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# IOE 265 Study Guide - Final Guide: Standard Deviation, Covariance, Central Limit Theorem

Department
Industrial And Operations Engineering
Course Code
IOE 265
Professor
Xiuli Chao
Study Guide
Final

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Confidence Intervals: Identify a plausible
range for point estimate
Ex. 95% confidence range implies 95% chance
that true value is in range
Find multiplier with invNorm(,0,1)
Use sample std. dev of
Note: Sum of independent normal random
variables leads to   and   
 
  
 
For probability  we have:
    
For one-sided CI (u):   
For all mean CI:   
Note: round-up, width is ½ of interval
Must know variance + normal dist.
Central Limit Theorem: When n is large
enough (, distribution of sample mean is
approx. normal (  )
Proportion Estimation: Bernoulli:
  
CI:     

Note: For n, use p and q as .5 to be safe
Chi-Squared Distribution: If population has
normal distribution then:
has normal dist. Mean Variance
Sample Variance 

is Chi-
Square Distributed with n-1 df
If Z is standard normal, and
is Chi-Square
dist. with n degrees freedom:
 is t-dist with n df
T-Distribution: Area under curve to the right
of t critical value is
 

Norm w/out mean and variance, small sample
CI:  

CI for variance


 


Hypothesis Testing: Make a claim
Reject or fail to reject that claim
Null hypothesis: 
Alternate hypothesis: 
1: Fix level of significance, acceptable prob of
rejecting null when it’s true
2: Compute prob. of observing sample, take z-
value and turn into p-value
3: If p-value is smaller than sig. level, reject
null hypo, otherwise fail to reject
You need:
Null and alternative hypothesis
Test statistic what decision is on
Rejection region (when to reject)
Significance level : probability, given
hypothesis is correct, of rejecting it
Type II error is Beta, fail to reject + H0 false
Reject the null if p-value < alpha (smaller p-
value is stronger evidence against H0)
Mu0 is actual and mu’ is test


Note: smallest level of signiﬁcance at which
you would be willing to reject the null
hypothesis is the P-value.
Definitions: Probability measure of
likelihood for something to occur. Statistics
concerned with learning from data. Types:
categorial (qualitative) numerical (quantitative)
Univariate: observations on 1 variable
Bivariate: observations on 2 variables
Discrete: Finite possible values
Continuous: any real value in interval
Dot plot: each datapoint is dot on a line
Stem&Leaf: First part of each piece of data on
left hand (stem) side. Remaining digits on
right-hand (leaf) side in order. Add a key
Frequency Analysis: # of frequencies of each
class/value in data set (relative, cumulative)
# of ranges is , 5-20
Histogram is graphical rep. for 1 variable
(exponential, normal, bi-modal, skewed)
Graph of rel. freq w/ width 1 is samp density
Density: 

 
Median: middle obs., further from skew
Trimmed mean: Eliminate % of values from
both ends, re-compute mean
Fourth spread (IQR): Upper 4th Lower 4th
Outlier is 1.5*IQR from closest 4th, Extreme
Outlier is 3*IQR away (open circle)
Mode: most common, Range: max-min

 sample uses N-1


Part 2 Mod 1: Sample space: set of all
possible outcomes. Events: any subset of
sample space. Probability: likelihood of event
A\B is 

N1 possible outcomes then N2 = N1*N2
Permutation (w/order):

Combination (w/out order):

P(A|B)
 Independence:
 

Discrete RV (pmf): 
Cumulative:    

Expected value: 
E(ax+b)=aE(x) +b, V(ax+b) = a2V(x)
V(x =   
Bernoulli: E(x) = p, V(x)=p(1-p)
Binomial: Bin(n,p): n=number of trials,
p=prob of success is

Multiple Bernoulli becomes binomial
E(x) = np, V(x)=np(1-p)
Geometric: Number of independent B trials
until success p(x)=(1-p)x-1p, E(x) = 1/p,
V(x) = (1-p)/p2, P(    
Hypergeometric: Batch has N products, M
good and N-M bad, pick N products, prob of x
good: h(x:n,M,N) =


E(x) = np, V(x)=
 , p=
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