Loading...

PPT – The Standard Normal Distribution PowerPoint presentation | free to download - id: 7b010a-MThjY

The Adobe Flash plugin is needed to view this content

The Standard Normal Distribution

- PSY440
- June 3, 2008

Outline of Class Period

- Article Presentation (Kristin M)
- Recap of two items from last time
- Using Excel to compute descriptive statistics
- Using SPSS to generate histograms
- Standardization (z-transformation) of scores
- The normal distribution
- Properties of the normal curve
- Standard normal distribution the unit normal

table - Intro to probability theory and hypothesis testing

Using Excel to Compute Mean SD

- Step 1 Compute mean of height with formula bar.
- Step 2 Create deviation scores by creating a

formula that subtracts the mean from each raw

score, and apply the formula to all of the cells

in a blank column next to the column of raw

scores. - Step 3 Square the deviations by creating a

formula and applying it to the cells in the next

blank column. - Step 4 Use the formula bar to add the squared

deviations, divide by (n-1) and take the square

root of the result. - Step 5 Check the result by computing the SD with

the formula bar.

Using SPSS to generate histograms

- Most common answer

- Most distinctive answer

How did this happen?

- The shape of the histogram will change depending

on the intervals used on the x axis. - For very large samples and truly continuous

variables, the shape will smooth out, but with

smaller samples, the shape can change

considerably if you change the size of the

intervals.

Make sure you are in charge of SPSS and not vice

versa!

- SPSS has default settings for many of its

operations that or may not be what you want. - You can tell SPSS how many intervals you want in

your histogram, or how large you want the

intervals to be.

Histogram with 16 intervals

In legacy dialogues, chose interactive and then

choose histogram. (see note) In chart builder,

choose histogram then choose element

properties then click on set parameters

The Z transformation

- If you know the mean and standard deviation

(sample or population we wont worry about

which one, since your text book doesnt) of a

distribution, you can convert a given score into

a Z score or standard score. This score is

informative because it tells you where that score

falls relative to other scores in the

distribution.

Locating a score

- Where is our raw score within the distribution?
- The natural choice of reference is the mean

(since it is usually easy to find). - So well subtract the mean from the score (find

the deviation score).

- The direction will be given to us by the negative

or positive sign on the deviation score - The distance is the value of the deviation score

Locating a score

X1 - 100 62

X1 162

X2 57

X2 - 100 -43

Locating a score

X1 - 100 62

X1 162

X2 57

X2 - 100 -43

Transforming a score

- The distance is the value of the deviation score
- However, this distance is measured with the units

of measurement of the score. - Convert the score to a standard (neutral) score.

In this case a z-score.

Transforming scores

- A z-score specifies the precise location of each

X value within a distribution. - Direction The sign of the z-score ( or -)

signifies whether the score is above the mean or

below the mean. - Distance The numerical value of the z-score

specifies the distance from the mean by counting

the number of standard deviations between X and ?.

X1 162

X2 57

Transforming a distribution

- We can transform all of the scores in a

distribution - We can transform any all observations to

z-scores if we know the distribution mean and

standard deviation. - We call this transformed distribution a

standardized distribution. - Standardized distributions are used to make

dissimilar distributions comparable. - e.g., your height and weight
- One of the most common standardized distributions

is the Z-distribution.

Properties of the z-score distribution

0

Properties of the z-score distribution

150

50

0

Xmean 100

1

Properties of the z-score distribution

150

50

1

0

Xmean 100

1

X1std 150

-1

Properties of the z-score distribution

- Shape - the shape of the z-score distribution

will be exactly the same as the original

distribution of raw scores. Every score stays in

the exact same position relative to every other

score in the distribution. - Mean - when raw scores are transformed into

z-scores, the mean will always 0. - The standard deviation - when any distribution

of raw scores is transformed into z-scores the

standard deviation will always 1.

From z to raw score

- We can also transform a z-score back into a raw

score if we know the mean and standard deviation

information of the original distribution. - Z (X - ?) --gt (Z)( ?) (X - ?) --gt X (Z)(

?) ? - ?

X (-0.60)( 50) 100

X 70

Lets try it with our data

- To transform data on height into standard scores,

use the formula bar in excel to subtract the mean

and divide by the standard deviation. - Can also choose standardize (x,mean,sd)
- Show with shoe size
- Observe how height and shoe size can be more

easily compared with standard (z) scores

Z-transformations with SPSS

- You can also do this in SPSS.
- Use Analyze . Descriptive Statistics.

Descriptives . - Check the box that says save standardized values

as variables.

The Normal Distribution

- Normal distribution

The 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
- The mean, median, and mode are all equal for this

distribution.

