1. BNAD 276: Statistical Inference in Management Winter, 2015
Welcome
Green sheet
Seating Chart

2. Daily group portfolios
Beginning of each lecture (first 5 minutes)
Meet in groups of 3 or 4
Quiz one another on class material
Discuss the questions and determine the correct answer for each question
Five copies (one for each group member – and typed) multiple choice questions based on lecture
Include 4 options (a, b, c, and d)
Include a name and describe a person in a certain situation
Margaret was interested in taking a Statistics course.
It is likely she was interested in studying which of the following? a. economic theories of communism
b. theological perspectives of life after death
c. musical compositions of the 12th century
d. statistical techniques and inference
They can be funny or serious, and must be clear and have only one correct answer.

3. Please start
portfolios

4. . Homework Assignment Go to D2L - Click on “Content”
Click on “Interactive Online Homework Assignments”

5. 66% chance of getting admitted
What is probability
1. Empirical probability: relative frequency approach
Number of observed outcomes
Number of observations
Probability of getting into an educational program
Number of people they let in
400
66% chance of getting admitted
Number of applicants
600
Probability of getting a rotten apple
Number of rotten apples 5
5% chance of
getting a rotten
apple
Number of apples
100

6. 1. Empirical probability: relative frequency approach
What is probability
1. Empirical probability: relative frequency approach
“There is a 20% chance that a new stock offered in an initial public offering (IPO) will reach or exceed
its target price on the first day.”
“More than 30% of the results from major search engines for the keyword phrase “ring tone” are fake
pages created by spammers.”
10% of people who buy a house with no pool build one. What is the likelihood that Bob will?
Number of observed outcomes
Number of observations
Probability of hitting the corvette
Number of carts that hit corvette
Number of carts rolled
182
= .91
200
91% chance of hitting a corvette

7. = = 2. Classic probability: a priori probabilities based on logic
rather than on data or experience.
All options are equally likely (deductive rather than inductive).
Likelihood get question right on multiple choice test
Chosen at random to be team captain
Lottery
Number of outcomes of specific event
Number of all possible events
In throwing a die what is the probability of getting a “2”
Number of sides with a 2 1
16% chance of getting a two =
Number of sides 6
In tossing a coin what is probability of getting a tail
Number of sides with a 1 1
50% chance
of getting a tail =
Number of sides 2

8. 3. Subjective probability: based on someone’s personal
judgment (often an expert), and often used when empirical
and classic approaches are not available.
Likelihood that company will invent new type of battery
Likelihood get a ”B” in the class
60% chance that Patriots will play at Super Bowl
There is a 5% chance that Verizon will
merge with Sprint
Bob says he is 90% sure he could swim across the river

9. Approach Example Empirical Classical Subjective
There is a 2 percent chance of twins in a randomly-chosen birth
Classical
There is a 50 % probability of heads on a coin flip.
Subjective
There is a 5% chance that
Verizon will merge with Sprint

10. The probability of event A [denoted P(A)], must lie
The probability of an event is the relative likelihood that the event will occur.
The probability of event A [denoted P(A)], must lie
within the interval from 0 to 1:
0 < P(A) < 1
If P(A) = 0, then the event cannot occur.
If P(A) = 1, then the event is certain to occur.

11. The probabilities of all simple events must sum to 1
P(S) = P(E1) + P(E2) + … + P(En) = 1
For example, if the following number of purchases were made by
credit card:
32%
debit card:
20%
cash:
35%
check:
13%
Sum =
100%
P(credit card) =
.32
P(debit card) =
.20
P(cash) =
.35
P(check) =
.13
Sum =
1.0
Probability

12. Probability of getting into an educational program
What is the complement of the probability of an event
The probability of event A = P(A).
The probability of the complement of the event A’ = P(A’)
A’ is called “A prime”
Complement of A just means probability of “not A”
P(A) + P(A’) = 100%
P(A) = 100% - P(A’)
P(A’) = 100% - P(A)
Probability of
getting a rotten apple
5% chance of “rotten apple”
95% chance of “not rotten apple”
100% chance of rotten or not
Probability of getting
into an educational program
66% chance of “admitted”
34% chance of “not admitted”
100% chance of admitted or not

13. Two mutually exclusive characteristics: if the occurrence of any one of them automatically implies the non-occurrence of the remaining characteristic
Two events are mutually exclusive if they cannot occur at the same time (i.e. they have no outcomes in common).
Two propositions that logically cannot both be true.
No Warranty
Warranty
For example, a car repair is either covered by the warranty (A) or not (B).

