M31- Dist of X-bars 1 Department of ISM, University of Alabama, 1992-2003 Sampling Distributions...
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Transcript of M31- Dist of X-bars 1 Department of ISM, University of Alabama, 1992-2003 Sampling Distributions...
M31- Dist of X-bars 1 Department of ISM, University of Alabama, 1992-2003
Sampling Distributions
Our Goal? Our Goal? To make a decisions.To make a decisions.
What are they?
Why do we care?
M31- Dist of X-bars 2 Department of ISM, University of Alabama, 1992-2003
Lesson Objectives
Know what is meant by the “sampling distribution” of a statistic, and the “population of all possible X-bar values.”
Know when the population of allpossible X-bar values IS normal.
Know when the population of allpossible X-bar values IS NOT normal.
M31- Dist of X-bars 3 Department of ISM, University of Alabama, 1992-2003
Descriptive
NumericalGraphical
Statistics
M31- Dist of X-bars 4 Department of ISM, University of Alabama, 1992-2003
Statistical Inference
Generalizing from a sample to a population,
by using a statisticto estimate
a parameter.
Goal: Goal: . .
ParameterParameterStatisticStatistic
Mean:
Standard deviation:
Proportion:
s
X
estimates
estimates
estimatesp
M31- Dist of X-bars 7 Department of ISM, University of Alabama, 1992-2003
A A statisticstatistic is a is a . .
Before we can make decisionsBefore we can make decisionsabout parameters about parameters and control and control the degree of riskthe degree of risk, we must know:, we must know:
the the of the statistic of the statistic and its and its values.values.
Objective of this section:
Original Population: 300 ST 260 students.X = Exam 2 score
= population mean (unknown) = population std deviation (unknown)
Calculate:
n = 4 x = mean s = std dev
Example 1:
PopulationSample
M31- Dist of X-bars 9 Department of ISM, University of Alabama, 1992-2003
Think of X as a random variable.
Fact:
Different samples of size “n” will produce different values of the sample mean.
The population mean is fixed as long as the population doesnot change.
M31- Dist of X-bars 10 Department of ISM, University of Alabama, 1992-2003
• Shape? (Skewed? Symmetric?)
• Center? (Mean? Median?)
• Spread? (Std. Deviation? IQR?)
• Is it one of our “special” distributions? (Normal? Exponential?)
For samples of size n, what is the distribution of the statistic X?
M31- Dist of X-bars 11 Department of ISM, University of Alabama, 1992-2003
For samples of size n, what is the distribution of p,
i.e, a sample proportion?
• Shape? (Skewed? Symmetric?)
• Center? (Mean? Median?)
• Spread? (Std. Deviation? IQR?)
• Possible values? (0/n, 1/n, 2/n, …, n/n)
• Is it one of our “special” distributions? (normal, exponential, binomial, Poisson)
^
M31- Dist of X-bars 12 Department of ISM, University of Alabama, 1992-2003
From pop. of all ST 260 students,randomly select n = 1 student.Record exam 2 grade:
Sampled value: 76
X = 76.0
How close can we expect this
estimate to be to the true mean ?How close can we expect this
estimate to be to the true mean ?
Example 1, continued:
M31- Dist of X-bars 13 Department of ISM, University of Alabama, 1992-2003
Sampled values: 64, 78, 94, 46
X = (64 + 78 + 94 + 46) / 4 = 70.5
From pop. of all ST 260 students,randomly select n = 4 students.Record exam 2 grades:
How close can we expect this
estimate to be to the true mean ?How close can we expect this
estimate to be to the true mean ?
Example 1, continued:
M31- Dist of X-bars 14 Department of ISM, University of Alabama, 1992-2003
x’s from samples tend to be to the true mean, than x ’s from smaller samples.
Fact:
M31- Dist of X-bars 15 Department of ISM, University of Alabama, 1992-2003
Sampling Distribution of X
is the distribution of all possible
sample means calculated from
all possible samples of size n.
Also calledAlso called “the population “the population
of all possible x-bars”. of all possible x-bars”.
Also calledAlso called “the population “the population
of all possible x-bars”. of all possible x-bars”.
M31- Dist of X-bars 16 Department of ISM, University of Alabama, 1992-2003
And so on, . . . ,
until we collect every possible sample of size n = 4.
How many samples of size 4 are there from a population of 300 members?
M31- Dist of X-bars 17 Department of ISM, University of Alabama, 1992-2003
Sampling Distribution of x for n = 4
Based on all samples of size n = 4
x
x
x-axis
And the shapelooks like aNormal dist.
M31- Dist of X-bars 18 Department of ISM, University of Alabama, 1992-2003
x
x
Definitions, from previous slide:
The subscripts identify
the population to which
the parameter refers.The subscripts identify
the population to which
the parameter refers.