The Normal Distribution

- This equation provides x and y coordinates on the

graph of the frequency distribution. You can plug

a given value of x into the formula to find the

corresponding y coordinate. Since the function

describes a symmetrical curve, note that the same

y (height) is given by two values of x

(representing two scores an equal distance above

and below the mean)

Y

The Normal Distribution

- As the distance between the observed score (x)

and the mean increases, the value of the

expression (i.e., the y coordinate) decreases.

Thus the frequency of observed scores that are

very high or very low relative to the mean, is

low, and as the difference between the observed

score and the mean gets very large, the frequency

approaches 0.

Y

The Normal Distribution

- As the distance between the observed score (x)

and the mean decreases (i.e., as the observed

value approaches the mean), the value of the

expression (i.e., the y coordinate) increases. - The maximum value of y (i.e., the mode, or the

peak in the curve) is reached when the observed

score equals the mean hence mean equals mode.

Y

The Normal Distribution

- The integral of the function gives the area under

the curve (remember this if you took calculus?) - The distribution is asymptotic, meaning that

there is no closed solution for the integral. - It is possible to calculate the proportion of the

area under the curve represented by a range of x

values (e.g., for x values between -1 and 1).

Y

The Unit Normal Table

- The normal distribution is often transformed into

z-scores.

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

- Gives the precise proportion of scores (in

z-scores) between the mean (Z score of 0) and any

other Z score in a Normal distribution - 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 .00 column corresponds to column (3) in Table

B of your textbook. - Note that for z0 (i.e., at the mean), the

proportion of scores to the left is .5 Hence,

meanmedian.

Using the Unit Normal Table

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

50-34-14 rule

Similar to the 68-95-99 rule

1

2

-1

-2

0

15.87 (13.59 and 2.28) of the

scores are to the right of the score 100-15.87

84.13 to the left

Using the Unit Normal Table

- Steps for figuring the percentage above or below

a particular raw or Z score

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

- 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 areas

percentage (using 50-34-14 rule)

Using the Unit Normal Table

- Steps for figuring the percentage above or below

a particular raw or Z score

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

- 4. Find exact percentage using unit normal table
- 5. If needed, subtract percentage from 100.
- 6. Check the exact percentage is within the

range of the estimate from Step 3

SAT Example problems

- The population parameters for the SAT are
- m 500, s 100, and it is Normally distributed

Suppose that you got a 630 on the SAT. What

percent of the people who take the SAT get your

score or lower?

From the table z(1.3) .9032

The Normal Distribution

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

percentages - 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 - 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

The Normal Distribution

- Example What z score is at the 75th percentile

(at or above 75 of the scores)? - 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 (between .5 and 1) - 3. Find the exact Z score using the unit normal

table (a little less than .7) - 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 using mean and standard

deviation info.

The Normal Distribution

- Finding the proportion of scores falling between

two observed scores - Convert each score to a z score
- Draw a graph of the normal distribution and shade

out the area to be identified. - Identify the area below the highest z score using

the unit normal table. - Identify the area below the lowest z score using

the unit normal table. - Subtract step 4 from step 3. This is the

proportion of scores that falls between the two

observed scores.

The Normal Distribution

- Example What proportion of scores falls between

the mean and .2 standard deviations above the

mean? - Convert each score to a z score (mean 0, other

score .2) - Draw a graph of the normal distribution and shade

out the area to be identified. - Identify the area below the highest z score using

the unit normal table - For z.2, the proportion to the left .5793
- Identify the area below the lowest z score using

the unit normal table. - For z0, the proportion to the left .5
- Subtract step 4 from step 3
- .5793 - .5 .0793
- About 8 of the observations fall between the

mean and .2 SD.

The Normal Distribution

- Example 2 What proportion of scores falls

between -.2 standard deviations and -.6 standard

deviations? - Convert each score to a z score (-.2 and -.6)
- Draw a graph of the normal distribution and shade

out the area to be identified. - Identify the area below the highest z score using

the unit normal table - For z-.2, the proportion to the left 1 - .5793

.4207 - Identify the area below the lowest z score using

the unit normal table. - For z-.6, the proportion to the left 1 -

.7257 .2743 - Subtract step 4 from step 3
- .4207 - .2743 .1464
- About 15 of the observations fall between -.2

and -.6 SD.

Hypothesis testing

- Example Testing the effectiveness of a new

memory treatment for patients with memory problems

- Our pharmaceutical company develops a new drug

treatment that is designed to help patients with

impaired memories. - Before we market the drug we want to see if it

works. - The drug is designed to work on all memory

patients, but we cant test them all (the

population). - So we decide to use a sample and conduct the

following experiment. - Based on the results from the sample we will make

conclusions about the population.

