Download The Normal Distribution - Illinois State University Department of

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 won’t worry
about which one, since your text book
doesn’t) 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 we’ll subtract the mean from the score (find the deviation score).
X 
– 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
Reference
point

  100
X1 = 162
X2 = 57

X 
X
1 - 100 = +62
X2 - 100 = -43
Direction
Locating a score
Reference
point
Below
X1 = 162
X2 = 57


  100
X 
X
1 - 100 = +62
X2 - 100 = -43
Above
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.
Raw score
z

X 

Population mean
Population standard deviation
Transforming scores
  100
  50

z
X   


X1 = 162
X1 - 100 = +1.20
50
X2 = 57
X2 - 100 = -0.86
50
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 .
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 Zdistribution.
Properties of the z-score distribution
  100
  50
0
z
X 

transformation
50
150
 

zmean 
Xmean = 100


100 100
50
=0
Properties of the z-score distribution
  100
  50
0
z
X 

transformation
50
150
 


100 100
50
150 100

50
Xmean = 100
zmean 
=0
X+1std = 150
z1std
= +1


+1
Properties of the z-score distribution
  100
  50
z
0
 1
X 



transformation
50
150
 
100 100
50
150 100
z1std 
50
50 100
z1std 
50
zmean 
Xmean = 100
X+1std = 150
X-1std = 50
-1



=0
= +1
= -1
+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 - ) --> (Z)( ) = (X - ) --> X = (Z)( ) + 

  100
  50
0
 1
X  Z  
transformation
50

X = 70

150
-1
X = (-0.60)( 50) + 100


 +1
Z = -0.60
Let’s 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
1
(X  ) 2 / 2 2
e
It is defined by the following equation:
2 2
•
• The mean, median, and mode are all equal for this distribution.

-2
-1
0
1
2
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)
1
Y =
-2
-1
0
1
2
2
2
e
(X  ) 2 / 2 2
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.
1
Y =
-2
-1
0
1
2
2
2
e
(X  ) 2 / 2 2
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.
1
Y =
-2
-1
0
1
2
2
2
e
(X  ) 2 / 2 2
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).
1
Y =
-2
-1
0
1
2
2
2
e
(X  ) 2 / 2 2
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
• The normal distribution is often
transformed into z-scores.
• Gives the precise proportion of scores (in zscores) 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 z=0 (i.e., at the mean), the
proportion of scores to the left is .5 Hence,
mean=median.
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
34.13%
13.59%
-2
-1
0
1
2.28%
2
At z = +1: 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
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
• Steps for figuring the
percentage above or below a
particular raw or Z score:
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)
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
• Steps for figuring the
percentage above or below a
particular raw or Z score:
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:
 = 500,  = 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?
z
X 


630  500
From the table:
1.3
100
z(1.3) =.9032
So 90.32% got your
score or lower
-2
-1
That’s 9.68%
above this score

