Download Central Limit Theorem & Confidence Intervals for the Mean

Inferential Statistics
• Descriptive statistics (mainly for samples)
• Our objective is to make a statement with reference to
a parameter describing a population
• Inferential statistics does this using a two-part process:
• (1) Estimation (of a population parameter)
• (2) Hypothesis testing
Inferential Statistics
• Estimation (of a population parameter) - The estimation
part of the process calculates an estimate of the parameter
from our sample (called a statistic), as a kind of “guess”
as to what the population parameter value actually is
• Hypothesis testing - This takes the notion of estimation a
little further; it tests to see if a sampled statistic is really
different from a population parameter to a significant
extent, which we can express in terms of the probability
of getting that result
Estimation
• Another term for a statistic is a point estimate, which is
simply an estimate of a population parameter
• The formula you use to compute a statistic is an
estimator, e.g.
i=n
Point
Estimate
S xi
x=
i=1
n
Estimator
• In this case, the sample mean is being used to estimate
m, the population mean
Estimation
• It is quite unlikely that our statistic will be exactly the
same as the population parameter (because we know
that sampling error does occur), but ideally it should be
pretty close to ‘right’, perhaps within some specified
range of the parameter
• We can define this in terms of our statistic falling within
some interval of values around the parameter value (as
determined by our sampling distribution)
• But how close is close enough?
Estimation and Confidence
• We can ask this question more formally:
• (1) How confident can we be that a statistic falls within
a certain distance of a parameter
• (2) What is the probability that the parameter is within
a certain range that includes our sample statistic
• This range is known as a confidence interval
• This probability is the confidence level
Confidence Interval & Probability
• A confidence interval is expressed in terms of a range
of values and a probability (e.g. my lectures are
between 60 and 70 minutes long 95% of the time)
• For this example, the confidence level that I used is the
95% level, which is the most commonly used
confidence level
• Other commonly selected confidence levels are 90%
and 99%, and the choice of which confidence level to
use when constructing an interval often depends on the
application
Central Limit Theorem
• We have now discussed both the notions of probability
and the way that the normal distribution
• By combining these two concepts, we can go further
and state some expectations about how the statistics
that we derive from a sample might relate to the
parameters that describe the population from which
the sample is drawn
• The approach that we use to construct confidence
intervals relies upon the central limit theorem
The Central Limit Theorem
• Suppose we draw a random sample of size n (x1,
x2, x3, … xn – 1, xn) from a population random
variable that is distributed with mean µ and standard
deviation σ
• Do this repeatedly, drawing many samples from the
population, and then calculate the
• We will treat the
x
x
of each sample
values as another distribution,
which we will call the sampling distribution of the
mean (
X
)
The Central Limit Theorem
• Given a distribution with a mean μ and variance σ2, the
sampling distribution of the mean approaches a
normal distribution with a mean (μ) and a variance
σ2/n as n, the sample size, increases
• The amazing and counter- intuitive thing about the
central limit theorem is that no matter what the shape
of the original (parent) distribution, the sampling
distribution of the mean approaches a normal
distribution
Central Limit Theorem
• A normal distribution is approached very quickly as n
increases
• Note that n is the sample size for each mean and not the
number of samples
• Remember in a sampling distribution of the mean the
number of samples is assumed to be infinite
• Foundation for many statistical procedures because the
distribution of the phenomenon under study does not
have to be normal because its average will be
Central Limit Theorem
• Three different components of the central limit
theorem
• (1) successive sampling from a population
• (2) increasing sample size
• (3) population distribution
• Keep in mind that this theorem applies only to the
mean and not other statistics
Central Limit Theorem – Example
• On the right are shown the resulting
frequency distributions each based on
500 means. For n = 4, 4 scores were
sampled from a uniform distribution
500 times and the mean computed
each time. The same method was
followed with means of 7 scores for n
= 7 and 10 scores for n = 10.
• When n increases:
1. The distributions becomes more and more normal
2. The spread of the distributions decreases
Source: http://davidmlane.com/hyperstat/A14461.html
Central Limit Theorem – Example
The distribution of an average tends to be normal,
even when the distribution from which the average is
computed is decidedly non-normal.
Source: http://www.statisticalengineering.com/central_limit_theorem.htm
Central Limit Theorem &
Confidence Intervals for the Mean
• The central limit theorem states that given a
distribution with a mean μ and variance σ2, the
sampling distribution of the mean approaches a
normal distribution with a mean (μ) and a variance
σ2/n as n, the sample size, increases
• Since we know something about the distribution of
sample means, we can make statements about how
confident we are that the true mean is within a given
interval about our sample mean
Standard Error
• The standard deviation of the sampling distribution
of the mean (X) is formulated as:
sX =
s
n
• This is the standard error (the unit of measurement
of a confidence interval, used to express the
closeness of a statistic to a parameter
• When we construct a confidence interval we are
finding how many standard errors away from the
mean we have to go to find the area under the
curve equal to the confidence level
99.7%
95%
68%
f(x)
-3σ
-2σ
P(Z>=2.0) = 0.0228
P(Z>=1.96) = 0.025
-1σ
μ
+1σ
+2σ
+3σ
P(-2<=Z<=+2) = 1 – 2*0.0228 = 0.9544
P(-1.96<=Z<=+1.96) = 1 – 2*0.025 = 0.95
Confidence Intervals for the Mean
• The sampling distribution of the mean roughly
follows a normal distribution
• 95% of the time, an individual sample mean should
lie within 2 (actually 1.96) standard deviations of the
mean
pr 1.96s   x    1.96s   0.95
Confidence Intervals for the Mean
s 
2
s
2
N
s
s
N
pr 1.96s   x    1.96s   0.95