= average of all possible X’s
(center of the sampling dist.)
= std. deviation of all pos. X’s
(spread of the sampling dist.)
M31- Dist of X-bars 19 Department of ISM, University of Alabama, 1992-2003
Compare parameters of theoriginal population of all scores and the parameters of the sampling dist. of all possible x’s
mean = & std. dev. = Original population:
x
x
(same mean as individual values)
n
(different std. dev., but related!)
M31- Dist of X-bars 20 Department of ISM, University of Alabama, 1992-2003
If = 75 and = 10,
Original Population: 300 ST 260 students.X = Exam 2 score.
then the population of all possible X-values for n = 4 will have
x
x n
M31- Dist of X-bars 21 Department of ISM, University of Alabama, 1992-2003
Questions
What is the probability that What is the probability that oneone randomly randomly selected Exam2 score will be selected Exam2 score will be within within 10 points10 points of the population mean, 75? of the population mean, 75?
X: 65 to 85X: 65 to 85
Z:Z:
= ,= ,
What is the probability that a What is the probability that a sample mean sample mean of n = 4of n = 4 randomly selected Exam 2 scores randomly selected Exam 2 scores will be will be within 10 pointswithin 10 points of the pop. mean? of the pop. mean?
= ,= ,XX
X: 65 to 85 X: 65 to 85
Z:Z:
M31- Dist of X-bars 23 Department of ISM, University of Alabama, 1992-2003
We now know the parametersof the population of all possible x-bar values.
What is the distribution?Look back at the plot exam 2 grades.
M31- Dist of X-bars 24 Department of ISM, University of Alabama, 1992-2003
If If XX ~ N ( ~ N ( ),),
then for samples of size n,then for samples of size n,
XX ~ ~ NN ( ( ,, ). ).
If original population has aNormal dist., then the distributionof X values is Normal also.
n
M31- Dist of X-bars 25 Department of ISM, University of Alabama, 1992-2003
1009080706050403020100X
Original PopulationOriginal Population: Normal (: Normal ( = 50, = 50, = 18) = 18)
= 18.00
M31- Dist of X-bars 26 Department of ISM, University of Alabama, 1992-2003
1009080706050403020100X
Original PopulationOriginal Population: Normal (: Normal ( = 50, = 50, = 18) = 18)
n = 36
n = 16
n = 4
n = 2
= 9.00x = 12.73x
= 4.50x
= 3.00x
= 18.00
M31- Dist of X-bars 27 Department of ISM, University of Alabama, 1992-2003
Bottle filling machine for soft drink.
Bottles should contain 20.00 ounces;assume actual contents follow a normal distribution with a mean of 20.18 oz. and a standard deviation of 0.12 oz.
X = contents of one randomly selected bottle
X ~ N( = 20.18, = 0.12)
Example 2:
This is the
original population.
M31- Dist of X-bars 28 Department of ISM, University of Alabama, 1992-2003
P( X < 20.00)
=
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
0
= P( Z < )
a. Find the proportion of individual bottles contain less than 20.00 oz?
20.1820.0
Z =
X = content of one bottle.X ~ N( = 20.18, = )
of the bottles will contain less than 20.00 ounces. Is this a problem?
of the bottles will contain less than 20.00 ounces. Is this a problem?
=
Z-axisX-axis
=
=
M31- Dist of X-bars 29 Department of ISM, University of Alabama, 1992-2003
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
0
= P( Z < )
a. Find the proportion of six-packs whose mean content is less than 20.00 oz?
20.1820.0
Z =
Only ________% of the six-packswill contain an average less than 20.00 ounces.
Only ________% of the six-packswill contain an average less than 20.00 ounces.
=
Z-axisX-axis
=
=
Is population of x-bars Normal?Yes; because original pop. is Normal.
X = mean of six-pack.X ~ N( = 20.18, = )X x
x = )
P( X < 20.00) =
Got this
from Excel
M31- Dist of X-bars 31 Department of ISM, University of Alabama, 1992-2003
New situation
Such as an . . .
Exponential Distribution?Such as an . . .
Exponential Distribution?
But what if the original population
is not normally distributed?
M31- Dist of X-bars 32 Department of ISM, University of Alabama, 1992-2003
Demonstration Demonstration of theof the
Central Limit TheoremCentral Limit Theorem
Page 289
1612840
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Original Population: Exponential (Original Population: Exponential ( = 4) = 4)
n = 1
= 4
4
1612840
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Original Population: Exponential (Original Population: Exponential ( = 4) = 4)
n = 30
n = 15
n = 5n = 2
= 4
Sampling distribution for X
= 1.789x = 2.828x
= 1.033x
= 0.730x
4
M31- Dist of X-bars 35 Department of ISM, University of Alabama, 1992-2003
If If XX ~ NOT Normal, ~ NOT Normal, then then for for largelarge samples of size n, samples of size n,
XX ~ ~ NN ( ( ,, ), ), approximately.approximately.