Hypothesis testing

- Example Testing the effectiveness of a new

memory treatment for patients with memory problems

55 errors

60 errors

- Is the 5 error difference
- A real difference due to the effect of the

treatment - Or is it just sampling error?

Testing Hypotheses

- Hypothesis testing
- Procedure for deciding whether the outcome of a

study (results for a sample) support a particular

theory (which is thought to apply to a

population) - 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 - If this probability is low, scenario of no effect

is rejected and the theory behind the

experimental procedure is supported

Basics of Probability

- Probability
- Expected relative frequency of a particular

outcome - Outcome
- The result of an experiment

Flipping a coin example

What are the odds of getting a heads?

n 1 flip

Flipping a coin example

What are the odds of getting two heads?

Number of heads

n 2

2

1

1

0

of outcomes 2n

This situation is known as the binomial

Flipping a coin example

What are the odds of getting at least one heads?

Number of heads

n 2

2

1

1

0

Flipping a coin example

Number of heads

n 3

HHH

3

HHT

2

HTH

2

HTT

1

2

THH

THT

1

TTH

1

TTT

0

23 8 total outcomes

2n

Flipping a coin example

Number of heads

Distribution of possible outcomes (n 3 flips)

3

2

X f p

3 1 .125

2 3 .375

1 3 .375

0 1 .125

2

1

2

1

1

0

Flipping a coin example

Can make predictions about likelihood of outcomes

based on this distribution.

Distribution of possible outcomes (n 3 flips)

.4

Whats the probability of flipping three heads in

a row?

.3

probability

.2

.1

p 0.125

.125

.125

.375

.375

0

1

2

3

Number of heads

Flipping a coin example

Can make predictions about likelihood of outcomes

based on this distribution.

Distribution of possible outcomes (n 3 flips)

.4

Whats the probability of flipping at least two

heads in three tosses?

.3

probability

.2

.1

p 0.375 0.125 0.50

.125

.125

.375

.375

0

1

2

3

Number of heads

Flipping a coin example

Can make predictions about likelihood of outcomes

based on this distribution.

Distribution of possible outcomes (n 3 flips)

.4

Whats the probability of flipping all heads or

all tails in three tosses?

.3

probability

.2

.1

p 0.125 0.125 0.25

.125

.125

.375

.375

0

1

2

3

Number of heads

Hypothesis testing

Can make predictions about likelihood of outcomes

based on this distribution.

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

Inferential statistics

- Hypothesis testing
- 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 - If this probability is low, scenario of no effect

is rejected and the theory behind the

experimental procedure is supported

- A five step program

- Step 1 State your hypotheses
- Step 2 Set your decision criteria
- Step 3 Collect your data
- Step 4 Compute your test statistics
- Step 5 Make a decision about your null hypothesis

Hypothesis testing

- Hypothesis testing a five step program

- Step 1 State your hypotheses as a research

hypothesis and a null hypothesis about the

populations - Null hypothesis (H0)
- Research hypothesis (HA)

- There are no differences between conditions (no

effect of treatment)

- Generally, not all groups are equal

- You arent out to prove the alternative

hypothesis - If you reject the null hypothesis, then youre

left with support for the alternative(s) (NOT

proof!)

Testing Hypotheses

- Hypothesis testing a five step program

- Step 1 State your hypotheses

One -tailed

- Our theory is that the treatment should improve

memory (fewer errors).

mTreatment gt mNo Treatment

H0

mTreatment lt mNo Treatment

HA

Testing Hypotheses

- Hypothesis testing a five step program

- Step 1 State your hypotheses

One -tailed

Two -tailed

- Our theory is that the treatment should improve

memory (fewer errors).

- Our theory is that the treatment has an effect

on memory.

mTreatment gt mNo Treatment

H0

mTreatment mNo Treatment

H0

mTreatment lt mNo Treatment

HA

mTreatment ? mNo Treatment

HA

One-Tailed and Two-Tailed Hypothesis Tests

- Directional hypotheses
- One-tailed test
- Nondirectional hypotheses
- Two-tailed test

Testing Hypotheses

- Hypothesis testing a five step program

- Step 1 State your hypotheses
- Step 2 Set your decision criteria

- Your alpha (?) level will be your guide for when

to reject or fail to reject the null hypothesis. - Based on the probability of making making an

certain type of error

Testing Hypotheses

- Hypothesis testing a five step program

- Step 1 State your hypotheses
- Step 2 Set your decision criteria
- Step 3 Collect your data

Testing Hypotheses

- Hypothesis testing a five step program

- Step 1 State your hypotheses
- Step 2 Set your decision criteria
- Step 3 Collect your data
- Step 4 Compute your test statistics

- Descriptive statistics (means, standard

deviations, etc.) - Inferential statistics (z-test, t-tests, ANOVAs,

etc.)

