, 7, 10, 11, 13, 14, 16, 18, 19, 26, 47, 48, 49, 53 Other: Excel functions, Excel Data Analysis Addin, Excel charts, Basic Stats Template Chapter 2 Probability Classical probability • • Expected value Subjective probability • Sets • Venn diagrams • • Factorals Permutations • Recommended Problems: 1, 2, 4, 5, 7, 8, 2 7, 28, 36, 37, 38, 52 Other: ? Chapter 3 Random Variables Random variable • • Probability distribution Cumulative distribution • • Discrete & continuous variables Expected values E(X) • Fair game • Functions of variables • • Bernoulli random variable Binomial distributions • • Negative binomial distributions Recommended Problems: 1, 2, 3, 8, 11, 12, 13, 14, 32, 33, 34, 36, 37, 43, 44, 45
Other: Binomial Template, Negative Binomial Template Chapter 4 The Normal Distribution • Characteristics Multiple variables • • Finding Z probabilities Z-transformations • • Inverse Transformation Recommended Problems: 1, 2, 3, 4, 5, 6, 2 2, 25, 27, 39, 40, 52, 53 Other: Using the z-table, Normal Distribution template Chapter 5 Sampling • Sample statistics Population parameters • • Random Number Table Random Number Generators • • Stratified sampling Non-response bias • • Sampling distributions Central Limit Theorem • • Degrees of freedom Recommended Problems: 2, 3, 4, 5, 12, 13, 18, 20, 21 Other: Excel RAND function, t Table, Z table, z Template Chapter 6 Confidence Intervals Point estimate • • Interval estimate CI for population proportions • • CI for population variance Determining sample size • Recommended Problems: 4, 5, 6, 7, 8, 18, 19, 20, 21, 40, 41, 42 Other: Chi-square table, Estimating mean template
Formulas – Section 1 Chapter 1 Descriptive Statistics Percentiles
( n + 1) P / 100 Mean of a sample
Mean of a population n
N
∑
∑ x
x j
x
=
j
i =1
µ =
n
Sample variance
∑ x ( x − x) j
2
=
N
i
j
2
i =1
σ =
n −1
s
∑
i
i =1
N
N
x j ( xi
−
x)
2
∑ x ( x − µ ) j
σ =
i =1
=
2
Population standard deviation
n
2
∑ x ( x − µ )
2
Sample standard deviation
s=
N
Population variance n
s
i =1
n −1
2
σ =
i
i =1
N
**also unbiased version page 16 (n-1) Sample variance (shortcut)
n
s2
∑ x =
n ∑ x j 2 − i 1
2
=
j
i =1
n
n −1
Chapter 2 Probability Probability of an event
P(A)=
Complement
n(A) n(S)
P( A) = 1 − P( A)
2
Intersection
Mutually exclusive
P( A ∩ B) =
P( A ∩ C ) = 0
n( A ∩ B ) n( S )
Union
Conditional probability
P( A ∪ B) =
n( A ∪ B)
P( A) + P( B) − P( A ∩ B)
=
n( S )
Permutations
P( A B) =
P ( A ∩ B) P( B )
, where P ( B) ≠ 0
Combinations
n r = n C r
n!
n Pr =
(n − r )!
n!
=
r!(n !(n − r)! r)!
Chapter 3 Random Variables Variance of a random variable σ
2
=
V (X )
=
E[( X
− µ)
2
]=
Standard deviation of a random variable
∑ ( x − µ )
2
P ( x)
all x
=
E( X ) − [ E( X ) ] = ∑ x2 P( x) − ∑ x P( x) al l x a ll x 2
2
σ =
SD( X ) = V ( X )
2
Binomial Distribution n P ( X = x) = p x(1 − p )( x
E( x) = np
Expected value of a random variable n− x)
µ =
=
V( x)= np(1 − p)
x − 1 s ( x −s ) x) = p (1 − p) s −1
( )x= µ = E
s p
V( )x=
∑ xP( x) all x
Negative Binomial Distribution
P( X
E( X) =
s(1-p) p
2
Chapter 4 Normal Distribution Normal Probability Density Function
Z-Transformation
f( x) =
1
−
2πσ
1 x − µ
2
σ
e
2
-∞
Z =
〈 x〈 ∞
x − µ σ
Inverse Transformation
x = µ + zσ
Chapter 5 Sampling Sample proportion
Variance of the sample mean 2
x
pˆ =
V ( X )
2 = σ X
=
n
n
Expected value of the sample mean
E ( X ) = µ
σ X
Standard deviation of the sample mean
SD( X ) = σ x
=
σ
n
When the population standard deviation is unknown, convert to a “t” value and use Student’s t table to find the probability
t =
X − µ s n
Chapter 6 Confidence Intervals Confidence Interval ( σ is known)
CI
=
x ± z
α
n
σ 2
Must use z-table Confidence Interval for population proportion
Confidence Interval ( σ is not known)
CI
=
s ± x t n 2, 1 n α
Must use Student’s t table
−
ˆˆ pq pˆ ± z / 2 n α /2
( )
ˆ pˆ -1 wher here q=
Must use z-table Minimum sample size for mean
Minimum sample size for proportion