1
2
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
1.
2.
3.
4.
5.
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.
-2
-1
0
1
2
The Normal Distribution
-2 -1 0
1
2
Example: What proportion of scores falls between the mean and .2
standard deviations above the mean?
1.
2.
3.
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
4.
Identify the area below the lowest z score using the unit normal table.
For z=0, the proportion to the left = .5
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
-2 -1 0
1
2
Example 2: What proportion of scores falls between -.2 standard
deviations and -.6 standard deviations?
1.
2.
3.
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
4.
Identify the area below the lowest z score using the unit normal table.
For z=-.6, the proportion to the left = 1 - .7257 = .2743
5.
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 can’t 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
Memory
patients
Memory
treatment
No Memory
treatment
Memory 55
Test
errors
Memory 60
errors
Test
• Is the 5 error difference:
– A “real” difference due to the effect of the treatment
– Or is it just sampling error?
5 error
diff
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
Possible successful outcomes
Probability 
All possible outcomes
• 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
Possible successful outcomes
Probability 
All possible outcomes
One outcome classified as heads
1
=
Total of two outcomes
= 0.5
2
Flipping a coin example
n=2
Number of heads
What are the odds of
getting two “heads”?
2
1
1
One 2 “heads”
outcome
Four total
outcomes
= 0.25
0
This situation is known as the binomial
# of outcomes = 2n
Flipping a coin example
n=2
Number of heads
2
1
1
0
What are the odds of
getting “at least one
heads”?
Three “at least one
heads” outcome
Four total
outcomes
= 0.75
Flipping a coin example
n=3
3=
n
=
2
2
8 total outcomes
HHH
Number of heads
3
HHT
2
HTH
2
HTT
1
THH
2
THT
1
TTH
1
TTT
0
Flipping a coin example
Number of heads
3
Distribution of possible outcomes
probability
(n = 3 flips)
.4
.3
.2
.1 .125
.375 .375 .125
0 1 2 3
Number of heads
2
X
f
p
3
1
.125
2
2
1
3
3
.375
.375
1
0
1
.125
1
2
1
0
(n = 3 flips)
Can make predictions about
likelihood of outcomes based on
this distribution.
probability
Flipping a coin example
What’s the probability of
flipping three heads in a
row?
Distribution of possible outcomes
.4
.3
.2
.1 .125
.375 .375 .125
0 1 2 3
Number of heads
p = 0.125
(n = 3 flips)
Can make predictions about
likelihood of outcomes based on
this distribution.
probability
Flipping a coin example
What’s the probability of
flipping at least two heads
in three tosses?
Distribution of possible outcomes
.4
.3
.2
.1 .125
.375 .375 .125
0 1 2 3
Number of heads
p = 0.375 + 0.125 = 0.50
(n = 3 flips)
Can make predictions about
likelihood of outcomes based on
this distribution.
probability
Flipping a coin example
What’s the probability of
flipping all heads or all tails
in three tosses?
Distribution of possible outcomes
.4
.3
.2
.1 .125
.375 .375 .125
0 1 2 3
Number of heads
p = 0.125 + 0.125 = 0.25
Hypothesis testing
Distribution of possible outcomes
(of a particular sample size, n)
Can make predictions about
likelihood of outcomes based on
this distribution.
• 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)
This is the one that you test
• There are no differences between conditions (no effect of treatment)
• Research hypothesis (HA)
• Generally, not all groups are equal
– You aren’t out to prove the alternative hypothesis
• If you reject the null hypothesis, then you’re left with
support for the alternative(s) (NOT proof!)
Testing Hypotheses
• Hypothesis testing: a five step program
– Step 1: State your hypotheses
In our memory example experiment:
One -tailed
– Our theory is that the
treatment should improve
memory (fewer errors).
H0: Treatment > No Treatment
HA:Treatment < No Treatment
Testing Hypotheses
• Hypothesis testing: a five step program
– Step 1: State your hypotheses
In our memory example experiment:
direction
One -tailed
specified
– Our theory is that the
treatment should improve
memory (fewer errors).
no direction
specified
Two -tailed
– Our theory is that the
treatment has an effect on
memory.
H0: Treatment > No Treatment
H0: Treatment = No Treatment
HA:Treatment < No Treatment
HA:Treatment ≠ No Treatment
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 isn’t.
– Sometimes called “significance level” or “alpha level”
– We try to minimize this (keep it low)
• Type II error (): concluding that there isn’t an
effect, when there really is.
– Related to the Statistical Power of a test (1-)
Error types
There really
isn’t an effect
Reject
H0
Experimenter’s
conclusions
Fail to
Reject
H0
Real world (‘truth’)
H0 is
correct
H0 is
wrong
There
really is
an effect
Error types
Real world (‘truth’)
I conclude that
there is an
effect
H0 is
correct
Reject
H0
Experimenter’s
conclusions
Fail to
Reject
H0
I can’t detect
an effect
H0 is
wrong
Error types
Real world (‘truth’)
H0 is
correct
Reject
H0
Experimenter’s
conclusions
Fail to
Reject
H0
H0 is
wrong
Type I
error

