Download Central Limit Theorem

Central Limit Theorem
General version
Statistics are
• Unbiased: on average, the calculated statistic
from a random sample equals in value to the
corresponding population parameter.
• Consistent: (ALL statistics contain error in
estimation—however,) the error from a random
sample in a statistic follows a normal distribution.
• Efficient: statistics from random samples have
the smallest possible error.
When? (not always)
• When the sample of data drawn from the
population is a randomly gathered data
sample.
Assume that
a sample is
collected at
random,
Sample mean
Unbiased
On average, a sample mean estimates
the actual value of a population mean
Consistent
The error in the sample mean estimate
follows a normal distribution
Efficient
The standard error of a sample mean
is equal to the population s, divided by
the square root of the sample size
x    x
sx 
s
n
Assume that
a sample is
collected at
random,
Sample proportions
(%)
Unbiased
On average, a sample proportions
estimates the actual value of a
population proportion
Consistent
The error in the sample proportion
estimate follows a normal distribution
Efficient
The standard error of a sample
proportion is equal to the product of the
sample proportion, times the
complement proportion, divided by the
sample size, all of it square rooted.
sp 
s
p(1  p)
n
Example: CLT for sample means
• A student group has taken over 10,000
exams during their lifetime. Assume that
you know that the population mean test
score for this group is 81 (), with a
population standard deviation of 6 (s).
• If you take a random sample of 144 exams
from the population of scores, then,
describe the sample mean of these 144
sampled scores by applying the CLT.
What should the sample mean look like?
What is the average amount of error in calculating a sample mean, s  s
x
In order to to estimate a population mean? 6/12 = 0.5 pt., n=144
n
What if n=64, then the
std. error is 6/8 = .75
pt …
The larger the
sample, the smaller
the statistical error
Negative errors, underestimates
79.5
80
80.5
Positive errors, overestimates
 = 81
81.5
82
82.5
Z
-3
-2
-1
0
1
2
3
Another example: sample
proportions (polling/surveys)
• Suppose you taste two colas and your job
is to properly identify each brand correctly.
Assume that you are equally likely to
identify the drinks correctly as you are to
identify the drinks incorrectly.
• Suppose you gather 400 randomly
selected answers to this taste test.
☻
☻’
P=.5
1-P=.5
What should the sample proportion look like?
What is the average amount of error in calculating a sample proportion,
When estimating a population proportion?
Std error for a sample proportion=(.5*.5/400)^.5=.025
Negative errors, underestimates
Positive errors, overestimates
p
.425
.45
.475
p = .5
.525
.55
.575
Z
-3
-2
-1
0
1
2
3
Sample exam 3 question
• A random sample of 256 persons was asked an opinion
question. The surveyed had two choices for a response:
Agree or Not Agree. The sample percentage of those
who Agreed was .25. construct a range that covers 95%
of possible population proportion values for the
percentage of all people who agree with this issue:
.25 ± 2*(.25*.75 / 256)^.5
•
(p
± 2*(p * (1-p) /n )^.5)
• .25 ± 2*0.027 = [25% ± 5.4%] = 95% confidence interval
estimate for the population proportion.