Transcript 2/24 - MegCherry.com

SP 225
Lecture 10
The Central Limit Theorem
Differences Between Statistics
and Parameters
Sample: 3 Randomly Selected
People
Population: All People
Parameter: 5 of 15 or 33% wear glasses
Statistic: 0 of 3 or 0% wear
glasses
Methods to for Better Samples
 Random sampling makes samples
representative
 Book term: probability sample
 EPSEM (Equal probability of selectino
method)
EPSEM Technique
 Begin with a list of all population
members
 Generate random numbers to identify
members of the list to be selected in the
sample
 Do everything possible to get selected
members to participate in the survey
Sampling Distributions
 The shape, measure of center and
measure of variation associated with
many sample statistics
 Unique and different from the population
distribution
Three Separate Distributions
 Sample distribution


Empirical and known
Intended to help learn about the population
 Population distribution


Empirical and unknown
Properties estimated using statistics
 Sampling distribution


Theoretical or non-empirical
Properties well-known and based on probabilities
Population Distribution
Number of Siblings
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
Population Distribution
Frequency
Population Distribution
Roll 1
Roll 2
Roll 3
Roll 4
Dice Face
Roll 5
Roll 6
Sample Distribution
600 Rolls of the Die
120
Frequency
100
80
60
40
20
0
Roll 1
Roll 2
Roll 3
Roll 4
Roll 5
Roll 6
Sampling Distribution of Mean
Histogram of C1
9
8
Frequency
7
6
5
4
3
2
1
0
3.35
3.40
3.45
3.50
C1
3.55
3.60
3.65
Central Limit Theorem
 For normally distributed populations
 If repeated samples of size N are drawn
from a normal population with mean µ
and standard deviation σ, then the
sampling distribution σ of sample means
will be normal with a mean of µ and a
standard deviation of σ/√N
Central Limit Theorem
 For any population
 If repeated samples of size N are drawn
from a normal population with mean µ
and standard deviation , then, as N
becomes large, the sampling distribution
σ of sample means will be normal with a
mean of µ and a standard deviation of
σ/√N
The Probability Distribution
 We can calculate probabilities for sample
means using the sampling distribution for
sample means
 Similar to calculating probabilities for an
individual using a population distribution
 Use standard deviation of the sampling
distribution instead of sampling
distribution of the population
Empirical Rule for Data with a
Bell-Shaped Distribution (3)
Example Problem (1)
 The average GPA at a particular school
is m=2.89 with a standard deviation
s=0.63. A random sample of 25 students
is collected. Find the probability that the
average GPA for this sample is greater
than 3.0.
Example Problem (2)
The time it takes students in a cooking school to
learn to prepare seafood gumbo is a random
variable with a normal distribution where the
average is 3.2 hours with a standard deviation
of 1.8 hours.
 Find the probability that the average time it
will take a class of 36 students to learn to
prepare seafood gumbo is less than 3.4
hours.
 Find the probability that it takes one student
between 3 and 4 hours to learn to prepare
seafood gumbo.
Confidence Interval
 Mathematical statement that says that
the parameter lies within a certain range
of values
 The average employee of XYZ
Automotive has been employed between
8 and 12 years
 95% confident that the mean length of
employment at XYZ automotive is
between 8 and 12 years
Probability Distribution for
Population Mean
 95% confident that the mean length of
employment at XYZ automotive is
between 8 and 12 years
(sample)
Confidence Level
 Percent of confidence intervals that contain the
population mean over the long run
 Probability this confidence interval contains the
population mean
 95% confident that the mean length of
employment at XYZ automotive is between 8
and 12 years
 99% confident that the mean length of
employment at XYZ automotive is between 8
and 12 years
Confidence Interval Formula
c.i.  X  Z (

N
)
Confidence Interval Estimate
s
c.i.  X  Z (
)
N 1
Confidence Interval Example
 A random sample of 100 television
programs contained an average of 2.37
acts of physical violence per program.
The standard deviation of the number of
acts of violence on television is 3. At the
99% level, what is your estimate of the
mean number of acts of violence for all
television programs?
Alpha
 Percent of confidence intervals that DO
NOT contain the population mean over
the long run
 Probability this confidence interval
DOES NOT contain the population mean
 Complement of confidence level
Efficiency
 Extent to which the confidence interval
clusters around the mean
 Width of the confidence interval
 Determined by population standard
deviation and sample size