s 
s 

pr    1.96
  x     1.96
  0.95
N
N 


Confidence Intervals for the Mean
• An individual sample mean should, 95% of the
time, lie within  1.96(s / n ) of the true mean, μ

s 
s 

pr    1.96
  x     1.96
  0.95
n
n 


• Rearrange the expression:

s 
s 

pr  x  1.96
     x  1.96
  0.95
n
n 


• This tells us that 95% of the time the true mean
should lie within  1.96(s / n ) of the sample mean
Confidence Intervals for the Mean
• More generally, a (1- α)*100% confidence interval
around the sample mean is:
margin of
Standard
error
error

s 
s 

pr  x  z
     x  z
  1  
n
n 


• Where zα is the value taken from the z-table that
is associated with a fraction α of the weight in the
tails (and therefore α/2 is the area in each tail)
Example
• Income Example: Suppose we take a sample of
75 students from UNC and record their family
incomes. Suppose the incomes (in thousands of
dollars) are:
28 29
35
42 ··· 158 167
235
x  89.96, s  51.68

s 
s 

pr  x  1.96
     x  1.96
  0.95
n
n 


Source: http://www.stat.wmich.edu/s160/book/node46.html
Example

s 
s 

pr  89.96  1.96
     89.96  1.96
  0.95
75 
75 


• We don't know σ so we can't use the interval! We
will replace σ by the sample standard deviation s

