probability & the normal distribution statistics for the social sciences psychology 340 spring...

Probability &

The Normal Distribution

Statistics for the Social SciencesPsychology 340

Spring 2010

PSY 340Statistics for the

Social Sciences Reminders

• Quiz 2 due Thursday

• Homework 3 due Tues, Feb 2

• Exam 1 Thurs Feb. 11


Social Sciences Basics of Probability

• Probability– Expected relative frequency of a particular outcome

• Outcome– The result of an experiment

Probability = Possible successful outcomes

All possible outcomes


Social Sciences Flipping a coin example

What are the odds of getting a “heads”?

One outcome classified as heads=

1

2= 0.5

Probability = Possible successful outcomes

All possible outcomes

Total of two outcomes

n = 1 flip



What are the odds of getting two “heads”?

Number of heads

2

1

1

0

One 2 “heads” outcome

Four total outcomes

= 0.25

This situation is known as the binomial # of outcomes = 2n

n = 2



What are the odds of getting “at least one heads”?

Number of heads

2

1

1

0

Four total outcomes

= 0.75

Three “at least one heads” outcome

n = 2



HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

Number of heads3

2

1

0

2

2

1

1

2n = 23 = 8 total outcomes

n = 3



Number of heads3

2

1

0

2

2

1

1

X f p

3 1 .125

2 3 .375

1 3 .375

0 1 .125Number of heads0 1 2 3

.1

.2

.3

.4

prob

abil

ity

.125 .125.375.375

Distribution of possible outcomes(n = 3 flips)



Number of heads0 1 2 3

.1

.2

.3

.4

prob

abil

ity

What’s the probability of flipping three heads in a row?

.125 .125.375.375 p = 0.125


Can make predictions about likelihood of outcomes based on this distribution.




.1

.2

.3

.4

prob

abil

ity

What’s the probability of flipping at least two heads in three tosses?

.125 .125.375.375 p = 0.375 + 0.125 = 0.50






.1

.2

.3

.4

prob

abil

ity

What’s the probability of flipping all heads or all tails in three tosses?

.125 .125.375.375 p = 0.125 + 0.125 = 0.25




Social Sciences Hypothesis testing


Distribution of possible outcomes(of a particular sample size, n)

• In hypothesis testing, we compare our observed samples with the distribution of possible samples (transformed into standardized distributions)

• This distribution of possible outcomes is often Normally Distributed


Social Sciences The Normal Distribution

• The distribution of days before and after due date (bin

width = 4 days).

0 14-14

Days before and after due date



• Normal distribution



• Normal distribution is a commonly found distribution that is symmetrical and unimodal. – Not all unimodal, symmetrical curves are Normal, so be careful

with your descriptions

• It is defined by the following equation:

€

1

2πσ 2e−(X −μ )2 / 2σ 2

1 2-1-2 0-3 3Z-scores


Social Sciences

Estimating Probabilities in a Normal Distribution

50%-34%-14% rule

1 2-1-2 0

50%

-3 3

Number of heads

0 1 2 3

.1

.2

.3

.4

prob

abili

ty

.125 .125.375.375

Same logic as before


Social Sciences


50%-34%-14% rule

50%

1 2-1-2 0

34.13%

13.59%

-3 3


Social Sciences


1 2-1-2 0

Similar to the 68%-95%-99% rule

13.59%2.28%

34.13%

13.59%2.28%

34.13%

68%

-3 3


Social Sciences


1 2-1-2 0

Similar to the 68%-95%-99% rule

13.59%2.28%

34.13%

13.59%2.28%

34.13%

-3 3

95%


Social Sciences The Unit Normal Table

z

0

:

:

1.00

:

:

2.31

2.32

• Gives the precise proportion of scores (in z-scores) above or below a given score in a Normal distribution

• There are many ways that this table gets organized

• Learn to understand what is in the table• What do the numbers represent?

• The normal distribution is often transformed into z-scores.

