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Cabinet

Cabinet

Lecturer’s desk

Table

Computer Storage Cabinet

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INTEGRATED LEARNING CENTER

ILC 120

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broken

desk

Introduction to Statistics for the Social SciencesSBS200, COMM200, GEOG200, PA200, POL200, or SOC200Lecture Section 001, Spring, 2013Room 120 Integrated Learning Center (ILC)10:00 - 10:50 Mondays, Wednesdays & Fridays.

Welcome

http://www.youtube.com/watch?v=oSQJP40PcGI

My last name starts with a

letter somewhere between

A. A – D

B. E – L

C. M – R

D. S – Z

Please hand in

your homework

study guide

By the end of lecture today2/25/13- Probability of an event
- Complement of an event; Union of two events
- Intersection of two events; Mutually exclusive events
- Collectively exhaustive events
- Conditional probability
- Connecting raw scores, z scores and probabilityConnecting probability, proportion and area of curve
- Law of Large Numbers
- Central Limit Theorem
- Three propositions
- True mean 2) Standard Error of Mean 3) Normal Shape
- Calculating Confidence Intervals

http://onlinestatbook.com/stat_sim/sampling_dist/index.html

http://www.youtube.com/watch?v=ne6tB2KiZuk

Before next exam (This Friday - March 1st)

Please read chapters 5, 6, & 8 in Ha & Ha

Please read Chapters 10, 11, 12 and 14 in Plous

Chapter 10: The Representativeness Heuristic

Chapter 11: The Availability Heuristic

Chapter 12: Probability and Risk

Chapter 14: The Perception of Randomness

Study Guide

is online

Homework due – Wednesday (February 27th)

- On class website:
- Please print and complete homework worksheet #14
- Confidence Intervals

∩

P(A B)

Union of two events means

Event A or Event B will happen

Intersection of two events means

Event A and Event B will happen

Also called a “joint probability”

P(A ∩ B)

The union of two events: all outcomes in the

sample space S that are contained either in event

Aor in event Bor both (denoted A B or “A or B”).

may be read as “or” since one or the other or both events may occur.

The union of two events: all outcomes contained either in event Aor in event Bor 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).

Probability of picking a Queen

Probability of picking a Red

26/52

4/52

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 + 26/52 – 2/52

= 28/52 = .5385 or 53.85%

∩

P(A B)

Union of two events means

Event A or Event B will happen

Intersection of two events means

Event A and Event B will happen

Also called a “joint probability”

P(A ∩ B)

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).

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)

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

P(A B) = P(A) +P(B)

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(poodle or lab) = P(poodle) + P(lab)

P(poodle or lab) = (.10) + (.15) = (.25)

Poodles and Labs:Mutually Exclusive

(assuming purebred)

Conditional Probabilities intersection is a null set (and we can use the “Special Law of Addition”)

Probability that A has occurred given that B has occurred

Denoted P(A | B):

The vertical line “ | ” is read as “given.”

P(A ∩ B)

P(A | B) =

P(B)

The sample space is restricted to B, an event that has occurred.

A B is the part of B that is also in A.

The ratio of the relative size of

A B to B is P(A | B).

Conditional Probabilities intersection is a null set (and we can use the “Special Law of Addition”)

Probability that A has occurred given that B has occurred

Of the population aged 16 – 21 and not in college:

P(U) = .1350

P(ND) = .2905

P(UND) = .0532

What is the conditional probability that a member of this

population is unemployed, given that the person has no diploma?

.0532

P(A ∩ B)

.1831

=

P(A | B) =

=

.2905

P(B)

or 18.31%

Conditional Probabilities intersection is a null set (and we can use the “Special Law of Addition”)

Probability that A has occurred given that B has occurred

Of the population aged 16 – 21 and not in college:

P(U) = .1350

P(ND) = .2905

P(UND) = .0532

What is the conditional probability that a member of this

population is unemployed, given that the person has no diploma?

.0532

P(A ∩ B)

.1831

=

P(A | B) =

=

.2905

P(B)

or 18.31%

Standard Error of the Mean (SEM) intersection is a null set (and we can use the “Special Law of Addition”)

Remember

confidence intervals?

Revisit 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.

- 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.

Greater confidence implies loss of precision.(95% confidence is most often used)

Can actually generate CI for any confidence level you want – these are just the most common

? intersection is a null set (and we can use the “Special Law of Addition”)

?

Mean = 50Standard deviation = 10

Find the scores for

the middle 95%

95%

x = mean ± (z)(standard deviation)

30.4

69.6

.9500

Please note:

We will be using this same logic for “confidence intervals”

.4750

.4750

?

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 30.4 - 69.6 capture the

middle 95% of the curve

? intersection is a null set (and we can use the “Special Law of Addition”)

?

Mean = 50Standard deviation = 10

n = 100

s.e.m. = 1

Confidence intervals

σ

95%

standard error

of the mean

=

Find the scores for

the middle 95%

n

√

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

.4750

?

10

=

100

√

x = mean ± (z)(s.e.m.)

x = 50 + (1.96)(1)

x = 51.96

x = 50 + (-1.96)(1)

x = 48.04

95% Confidence Interval

is captured by the scores 48.04 – 51.96

Confidence intervals intersection is a null set (and we can use the “Special Law of Addition”)

?

?

σ

standard error

of the mean

95%

=

n

√

Mean = 50 Standard error mean = 10

Hint always draw a picture!

Tell me the scores associated that border exactly

the middle 95% of the curve

We know this

raw score = mean ± (z score)(standard deviation)

Construct a 95 percent confidence interval around the mean

Similar, but uses

standard error the mean

raw score = mean ± (z score)(standard error of the mean)

Law of large numbers: As the number of measurements intersection is a null set (and we can use the “Special Law of Addition”)

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.

Law of large numbers: As the number of measurements intersection is a null set (and we can use the “Special Law of Addition”)

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)

http://www.youtube.com/watch?v=ne6tB2KiZuk

Central Limit Theorem intersection is a null set (and we can use the “Special Law of Addition”)

Sampling distributions of sample means intersection is a null set (and we can use the “Special Law of Addition”)

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

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- Frequency distributions of individual scores
- derived empirically
- we are plotting raw data
- this is a single sample

Melvin

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Take a single score

x

Repeat

over and over

x

x

x

Population

x

x

x

x

- 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

- 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

- 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

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Melvin

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- Sampling distributions sample means
- theoretical distribution
- we are plotting means of samples

23rd sample

2nd sample

X distribution of

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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

23rd sample

Eugene

X

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2nd sample

Thank you! distribution of

See you next time!!

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