Testing Hypotheses

- Hypothesis testing a five step program

- Step 1 State your hypotheses
- Step 2 Set your decision criteria
- Step 3 Collect your data
- Step 4 Compute your test statistics
- Step 5 Make a decision about your null hypothesis

- Based on the outcomes of the statistical tests

researchers will either - Reject the null hypothesis
- Fail to reject the null hypothesis
- This could be correct conclusion or the incorrect

conclusion

Error types

- Type I error (?) concluding that there is a

difference between groups (an effect) when

there really isnt. - Sometimes called significance level or alpha

level - We try to minimize this (keep it low)
- Type II error (?) concluding that there isnt an

effect, when there really is. - Related to the Statistical Power of a test (1-?)

Error types

Real world (truth)

H0 is correct

H0 is wrong

Reject H0

Experimenters conclusions

Fail to Reject H0

Error types

Real world (truth)

H0 is correct

H0 is wrong

Reject H0

Experimenters conclusions

Fail to Reject H0

Error types

Real world (truth)

H0 is correct

H0 is wrong

Type I error

Reject H0

Experimenters conclusions

Fail to Reject H0

Type II error

Performing your statistical test

- What are we doing when we test the hypotheses?

Real world (truth)

H0 is true (no treatment effect)

H0 is false (is a treatment effect)

Performing your statistical test

- What are we doing when we test the hypotheses?
- Computing a test statistic Generic test

Generic statistical test

- The generic test statistic distribution (think of

this as the distribution of sample means) - To reject the H0, you want a computed test

statistics that is large - Whats large enough?
- The alpha level gives us the decision criterion

?-level determines where these boundaries go

Generic statistical test

- The generic test statistic distribution (think of

this as the distribution of sample means) - To reject the H0, you want a computed test

statistics that is large - Whats large enough?
- The alpha level gives us the decision criterion

Generic statistical test

- The alpha level gives us the decision criterion

One -tailed

Two -tailed

Generic statistical test

- The alpha level gives us the decision criterion

One -tailed

Two -tailed

Reject H0

Reject H0

Fail to reject H0

Fail to reject H0

Generic statistical test

- The alpha level gives us the decision criterion

One -tailed

Two -tailed

all of it in one tail

Reject H0

Reject H0

Fail to reject H0

Fail to reject H0

Generic statistical test

- An example One sample z-test

- Step 1 State your hypotheses

- We give a n 16 memory patients a memory

improvement treatment.

mTreatment mpop 60

- How do they compare to the general population

of memory patients who have a distribution of

memory errors that is Normal, m 60, s 8?

mTreatment ? mpop ? 60

Generic statistical test

- An example One sample z-test

H0 mTreatment mpop 60

HA mTreatment ? mpop ? 60

- We give a n 16 memory patients a memory

improvement treatment.

- Step 2 Set your decision criteria

a 0.05

- How do they compare to the general population

of memory patients who have a distribution of

memory errors that is Normal, m 60, s 8?

Generic statistical test

- An example One sample z-test

H0 mTreatment mpop 60

HA mTreatment ? mpop ? 60

- We give a n 16 memory patients a memory

improvement treatment.

a 0.05

One -tailed

- Step 3 Collect your data

- How do they compare to the general population

of memory patients who have a distribution of

memory errors that is Normal, m 60, s 8?

Generic statistical test

- An example One sample z-test

H0 mTreatment mpop 60

HA mTreatment ? mpop ? 60

- We give a n 16 memory patients a memory

improvement treatment.

a 0.05

One -tailed

- Step 4 Compute your test statistics

- How do they compare to the general population

of memory patients who have a distribution of

memory errors that is Normal, m 60, s 8?

-2.5

Generic statistical test

- An example One sample z-test

H0 mTreatment mpop 60

HA mTreatment ? mpop ? 60

- We give a n 16 memory patients a memory

improvement treatment.

a 0.05

One -tailed

- Step 5 Make a decision about your null hypothesis

- How do they compare to the general population

of memory patients who have a distribution of

memory errors that is Normal, m 60, s 8?

Reject H0

Generic statistical test

- An example One sample z-test

H0 mTreatment mpop 60

HA mTreatment ? mpop ? 60

- We give a n 16 memory patients a memory

improvement treatment.

a 0.05

One -tailed

- Step 5 Make a decision about your null hypothesis

- Reject H0

- How do they compare to the general population

of memory patients who have a distribution of

memory errors that is Normal, m 60, s 8?

- Support for our HA, the evidence suggests that

the treatment decreases the number of memory

errors