21/11/2018
1
Introduction to
t
‐
distributions;
Estimation; Confidence intervals
Week 6
1
Distributions so far…
•
Binomial distribution
Discrete random variable
•
X
is count of successes out of
n
trials
•
ܺ~ܤ݅݊ሺ݊, ሻ
•
Normal distribution
Continuous random variable
•
X
is (usually) a measurement
•
ܺ~ܰሺߤ, ߪ
ଶ
ሻ
•
ܼ~ܰሺ0,1ሻ
for standard Normal distribution
•
Normal dist is symmetric, very few outliers
New:
•
t
‐
distribution
Continuous random variable
•
ܶ~ݐ
υ
(
υ
= degrees of freedom, pronounced ‘nu’)
•
Symmetric, very few outliers
2
t
‐
distributions
•
t
‐
distributions are used for continuous data (measurements)
e.g.
•
Based on Normal distribution – symmetric, very few outliers
•
Typically used for formal inference about a population with a Normal distribution
when:
sample size is ‘small’ and
σ
is unknown
we estimate
σ
with its point estimate, s
•
t
‐
distribution gets closer to the Normal distribution as sample size increases
•
for large enough sample sizes,
t
‐
distribution and Normal distribution give almost identical
results
3
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2
t
‐
distributions are the same basic ‘bell shape’ as the Normal distribution
The smaller the sample size, the flatter the curve
and the ‘heavier’ the tails
sample sizes
T
݊ ൌ 11
݊ ൌ 5
݊ ൌ 3
݊ ൌ 2
4
sample sizes
T
݊ ൌ 11
݊ ൌ 5
݊ ൌ 3
݊ ൌ 2
•
The diagram shows that with
small values of
n
it is important
to use the
correct
t
‐
distribution
as they differ substantially
•
need the assumption of Normally
distributed data in order to use a
t
‐
distribution
•
Each
t
‐
distribution is
characterised by its
degrees of
freedom
– this is the
sample size
minus one
•
݂݀ ൌ ݊ െ 1
•
Notice that the mean of a
t
‐
distribution is zero
5
Degrees of freedom
T
݊ ൌ 11
݊ ൌ 5
݊ ൌ 3
݊ ൌ 2
T
6
Sample size
ሺ݊ሻ
and degrees of freedom
ሺ݂݀
or
υ
)
21/11/2018
3
Estimation in Statistics
7
Point estimates and interval estimates
Estimates
•
Numerical summaries of sample data (sample statistics) give us
estimates
of population parameters
a)
Point estimates
b) Interval estimates
•
We generally get ‘good’ estimates if
•
sample is randomly chosen
•
‘
n
’ is large
8
Point estimates
A
point estimate
of a population parameter (such as
ߤ
, the population mean)
is a single number that is our best guess of the value of that parameter
This estimate will usually be ‘wrong’, as it will not generally be the true value
of the population median/mean, but
it is a very good starting point.
In fact, it
is the most sensible starting point in our quest to find information about the
whole population
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4
Some population parameters and their point estimates
Population parameter (words)
Population parameter (symbol)
Point estimate of the parameter
Population mean
ߤ
(mu)
ݔ̅
Population standard deviation
ߪ
(sigma)
ݏ
Population proportion
̂
10
Confidence intervals
(Interval estimates)
11
Interval estimates
•
An
interval estimate
of a population parameter is a range of plausible values
that we think (with some specified level of confidence) contains the (unknown)
population parameter
•