14. Collectively Exhaustive Events
Events are collectively exhaustive if their union is the entire sample space S.
Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events.
For example, a car repair is either covered by the warranty (A) or not (B).
No Warranty
Warranty

15. Satirical take on being “mutually exclusive”
No Warranty
Warranty
Recently a public figure in the heat of the moment inadvertently made a statement that reflected extreme stereotyping that many would find highly offensive. It is within this context that comical satirists have used the concept of being “mutually exclusive” to have fun with the statement.
Decent ,
family man
Arab
Transcript:
Speaker 1:
“He’s an Arab”
Speaker 2:
“No ma’am, no ma’am.
He’s a decent, family
man, citizen…”

16. ∩ Union versus Intersection Union of two events means
Event A or Event B will happen ∩
P(A B)
Intersection of two events means
Event A and Event B will happen
Also called a “joint probability”
P(A ∩ B)

17. The union of two events: all outcomes in the
sample space S that are contained either in event
A or in event B or both (denoted A  B or “A or B”).
 may be read as “or” since one or the other or both events may occur.

18. The union of two events: all outcomes contained either in event A or in event B or both (denoted A  B or “A or B”).
What is probability of drawing a red card or a queen?
what is Q  R?
It is the possibility of drawing
either a queen (4 ways) or a red card (26 ways) or both (2 ways).

19. P(Q) = 4/52 (4 queens in a deck) 2/52
Probability of picking a Queen
Probability of picking a Red
4/52
26/52
P(Q) = 4/52 (4 queens in a deck)
2/52
P(R) = 26/52
(26 red cards in a deck)
P(Q  R) = 2/52
(2 red queens in a deck)
Probability of picking both
R and Q
When you add the P(A) and P(B) together, you count the P(A and B) twice. So, you have to subtract P(A  B) to avoid over-stating the probability.
P(Q  R) = P(Q) + P(R) – P(Q  R)
= 4/ /52 – 2/52
= 28/52 = or 53.85%

20. ∩ Union versus Intersection Union of two events means
Event A or Event B will happen ∩
P(A B)
Intersection of two events means
Event A and Event B will happen
Also called a “joint probability”
P(A ∩ B)

21. The intersection of two events: all outcomes contained in both event A and event B (denoted A  B or “A and B”)
What is probability of drawing red queen?
what is Q  R?
It is the possibility of drawing both a queen and a red card (2 ways).

22. Poodles and Labs: Mutually Exclusive
If two events are mutually exclusive (or disjoint) their intersection is a null set (and we can use the “Special Law of Addition”)
P(A ∩ B) = 0
Intersection of two events means
Event A and Event B will happen
Examples:
mutually
exclusive
If A = Poodles
If B = Labradors
Poodles and Labs: Mutually Exclusive
(assuming purebred)

23. Poodles and Labs: Mutually Exclusive
If two events are mutually exclusive (or disjoint) their intersection is a null set (and we can use the “Special Law of Addition”)
P(A ∩ B) = 0
Dog Pound
Intersection of two events means
Event A and Event B will happen
Examples:
If A = Poodles
If B = Labradors
(let’s say 10% of dogs are poodles)
(let’s say 15% of dogs are labs)
What’s the probability of picking a
poodle or a lab at random from pound? ∩
P(A B) = P(A) +P(B)
P(poodle or lab) = P(poodle) + P(lab)
P(poodle or lab) = (.10) + (.15) = (.25)
Poodles and Labs: Mutually Exclusive
(assuming purebred)

24. Practice 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)

25. Practice
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”)

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

27. .
Homework Assignment

28. Raw scores, z scores & probabilities
Distance from the mean
(z scores)
convert
convert
Raw Scores
(actual data)
Proportion of curve
(area from mean)
68%
z = -1
z = 1
We care about this!
What is the actual number on this scale? “height” vs “weight”
“pounds” vs “test score”
We care about this!
“percentiles”
“percent of people”
“proportion of curve” “relative position”
68%
Raw Scores
(actual data)
Proportion of curve
(area from mean)
z = -1
z = 1
Distance from the mean
(z scores)
convert
convert