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)
One
population
Two
populations
XA
the memory treatment sample are the
same as those in the population of
memory patients.
XA
they aren’t the same as those in the
population of memory patients
Performing your statistical test
• What are we doing when we test the hypotheses?
– Computing a test statistic: Generic test
Could be difference between a sample and a
population, or between different samples
observed difference
test statistic 
difference expected by chance
Based on standard error or an
estimate of the standard error
“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
– What’s large enough?
• The alpha level gives us the decision criterion
Distribution of the test statistic
-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
– What’s large enough?
• The alpha level gives us the decision criterion
Distribution of the test statistic
If test statistic is
here Reject H0
If test statistic is here
Fail to reject H0
“Generic” statistical test
• The alpha level gives us the decision criterion
Two -tailed
One -tailed
 = 0.05
Reject H0
Reject H0
0.025
split up
into the
two tails
0.025
Fail to reject H0
Reject H0
Fail to reject H0
Fail to reject H0
“Generic” statistical test
• The alpha level gives us the decision criterion
Two -tailed
One -tailed
 = 0.05
all of it in
one tail
Reject H0
Reject H0
0.05
Fail to reject H0
Reject H0
Fail to reject H0
Fail to reject H0
“Generic” statistical test
• The alpha level gives us the decision criterion
Two -tailed
One -tailed
 = 0.05
Reject H0
all of it in
one tail
Reject H0
0.05
Fail to reject H0
Reject H0
Fail to reject H0
Fail to reject H0
“Generic” statistical test
An example: One sample z-test
Memory example experiment:
• We give a n = 16 memory patients a
memory improvement treatment.
• After the treatment they have an
average score of X = 55 memory errors.
• How do they compare to the general
population of memory patients who have
 of memory errors that is
a distribution
Normal,  = 60,  = 8?
•
Step 1: State your hypotheses
H0: the memory treatment
sample are the same as
those in the population of
memory patients.
Treatment = pop = 60
HA: they aren’t the same as
those in the population of
memory patients
Treatment ≠ pop ≠ 60
“Generic” statistical test
An example: One sample z-test
H0: Treatment = pop = 60
HA: Treatment ≠ pop ≠ 60
Memory example experiment:
• We give a n = 16 memory patients a
memory improvement treatment.
• After the treatment they have an
average score of X = 55 memory errors.
• How do they compare to the general
population of memory patients who have
 of memory errors that is
a distribution
Normal,  = 60,  = 8?
•
Step 2: Set your decision
criteria
 = 0.05
One -tailed
“Generic” statistical test
An example: One sample z-test
H0: Treatment = pop = 60
HA: Treatment ≠ pop ≠ 60
Memory example experiment:
• We give a n = 16 memory patients a
memory improvement treatment.
• After the treatment they have an
average score of X = 55 memory errors.
• How do they compare to the general
population of memory patients who have
 of memory errors that is
a distribution
Normal,  = 60,  = 8?
One -tailed
•
 = 0.05
Step 3: Collect your data
“Generic” statistical test
An example: One sample z-test
H0: Treatment = pop = 60
HA: Treatment ≠ pop ≠ 60
Memory example experiment:
• We give a n = 16 memory patients a
memory improvement treatment.
• After the treatment they have an
average score of X = 55 memory errors.
• How do they compare to the general
population of memory patients who have
 of memory errors that is
a distribution
Normal,  = 60,  = 8?

 = 0.05
One -tailed
•
Step 4: Compute your test
statistics
zX 
X  X
X
= -2.5


55  60
 8



 16 
“Generic” statistical test
An example: One sample z-test
Memory example experiment:
• We give a n = 16 memory patients a
memory improvement treatment.
H0: Treatment = pop = 60
HA: Treatment ≠ pop ≠ 60
 = 0.05
One -tailed
zX  2.5
• Step 5: Make a decision
• After the treatment they have an
about your null hypothesis
average score of X = 55 memory errors.
• How do they compare to the general
population of memory patients who have 5%
 of memory errors that is
a distribution
Normal,  = 60,  = 8?
-2
-1

Reject H0
1
2
“Generic” statistical test
An example: One sample z-test
H0: Treatment = pop = 60
HA: Treatment ≠ pop ≠ 60
Memory example experiment:
• We give a n = 16 memory patients a
memory improvement treatment.
• After the treatment they have an
average score of X = 55 memory errors.
• How do they compare to the general
population of memory patients who have
 of memory errors that is
a distribution
Normal,  = 60,  = 8?
One -tailed
 = 0.05
zX  2.5
•
Step 5: Make a decision
about your null hypothesis
- Reject H0
- Support for our HA, the
evidence suggests that the
treatment decreases the
number of memory errors