1. Physics- atmospheric Sciences (PAS) - Room 201
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Physics- atmospheric Sciences (PAS) - Room 201

2. MGMT 276: Statistical Inference in Management Fall 2015
Welcome

3. Just for Fun Assignments Go to D2L - Click on “Content”
Click on “Interactive Online Just-for-fun Assignments”
Please note: These are not worth any class points
and are different from the required homeworks

4. Please re-register your clicker

5. Schedule of readings Before our next exam (October 20th)
OpenStax Chapters 1 – 11
Plous (10, 11, 12 & 14)
Chapter 10: The Representativeness Heuristic
Chapter 11: The Availability Heuristic
Chapter 12: Probability and Risk
Chapter 14: The Perception of Randomness

6. Homework
On class website:
Please print and complete homework worksheet #10
Due Thursday October 13th
Dan Gilbert Reading and Law of Large Numbers

7. By the end of lecture today 10/8/15
Use this as your
study guide
By the end of lecture today 10/8/15
Law of Large Numbers
Central Limit Theorem

8. Review of Homework Worksheet just in case of questions

9. Homework review 2 5
= .40
Based on apriori probability – all options equally likely – not based on previous experience or data
Based on expert opinion - don’t have previous data for these two companies merging together
Based on frequency data (Percent of rockets that successfully launched)

10. Homework review
Based on apriori probability – all options equally likely – not based on previous experience or data 30
100
= .30
Based on frequency data (Percent of times at bat that successfully resulted in hits)
Based on frequency data (Percent of times that pages that are “fake”)

11. Homework review 5 50
= .10
Based on frequency data (Percent of students who
successfully chose to be Economics majors)

12. .
.8276
.1056
.2029
.1915
.3944
.4332
.3944
.3944 44 50 55 50 55 52 55 4 =
-1.5 4
+.5 4 = =
+1.25
z of 1.5 = area of .4332
z of .5 = area of .1915
1.25 = area of .3944 4 4 =
+1.25 =
+1.25
= .1056
z of 1.25 = area of .3944
z of 1.25 = area of .3944
= .8276
= .2029

13. Homework review
.3264
.2152
.5143
.1255
.3888
.1736
.1736
.3888
3,000
3,000 3,500
2, ,500
650
650 =
0.45
650 =
0.45 =
-.32
z of 0.45 = area of .1736
z of = area of .1255
z of 0.45 = area of .1736
650
650 =
1.22 =
1.22
= .3264
z of 1.22 = area of .3888
z of 1.22 = area of .3888
= .2152
= .5143

14. Homework review
.0764
.9236
.1185
.4236
.4236
.4236
.3051
10 12 20 20
3.5
3.5 =
1.43
3.5
1.43 =
-1.43 =
z of 1.43 = area of .4236
z of = area of .4236
z of 1.43 = area of .4236
3.5 =
-0.86
= .9236
= .0764
z of -.86 = area of .3051
.4236 – = .1185

15. Comments on Dan Gilbert Reading

16. Law of large numbers: As the number of measurements
increases the data becomes more stable and a better
approximation of the true (theoretical) probability
As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.

17. Law of large numbers: As the number of measurements
increases the data becomes more stable and a better
approximation of the true signal (e.g. mean)
As the number of observations (n) increases or the number of times the experiment is performed, the signal will become more clear (static cancels out)
With only a few people any little error is noticed
(becomes exaggerated when we look at whole group)
With many people any little error is corrected
(becomes minimized when we look at whole group)

18. Sampling distributions of sample means
versus frequency distributions of individual scores
Distribution of raw scores: is an empirical probability distribution
of the values from a sample of raw scores from a population
Eugene X X X X X X
Frequency distributions of individual scores
derived empirically
we are plotting raw data
this is a single sample X
Melvin X X X X X X
Take a single score
Repeat
over and over x x x
Population x x
Preston x x x

19. important note: “fixed n”
Sampling distribution: is a theoretical probability distribution of
the possible values of some sample statistic that would
occur if we were to draw an infinite number of same-sized
samples from a population
important note: “fixed n”
Sampling distributions of sample means
theoretical distribution
we are plotting means of samples
Take sample – get mean
Repeat over and over
Population
Mean for 1st sample

20. important note: “fixed n”
Sampling distribution: is a theoretical probability distribution of
the possible values of some sample statistic that would
occur if we were to draw an infinite number of same-sized
samples from a population
important note: “fixed n”
Sampling distributions of sample means
theoretical distribution
we are plotting means of samples
Take sample – get mean
Repeat over and over
Population
Distribution of means of samples

21. Sampling distribution: is a theoretical probability distribution of
the possible values of some sample statistic that would
occur if we were to draw an infinite number of same-sized
samples from a population
Eugene
Frequency distributions of individual scores
derived empirically
we are plotting raw data
this is a single sample X X
Melvin X X X X X X X X X X X
Sampling distributions sample means
theoretical distribution
we are plotting means of samples
23rd sample
2nd sample

22. Sampling distribution for continuous distributions
Central Limit Theorem: If random samples of a fixed N are drawn
from any population (regardless of the shape of the
population distribution), as N becomes larger, the
distribution of sample means approaches normality, with
the overall mean approaching the theoretical population
mean.
Distribution of Raw Scores
Sampling Distribution of Sample means
Melvin X
Eugene
23rd sample X
2nd sample

23. Sampling distribution: is a theoretical probability distribution of
the possible values of some sample statistic that would
occur if we were to draw an infinite number of same-sized
samples from a population
Notice: SEM is smaller than SD
– especially as n increases
Eugene X X X X X X X
Melvin X X
µ= 100 X X
Mean = 100 X X
σ = 3
100
Standard Deviation = 3
23rd sample
An example of a
sampling distribution of sample means
2nd sample
µ = 100
Mean = 100
Standard Error
of the Mean = 1
= 1
100