29. z table Formula Normal distribution Raw scores z-scores probabilities
Have z
Find raw score Z
Scores
Have z
Find area
z table
Formula
Have area
Find z
Area &
Probability
Have raw score
Find z
Raw Scores

30. Raw scores, z scores & probabilities
Notice:
3 types of numbers
raw scores
z scores
probabilities
Mean = 50
Standard deviation = 10
z = -2
z = +2
If we go up two standard deviations
z score = +2.0 and raw score = 70
If we go down two standard deviations
z score = -2.0 and raw score = 30

31. Always draw a picture!
Homework worksheet

32. 1 .6800 1 sd 1 sd 28 30 32 Homework worksheet .6800 also fine: 68%
z =-1
z = 1 28 30 32

33. 2 .9500 2 sd 2 sd 26 28 30 32 34 Homework worksheet .9500
also fine: 95%
also fine:
.9500
2 sd
2 sd
z =-2
z = 2 26 28 30 32 34

34. 3 .9970 3 sd 3 sd 24 26 28 30 32 34 36 Homework worksheet .9970
also fine: 99.7%
also fine:
.9970
3 sd
3 sd
z =-3
z = 3 24 26 28 30 32 34 36

35. 4 .5000 24 26 28 30 32 34 36 Homework worksheet .5000 also fine: 50%
z = 0 24 26 28 30 32 34 36

36. 5 .4332 24 26 28 30 32 34 36 Homework worksheet z = 33-30 z = 1.5
Go to
table
.4332 2 5
also fine: %
.4332
z = 1.5 24 26 28 30 32 34 36

37. z =
33-30 2
z = 1.5
Go to
table
.4332
Add area
Lower half
= .9332 6
also fine: %
.9332
.4332
.5000
z = 1.5 24 26 28 30 32 34 36

38. 7 .4332 .0668 24 26 28 30 32 34 36 Homework worksheet z = 33-30 = 1.5
= 1.5
Go to
table
.4332
Subtract
from .5000
= .0668 2 7
also fine: 6.68%
.4332
.0668
z = 1.5 24 26 28 30 32 34 36

39. z =
29-30 2
= -.5
Go to
table
.1915
Add to upper
Half of curve
= .6915 8
also fine: %
.6915
.1915
.5000
z = -.5 24 26 28 30 32 34 36

40. =
25-30 2
= -2.5
.4938
Go to table =
31-30 2
=.5
.1915
Go to table
= .6853 9
also fine: %
.6853
.1915
.4938
z =-2.5
z = .5 24 26 28 30 32 34 36

41. z =
27-30 2
= -1.5
Go to
table
.4332
Subtract
From .5000
= .0668 10
also fine: 6.68%
.5000
.0668
.4332
z =-1.5 24 26 28 30 32 34 36

42. z =
27-30 2
= -1.5
Go to
table
.4332
Add lower
Half of curve
= .9932 11
also fine: %
.9332
.5000
.4332
z =-1.5 24 26 28 30 32 34 36

43. z =
32-30 2
= 1.0
Go to
table
.3413
Subtract
from .5000
= .0068 12
.5000
also fine: %
.1587
.3413
z =1 24 26 28 30 32 34 36

44. 13 24 26 28 30 32 34 36 50th percentile = median 30 In a normal curve
Median= Mean = Mode
z =0 24 26 28 30 32 34 36

45. 28 32 14
.6800
1 sd
1 sd
z =-1
z = 1 24 26 28 30 32 34 36

46. z table provides area from mean to score
x = mean + z σ = 30 + (.74)(2) = 31.48
77th percentile
Find area
of interest
= .2700
Find nearest z = .74 15
.2700
.7700
z table provides area from mean to score
.5000
31.48
z =.74 24 ? 30 36

47. z table provides area from mean to score
13th percentile
Find area
of interest
= .3700
Find nearest z = -1.13
x = mean + z σ = 30 + (-1.13)(2) = 27.74 16
Note:
=.50
z table provides area from mean to score
.3700
.1300
z =-1.13
27.74 ? 24 30 36

48. Please use the following distribution with
a mean of 200 and a standard deviation of 40. 80
120
160
200
240
280
320

49. 17 .6800 1 sd 1 sd 160 200 240 .6800 also fine: 68% also fine: .6826
z =-1
z = 1
160
200
240

