Normal distribution, Mean & Standard deviation
Normal distribution, Mean & Standard deviation
Mean most commonly used measure of central tendency
influenced by every value in a sample
[pic]
µ is population mean
[pic] is sample mean
Standard deviation measure of variability
[pic]
if µ is unknown, use [pic]
correct for smaller SS bias by dividing by n-1
[pic]
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 entire collection of units of interest
sample 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
central limit theorem distribution of random samples of mean tends towards a normal distribution, even if parent population isn’t normal
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 ([pic]- µ)
as n increases, variability decreases
[pic]
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
[pic]
normal distribution with mean 0, standard deviation 1 (cf central limit theorem)
e.g.
[pic]= 104.0
H0 : µ = 100
[pic]= 3
z = (104 – 100) / 3 = 1.33
α = 0.05
therefore retain H0
[pic]
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
[pic]= 120
n = 25
sx = 35.5
[pic]
[pic]
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:
[pic]
(‘mean square within’)
estimates inherent variance
Between-groups variance estimate:
[pic]
(‘mean square between’)
estimates inherent variance + treatment effect
[pic]
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