24. Central Limit Theorem x will approach µ
Proposition 1: If sample size (n) is large enough (e.g. 100)
The mean of the sampling distribution will
approach the mean of the population
As n ↑
x will approach µ
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population
As n ↑ curve
will approach normal shape
Proposition 3: The standard deviation of the sampling
distribution equals the standard deviation of
the population divided by the square root of the
sample size. As n increases SEM decreases.
As n ↑ curve
variability gets smaller X

25. Central Limit Theorem
Proposition 1: If sample size (n) is large enough (e.g. 100)
The mean of the sampling distribution will
approach the mean of the population
Law of large numbers: As the number of measurements
increases the data becomes more stable and a better
approximation of the true (theoretical) probability.
Larger sample sizes tend to be associated with stability.
As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.

26. population population population n = 2 n = 5 n = 4 n = 30 n = 5 n = 25
Take sample
(n = 5) – get mean
Proposition 2: If sample size (n) is large
enough (e.g. 100), the sampling distribution
of means will be approximately normal,
regardless of the shape of the population
Repeat
over and over
Population
population
population
population
sampling distribution
n = 2
sampling distribution
n = 5
sampling distribution
n = 4
sampling distribution
n = 30
sampling distribution
n = 5
sampling distribution
n = 25

27. Central Limit Theorem
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population

28. Central Limit Theorem
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population

29. Central Limit Theorem
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population

30. Central Limit Theorem
Proposition 3: The standard deviation of the sampling
distribution equals the standard deviation of
the population divided by the square root of the
sample size. As n increases SEM decreases. X

31. Central Limit Theorem x will approach µ
Proposition 1: If sample size (n) is large enough (e.g. 100)
The mean of the sampling distribution will
approach the mean of the population
As n ↑
x will approach µ
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population
As n ↑ curve
will approach normal shape
Proposition 3: The standard deviation of the sampling
distribution equals the standard deviation of
the population divided by the square root of the
sample size. As n increases SEM decreases.
As n ↑ curve
variability gets smaller X

32. Central Limit Theorem
Proposition 3: The standard deviation of the sampling
distribution equals the standard deviation of
the population divided by the square root of the
sample size. As n increases SEM decreases. X

33. Central Limit Theorem x will approach µ
Proposition 1: If sample size (n) is large enough (e.g. 100)
The mean of the sampling distribution will
approach the mean of the population
As n ↑
x will approach µ
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population
As n ↑ curve
will approach normal shape
Proposition 3: The standard deviation of the sampling
distribution equals the standard deviation of
the population divided by the square root of the
sample size. As n increases SEM decreases.
As n ↑ curve
variability gets smaller X

34. Animation for creating sampling distribution of sample means
Central Limit Theorem: If random samples of a fixed N are drawn from any population (regardless of the shape of the population distribution), as
N becomes larger, the distribution of sample means approaches normality, with the overall mean approaching the theoretical population mean.
Distribution of Raw Scores
Animation for creating
sampling distribution
of sample means
Distribution of single sample
Eugene
Melvin
Sampling Distribution of Sample means
Sampling Distribution of Sample means
Mean for sample 12
Mean for sample 7

35. Sampling distribution for continuous distributions
Central Limit Theorem: If random samples of a fixed N are drawn
from any population (regardless of the shape of the
population distribution), as N becomes larger, the
distribution of sample means approaches normality, with
the overall mean approaching the theoretical population
mean.
Distribution of Raw Scores
Sampling Distribution of Sample means
Melvin X
Eugene
23rd sample X
2nd sample

36. Central Limit Theorem x will approach µ
Proposition 1: If sample size (n) is large enough (e.g. 100)
The mean of the sampling distribution will
approach the mean of the population
As n ↑
x will approach µ
Proposition 2: If sample size (n) is large enough (e.g. 100)
The sampling distribution of means will
be approximately normal, regardless of the
shape of the population
As n ↑ curve
will approach normal shape
Proposition 3: The standard deviation of the sampling
distribution equals the standard deviation of
the population divided by the square root of the
sample size. As n increases SEM decreases.
As n ↑ curve
variability gets smaller X

37. Writing Assignment: Writing a letter to a friend.
Writing Assignment: Writing a letter to a friend
Imagine you have a good friend (pick one). This is a good friend whom you consider to be smart and interested in stuff generally. They are teaching themselves stats (hoping to test out of the class) but need your help on a couple ideas. For this assignment please write your friend/mom/dad/ favorite cousin a letter answering these five questions: (Feel free to use diagrams and drawings if you think that can help)
Dear Friend,
1. I’m struggling with this whole Central Limit Theorem idea. Could you
describe for me the difference between a distribution of raw scores, and a
distribution of sample means?
2. I also don’t get the “three propositions of the Central Limit Theorem”. They all
seem to address sample size, but I don’t get how sample size could affect
these three things. If you could help explain it, that would be really helpful.

38. distribution of sample means?
Imagine you have a good friend (pick one). This is a good friend whom you consider to be smart and interested in stuff generally. They are teaching themselves stats (hoping to test out of the class) but need your help on a couple ideas. For this assignment please write your friend/mom/dad/ favorite cousin a letter answering these five questions: (Feel free to use diagrams and drawings if you think that can help)
Dear Friend,
1. I’m struggling with this whole Central Limit Theorem idea. Could you
describe for me the difference between a distribution of raw scores, and a
distribution of sample means?
2. I also don’t get the “three propositions of the Central Limit Theorem”. They all
seem to address sample size, but I don’t get how sample size could affect
these three things. If you could help explain it, that would be really helpful. .

39. .

40. Thank you!
See you next time!!