50. 18 .9500 2 sd 2 sd 120 200 280 .9500 also fine: 95% also fine: .9544
z =-2
z = 2
120
200
280

51. 19 .9970 3 sd 3 sd 80 200 320 .9970 also fine: 99.7% also fine: .9974
z =-3
z = 3 80
200
320

52.
Go to
table =
= .75
.2734 40 20
also fine: %
.2734
z =.75 80
120
160
200
240
280
320

53. Go to
table
Add to upper
Half of curve
z = 40
= -.5
.1915
= .6915 22
also fine: %
.5000
.1915
.6915
z =-.5 80
120
160
200
240
280
320

54. z = 40
= 0.9
Go to
table
.3159
Subtract
from .5000
= .1841 23
.3159
also fine: %
.1841
z =.9 80
120
160
200
240
280
320

55. z = 40
= -.2
.0793
Go to table
= .2881
z = 40
=.55
.2088
Go to table 24
also fine: %
.2881
.2088
.0793
z =-.2
z =.55 80
120
160
200
240
280
320

56. = .9693
z =
= 1.875
Go to
table
Add area
Lower half
.4693 or
.4699
= .9699 40 25
Please note:
If z-score rounded to 1.88,
then percentile = 96.99%
also fine: %
.9693
.4693
.5000
z =1.875 80
120
160
200
240
280
320

57. Add area
Lower half
= .0089
= .0087
z = 40
z = 2.375
Go to
table
.4911 or
.4913 26
Please note:
If z-score rounded to 2.38,
then area = .0087
also fine: 0.89%
.4911
.0089
z =2.375 80
120
160
200
240
280
320

58. z = 40
= -1.75
.4599
Add to upper
Half of curve
Go to
table
= .9599 27
also fine: %
.9599
.5000
.4599
z =-1.75 80
120
160
200
240
280
320

59. 40
z =
= -1.75
.4599
Subtract
from .5000
= .0401
Go to
table 28
.0401
.4599
.5000
also fine: 4.01%
z =-1.75 80
120
160
200
240
280
320

60. z table provides area from mean to score
x = mean + z σ = (2.33)(40) = 293.2
99th percentile
Find area
of interest
= .4900
Find nearest z = 2.33 29
.4900
.9900
z table provides area from mean to score
.5000
293.2
z =2.33 80 ?
120
160
200
240

61. z table provides area from mean to score
33rd percentile
Find area
of interest
= .1700
Find nearest z = .44
x = mean + z σ = (-.44)(40) = 182.4 30
z table provides area from mean to score
Note:
=.50
.3300
.1700
182.4
z =-.44 ? 80
200
240
280
320

62. z table provides area from mean to score
40th percentile
Find area
of interest
= .1000
Find nearest z = -.25
x = mean + z σ = (-.25)(40) = 190 31
z table provides area from mean to score
Note:
=.50
182.4
.1000
.4000
z =-.25 ? 80
200
240
280
320

63. z table provides area from mean to score
67th percentile
Find area
of interest
= .1700
Find nearest z = .44
x = mean + z σ = (.44)(40) = 217.6 32
z table provides area from mean to score
.1700
z =.44 80
200
217.6 ?
320

64. Mean of 30 and standard deviation of 2.
Try this one:
Please find the (2) raw scores that border exactly the middle 95% of the curve
Mean of 30 and standard deviation of 2
Go to
table
.4750
nearest z = 1.96
mean + z σ = 30 + (1.96)(2) = 33.92
Go to
table
.4750
nearest z = -1.96
mean + z σ = 30 + (-1.96)(2) = 26.08
.9500
.475
.475
Additional practice
26.08
33.92 24 ? 28 30 32 ? 36

65. Mean of 30 and standard deviation of 2.
Try this one:
Please find the (2) raw scores that border exactly the middle 99% of the curve
Mean of 30 and standard deviation of 2
Go to
table
.4750
nearest z = 2.58
mean + z σ = 30 + (2.58)(2) = 35.16
Go to
table
.4750
nearest z = -2.58
mean + z σ = 30 + (-2.58)(2) = 24.84
.9900
.495
.495
Additional practice
24.84 ?
35.16 28 32 ? 30

66. Finding the interval that borders the middle of the curve.
Finding the interval that borders the middle of the curve
Which is wider?
Please find the raw scores that
border the middle 95% of the curve
Please find the raw scores that
border the middle 99% of the curve

67. z table Formula Normal distribution Raw scores z-scores probabilities
Have z
Find raw score Z
Scores
Have z
Find area
z table
Formula
Have area
Find z
Area &
Probability
Have raw score
Find z
Raw Scores

68. Comments on Dan Gilbert Reading

69. 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.

