Normal-distribution,-Mean-&-Standard-deviation

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					Normal distribution, Mean & Standard deviation Mean most commonly used measure of central tendency influenced by every value in a sample


X
N

µ is population mean X is sample mean Standard deviation measure of variability


(X  )
N

2

if µ is unknown, use X correct for smaller SS bias by dividing by n-1


(X  X )
n 1

2

Normal distribution

bell shaped curve reasonably accurate description of many distributions properties : unimodal symmetrical points of inflection at µ ± σ tails approach x-axis completely defined by mean and SD

Sampling & population inference population sample entire collection of units of interest collection of observations from a well defined population random sample each unit in a population has an equal chance of being sampled and each unit is independent of each other population inference to form a conclusion about a population from a sample distribution of random samples of mean tends towards a normal distribution, even if parent population isn’t normal

central limit theorem

normal distribution is model for distribution of sample stats approximation to normal distribution improves as n increases

standard error of mean

standard deviation of sampling error of different samples of a given sample size how great is sampling error of ( X - µ) as n increases, variability decreases
X  
n

z, t & F distributions hypothesis testing: often want to know the likelihood that a given sample has come from a population with known characteristic(s) 1. define H0 2. test likelihood of H0
z X 

X

normal distribution with mean 0, standard deviation 1 (cf central limit theorem) e.g.
X = 104.0

H0 : µ = 100
X= 3

z = (104 – 100) / 3 = 1.33 α = 0.05 therefore retain H0

t

X  sX

for a given mean and sd, normal distribution is completely defined there are a family of t curves, depending on degrees of freedom n – 1 degrees of freedom associated with deviations from a single mean with infinite degrees of freedom, t = z H0 : µ = 100
X = 120

n = 25 sx = 35.5
sX  sx n  35 .5 25  7.1

t

X   120  100   2.82 sX 7.1

df = 24 α = 0.05 tcrit = 2.06 therefore reject H0 t values may be converted to z values via p values

Analysis of variance SStotal = SSwithin + SSbetween dfwithin = ntotal – k dfbetween = k – 1 Within-groups variance estimate:
2 sw 

SS w df w

(‘mean square within’) estimates inherent variance

Between-groups variance estimate:
2 sbet 

SS bet df bet

(‘mean square between’) estimates inherent variance + treatment effect

F

2 s bet 2 sw


				
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posted:11/29/2009
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Description: Normal-distribution,-Mean-&-Standard-deviation