Understand your table

z0



– Contains the proportions of a Normal distribution

– Proportion between the z-score and left side of the distribution

– Proportion in the tail to the right of corresponding z-scores

– Proportion between the z-score and the mean

• Note: This means that this table lists only positive Z scores


In tail


Z Prop in Body

Prop in tail

Prop btwn mean and z

0.000.010.02

::

1.0:

1.3::

4.00

.5000

.5040

.5080::

.8413:

.9032::

.99997

.5000

.4960

.4920::

.1587:

.0968::

.00003

.0000

.0040

.0080::

.3413:

.4032::

.49997

From the left side of the dist.

z0



z .00 .01

0

:

:

1.0

:

:

2.3

2.4

:

0.5000

:

:

0.1587

:

:

0.0107

0.0082

:

0.4960

:

:

0.1562

:

:

0.0104

0.0080

:

– Contains the proportions in the tail to the left of corresponding z-scores of a Normal distribution

• This means that the table lists only positive Z scores

• The different columns give the second decimal place of the z-score


In tail

The unit normal table I have provided on-line (see ‘statistical tables’ link at top of labs)


z0



z Mean to Z In tail

0

:

:

1.00

:

:

2.31

2.32

:

0.0000

:

:

0.3413

:

:

0.4896

0.4898

:

0.5000

:

:

0.1587

:

:

0.0104

0.0102

:

– Contains the proportions– Proportion between the z-score and the

mean– Proportion in the tail to the left of

corresponding z-scores of a Normal distribution

• Note: This means that this table lists only positive Z scores


Mean to Z

In tail


z0



z .00 .01

-3.4

-3.3

:

:

0

:

:

1.0

:

:

3.3

3.4

0.0003

0.0005

:

:

0.5000

:

:

0.8413

:

:

0.9995

0.9997

0.0003

0.0005

:

:

0.5040

:

:

0.8438

:

:

0.9995

0.9997

– Contains the proportions to the left of corresponding z-scores of a Normal distribution

• This table lists both positive and negative Z scores


From the left side of the dist.

Another common way the unit normal table is presented in textbooks


z0


Social Sciences Using the Unit Normal Table

1. Convert raw score to Z score (if necessary)

2. Draw normal curve, where the Z score falls on it, shade in the area for which you are finding the percentage

3. Make rough estimate of shaded area’s percentage (using 50%-34%-14% rule)

• Steps for figuring the percentage below a particular raw or Z score:

Z Prop in Body

Prop in tail


0.000.010.02

::

1.0:

1.3::

4.00

.5000

.5040

.5080::

.8413:

.9032::

.99997

.5000

.4960

.4920::

.1587:

.0968::

.00003

.0000

.0040

.0080::

.3413:

.4032::

.49997

z =X−M

SD


Social Sciences Using the Unit Normal Table

4. Find exact percentage using unit normal table– Use your sketch and

understanding of the table

5. Check the exact percentage is within the range of the estimate from Step 3

• Steps for figuring the percentage below a particular raw or Z score:

Z Prop in Body

Prop in tail


0.000.010.02

::

1.0:

1.3::

4.00

.5000

.5040

.5080::

.8413:

.9032::

.99997

.5000

.4960

.4920::

.1587:

.0968::

.00003

.0000

.0040

.0080::

.3413:

.4032::

.49997


Social Sciences

Suppose that you got a 630 on the SAT. What percent of the people who take the SAT get your score or worse?

SAT Example problems

• The population parameters for the SAT are: μ = 500, σ = 100, and it is Normally distributed

€

z =X − μ

σ=

630 − 500

100=1.3 From the table:

z(1.3) =.0968 μ-1-2 1 2

That’s 9.68% above this score

So 90.32% got your score or worse



• You can go in the other direction too– Steps for figuring Z scores and raw scores from

percentages (or proportions):1. Draw normal curve, shade in approximate area for the percentage (using the 50%-34%-14% rule)

2. Make rough estimate of the Z score where the shaded area starts

3. Find the exact Z score using the unit normal table- So now you’re looking for a percentage/proportion in the body of the table,

and then looking to see what z-score it corresponds to

4. Check that your Z score is similar to the rough estimate from Step 2

5. If you want to find a raw score, change it from the Z score


Social Sciences Testing Hypotheses

• Looking ahead:– Core logic of hypothesis testing

• Considers the probability that the result of a study could have come about if the experimental procedure had no effect

€

test statistic =observed difference

difference expected by chance

How do we determine this?

Based on standard error or an estimate of the standard error

Z =(X−μX )

σ X

• “Studies” typically look not at single scores, but rather samples of scores. So we need to think about the probability of getting samples with particular characteristics (means).

• Next time:– The distribution of sample means

probability & the normal distribution statistics for the social sciences psychology 340 spring...

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