70. 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)

71. 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

72. 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

73. 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

74. 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

75. 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

76. 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: Standard Error of the Mean (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

77. 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

78. 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.
Related but not identical
Related but not identical
Proposition 1: If sample size (n) is large enough (e.g. ~30) 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.

79. 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

80. 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

81. 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

82. 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

83. 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

84. 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

85. 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

86. 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

87. 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

88. 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

89. 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

90. 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 two 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.

91. 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 two 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. .

92. .

93. Comments on Dan Gilbert Reading

94. Standard Error of the Mean (SEM).
Standard Error of the Mean (SEM)
Building towards
Confidence Intervals
Reviewing the components for
calculating confidence intervals:
How to calculate confidence intervals
Simply finding the raw score that is a certain distance from the mean that is associated with an area under the curve
When to use confidence intervals:
when you are estimating (guessing) a single number by providing likely range that the number appears in
The relevance of the Central Limit Theorem
When we are predicting a value we will use the standard error of the mean (rather than the standard deviation)

95. Building towards confidence intervals
Normal distribution
Raw scores z-scores probabilities
Have z
Find raw score Z
Scores
Have z
Find area
z table
Formula
Have area
Find z
Area &
Probability
Have raw score
Find z
Raw Scores
Building towards confidence intervals
Part 1

96. .
Writing Assignment - Try this one:
Please find the (2) raw scores that border exactly the middle 95% of the curve
Mean of 100 and standard deviation of 5
Go to
table
.4750
nearest z = 1.96
mean + z σ = (1.96)(5) =
Go to
table
.4750
nearest z = -1.96
mean + z σ = (-1.96)(5) = 90.20
.9500
.475
.475
Building towards confidence intervals
90.2
109.8 85 ? 95
105 ?
100
115
Part 1

97. .
Try this one:
Please find the (2) raw scores that border exactly the middle 95% of the curve
Mean of 30 and standard deviation of 2
Go to
table
.4750
nearest z = 1.96
mean + z σ = 30 + (1.96)(2) = 33.92
Go to
table
.4750
nearest z = -1.96
mean + z σ = 30 + (-1.96)(2) = 26.08
.9500
.475
.475
Building towards confidence intervals
26.08 ?
33.92 28 32 ? 30
Part 1

98. .
Try this one:
Please find the (2) raw scores that border exactly the middle 99% of the curve
Mean of 30 and standard deviation of 2
Go to
table
.4950
nearest z = 2.58
mean + z σ = 30 + (2.58)(2) = 35.16
Go to
table
.4950
nearest z = -2.58
mean + z σ = 30 + (-2.58)(2) = 24.84
.9900
.495
.495
Building towards confidence intervals
24.84 ?
35.16 28 32 ? 30
Part 1

99. Finding the interval that borders the middle of the curve.
Finding the interval that borders the middle of the curve
Which is wider?
Please find the raw scores that
border the middle 95% of the curve
Building towards confidence intervals
Please find the raw scores that
border the middle 99% of the curve
Part 1

100. Remember Confidence Intervals.
Remember
Confidence Intervals
95% Confidence Interval:
We can be 95% confident that the
estimated score really does fall
between these two scores
Please find the raw scores that
border the middle 95% of the curve
99% Confidence Interval:
We can be 99% confident that the
estimated score really does fall
between these two scores
Building towards confidence intervals
Please find the raw scores that
border the middle 99% of the curve
Part 1

101. Building towards confidence intervals.
Building towards confidence intervals
Part 2

102. Remember Confidence Intervals
Standard Error of the Mean (SEM)
Remember
Confidence Intervals
Confidence Intervals (based on z):
We are using this to estimate a value such as a population mean,
with a known degree of certainty with a range of values
The interval refers to possible values of the population mean.
Subjective vs
Empirical
We can be reasonably confident that the population mean
falls in this range (90%, 95%, or 99% confident)
In the long run, series of intervals, like the one we
figured out will describe the population mean about 95%
of the time.
Can actually generate CI for any confidence level you want – these are most common
Building towards confidence intervals
Part 3