51.68 
51.68 

pr  89.96  1.96
     89.96  1.96
  0.95
75 
75 


Example

51.68 
51.68 

pr  89.96  1.96
     89.96  1.96
  0.95
75 
75 


(89.96 – 1.96*5.97, 89.96 + 1.96*5.97)
(78.26, 101.66)
pr78.26    101.66  0.95
Constructing a Confidence Interval
• 1. Select our desired confidence level (1-α)*100%
• 2. Calculate α and α/2
• 3. Look up the corresponding z-score in a
standard normal table
• 4. Multiply the z-score by the standard error to
find the margin of error
• 5. Find the interval by adding and subtracting this
product from the mean
Constructing a Confidence
Interval - Steps
1. Select our desired level of confidence
• Let’s suppose we want to construct an interval
using the 95% confidence level
2. Calculate α and α/2
•
(1-α)*100% = 95%  α = 0.05, α/2 = 0.025
3. Look up the corresponding z-score
•
α/2 = 0.025  a z-score of 1.96
Constructing a Confidence
Interval - Steps
4. Multiply the z-score by the standard error to find
the margin of error
Z / 2 
s
n
 1.96 
s
n
5. Find the interval by adding and subtracting this
product from the mean
( x  Z / 2  std.error , x  Z / 2  std.error )
( x  1.96 
s
n
, x  1.96 
s
n
)
Common Confidence Levels and 
values
• For your convenience, here is a table of commonly
used confidence levels, α and α/2 values, and
corresponding z-scores:
α
α/2
Zα/2
90%
0.1
0.05
1.645
95%
0.05
0.025
1.96
99%
0.01
0.005
2.58
(1 - a)*100%
Constructing a Confidence
Interval - Example
• Suppose we conduct a poll to try and get a sense of the
outcome of an upcoming election with two candidates.
We poll 1000 people, and 550 of them respond that they
will vote for candidate A
• How confident can we be that a given person will cast
their vote for candidate A?
1. Select our desired levels of confidence
• We’re going to use the 90%, 95%, and 99% levels
Constructing a Confidence
Interval - Example
2. Calculate α and α/2
• Our  values are 0.1, 0.05, and 0.01 respectively
• Our /2 values are 0.05, 0.025, and 0.005
3. Look up the corresponding z-scores
• Our Z/2 values are 1.645, 1.96, and 2.58
5. Multiply the z-score by the standard error to find
the margin of error
• First we need to calculate the standard error
Constructing a Confidence
Interval - Example
5. Find the interval by adding and subtracting this
product from the mean
• In this case, we are working with a distribution we have
not previously discussed, a normal binomial
distribution (i.e. a vote can choose Candidate A or B, a
binomial function)
• We have a probability estimator from our sample, where
the probability of an individual in our sample voting for
candidate A was found to be 550/1000 or 0.55
• We can use this information in a formula to estimate the
standard error for such a distribution:
Constructing a Confidence
Interval - Example
5. Multiply the z-score by the standard error cont.
• For a normal binominal distribution, the standard
error can be estimated using:
sX =
s
n
=
(p)(1-p)
n
=
(0.55)(0.45)
1000
= 0.0157
• We can now multiply this value by the z-scores to
calculate the margins of error for each conf. level
Constructing a Confidence
Interval - Example
5. Multiply the z-score by the standard error cont.
• We calculate the margin of error and add and subtract
that value from the mean (0.55 in this case) to find the
bounds of our confidence intervals at each level of
confidence:
CI
Z/2
Margin
of error
Bounds
Lower
Upper
90%
1.645
0.026
0.524
0.576
95%
1.96
0.031
0.519
0.581
99%
2.58
0.041
0.509
0.591
t-distribution
• The central limit theorem applies when the sample size
is “large”, only then will the distribution of means
possess a normal distribution
• When the sample size is not “large”, the frequency
distribution of the sample means has what is known as
the t-distribution
• t-distribution is symmetric, like the normal distribution,
but has a slightly different shape
• The t distribution has relatively more scores in its tails
than does the normal distribution. It is therefore
leptokurtic
t-distribution
• The t-distribution or Student's t-distribution is a
probability distribution that arises in the problem of
estimating the mean of a normally distributed population
when the sample size is small
• It is the basis of the popular Student's t-tests for the
statistical significance of the difference between two
sample means, and for confidence intervals for the
difference between two population means
t-distribution
• The derivation of the t-distribution was first published
in 1908 by William Sealy Gosset. He was not allowed to
publish under his own name, so the paper was written
under the pseudonym Student
• The t-test and the associated theory became well-known
through the work of R.A. Fisher, who called the
distribution "Student's distribution"
• Student's distribution arises when (as in nearly all
practical statistical work) the population standard
deviation is unknown and has to be estimated from the
data
Confidence intervals & t-distribution
• The areas under the t-distribution are given in Table
A.3 in Appendix A
• e.g., with a sample size of n = 30, 95% confidence
intervals are constructed using t = 2.045, instead of the
value of z = 1.96 used above for the normal distribution
• For the commuting data (textbook):

14.43 
14.43 

pr  21.93  2.045
     21.93  2.045
  0.95
30 
30 


Confidence intervals & t-distribution

14.43 
14.43 

pr  21.93  2.045
     21.93  2.045
  0.95
30 
30 


• We are 95% sure that the true mean is within the
interval (16.54, 27.32)
• More precisely, 95% of confidence intervals
constructed from samples in this way will contain the
true mean