If original population doesNOT have a Normal dist.,
n
Central Limit Theorem
Page 289Central Limit Theorem
Page 289
the X values are approximately Normal IF n is large.
M31- Dist of X-bars 36 Department of ISM, University of Alabama, 1992-2003
How big is BIG?
Bigger is better, but
is enough!
This same phenomena will happenfor ANY non-normal distribution,IF “n” is BIG!
M31- Dist of X-bars 37 Department of ISM, University of Alabama, 1992-2003
“Investment opportunity”
Earnings: x -80 0 +60 P(X=x) .40 .10 .50
P(player looses) = .40
Expected value:
= -80 (.40) + 0 (.10) + 60 (.50)
Also,Also, = = 66.066.0= = -2.00.
Example 3 (C.L.T.)
This is
definitely
NOT normal!
M31- Dist of X-bars 38 Department of ISM, University of Alabama, 1992-2003
P( X < 0.0) = ?
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
0 Z-axis
After 36 plays, what is the probability that the average earnings is negative?
-2.0
X = earning for one playX ~ NOT Normal
= x
X ~ N( = -2.0, = )x
X = Avg. earnings, 36 plays X-axis
X-bar pop. is Normal because n is BIG.
M31- Dist of X-bars 39 Department of ISM, University of Alabama, 1992-2003
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
Z-axis
After 6400 plays, what
is the probability that the average earnings is negative?
-2.0
Same as previous, BUT . . . .
= x
X ~ N( = -2.0, = )x
X-axis
P( X < 0.0) = ?
M31- Dist of X-bars 40 Department of ISM, University of Alabama, 1992-2003
Summary different values of”n”
Number of plays P( you lose)
Expected Total Amount ofearnings
1 .4000 -2
36 .5714 -72100 .6179 -200
6400 .9922 -12,800
12,000 .9995 -24,000
M31- Dist of X-bars 41 Department of ISM, University of Alabama, 1992-2003
The house never
The house never
loses!loses!The house never
The house never
loses!loses!
M31- Dist of X-bars 42 Department of ISM, University of Alabama, 1992-2003
X = number of accidents in one week
On-the-job accidents in a company.
X ~ Poisson ( = 2.2 acc/wk )
Example 4 (C.L.T.)
a. Find the probability of having two or fewer accidents in one randomly selected week.
P(X < 2) = , from Table A.4. This is a Chapter 6 problem.
The probability of being two or less is greater than .5, but the mean is 2.2! How is this possible?
The probability of being two or less is greater than .5, but the mean is 2.2! How is this possible?
M31- Dist of X-bars 43 Department of ISM, University of Alabama, 1992-2003
Example 4 continued
X = number of accidents in one weekOn-the-job accidents in a company.
X ~ Poisson ( = 2.2 acc/wk )Poisson, mean = 2.2
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Number of Accidents
M31- Dist of X-bars 44 Department of ISM, University of Alabama, 1992-2003
Example 4 continued
Ori. pop. is definitely NOT normal; BUT n is large!
b. What is the probability that the average number of accidents for next 52 weeks will be 2.0 or less?
X = mean for 52 weeks; n = 52.What is the sampling distribution?
X = number of accidents in one week
On-the-job accidents in a company.
X ~ Poisson ( = 2.2 acc/wk )
M31- Dist of X-bars 45 Department of ISM, University of Alabama, 1992-2003
What is the sampling distribution of X ?
XXX ~ N ( = , = )2.2 1.483/ 52
By the C.L.T., it is approximately Normal.
Recall: for Poisson the mean is , the standard deviation is the square root of .
= 0.2057
Example 4 continued
X = number of accidents in one week
On-the-job accidents in a company.
X ~ Poisson ( = 2.2 acc/wk )
M31- Dist of X-bars 46 Department of ISM, University of Alabama, 1992-2003
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
0
b. What is the probability that the average number of accidents for next 52 weeks
will be 2.0 or less? 2.2
It is much less likely that the average number of accidents per week will be two or less, than any one specific week.
It is much less likely that the average number of accidents per week will be two or less, than any one specific week.
Z-axisX- axis
X = mean of accidents.X ~ N( = 2.2, = _______)x x
P( X < 2.0) =
Example 4 cont.
M31- Dist of X-bars 47 Department of ISM, University of Alabama, 1992-2003
Anytime the original pop. is Normal (true for any n).
Anytime the original pop. is not Normal, but n is BIG (n > 30).
Reminder
When is the population of all possible X values Normal?