103. Building towards confidence intervals
Confidence Intervals (based on z): A range of values that,
with a known degree of certainty, includes an unknown
population characteristic, such as a population mean
How can we make our confidence interval smaller?
Decrease variability by increasing sample size
Decrease variability through more careful assessment
and measurement practices (minimize noise)
Decrease level of confidence .
95%
Building towards confidence intervals
95%
Part 3

104. Standard Error of the Mean (SEM).
Standard Error of the Mean (SEM)
Building towards
Confidence Intervals
We now know all components of actually
calculating confidence intervals:
When to use confidence intervals:
when you are estimating (guessing) a single number by providing likely range that the number appears in
How to calculate confidence intervals
Simply finding the raw score that is a certain distance from the mean that is associated with an area under the curve
The relevance of the Central Limit Theorem
When we are predicting a value we will use the standard error of the mean (rather than the standard deviation)

105. We will be using this same logic for “confidence intervals”
Mean = 50 Standard deviation = 10
Find the scores for
the middle 95% ? ?
95%
x = mean ± (z)(standard deviation)
30.4
69.6 ?
.9500
.4750
Please note:
We will be using this same logic for “confidence intervals”
1) Go to z table - find z score for
for area .4750
z = 1.96
2) x = mean + (z)(standard deviation)
x = 50 + (-1.96)(10)
x = 30.4
30.4
3) x = mean + (z)(standard deviation)
x = 50 + (1.96)(10)
x = 69.6
69.6
Scores capture the
middle 95% of the curve

106. is captured by the scores 48.04 – 51.96
Confidence intervals
Mean = 50 Standard deviation = 10
n = 100
s.e.m. = 1 ? ?
95%
Find the scores for
the middle 95%
48.04
51.96
For “confidence intervals”
same logic – same z-score
But - we’ll replace standard deviation with the standard error of the mean ?
.9500
.4750
standard error
of the mean σ √ n = 10 = √
100
x = mean ± (z)(s.e.m.)
x = 50 + (1.96)(1)
x =
x = 50 + (-1.96)(1)
x =
95% Confidence Interval
is captured by the scores – 51.96

107. σ n Find a 95% Confidence Interval for this distribution = √ √
mean = 121
standard deviation= 15
n = 25
Find a 95% Confidence
Interval for this distribution
raw score = mean + (z score)(standard error)
standard error
of the mean σ √ n = 15 = = 3 √ 25
raw score = mean ± (z score)(sem)
x = x ± (z)(σx)
X = 121 ± (1.96)(3) = 121 ± 5.88
Please notice:
We know the standard deviation and can calculate the standard error of the mean from it
(115.12, )
confidence interval

108. Confidence intervals ? ? Tell me the scores that border exactly
95% ?
Tell me the scores that border exactly
the middle 95% of the curve
We know this
raw score = mean ± (z score)(s.d.)
Construct a 95 percent confidence interval around the mean
Similar, but uses
standard error the mean
based on population s.d.
raw score = mean ± (z score)(s.e.m.)

109. Confidence intervals Level of Alpha 1.96 = .05 1.64 = .10 2.58 = .01
Construct a 95 percent confidence interval around the mean
Tell me the scores that border exactly
the middle 95% of the curve
- use z score of 1.96
Level of Alpha
1.96
= .05
1.64
= .10
90%
2.58
= .01
z scores for different
levels of confidence
How do we know which z score to use?

110. Confidence Interval of 95% Has and alpha of 5% α = .05
Critical z
-2.58
Critical z
2.58
Confidence Interval of 99%
Has and alpha of 1%
α = .01
99%
Area outside confidence interval is alpha
Critical z
-1.96
Critical z
1.96
Confidence Interval of 95% Has and alpha of 5% α = .05
95%
Area in the tails is called alpha
Critical z
-1.64
Critical z
1.64
Confidence Interval of 90%
Has and alpha of 10%
α = . 10
90%
Area associated with most extreme scores is called alpha

111. Thank you!
See you next time!!