Outline:
 probabilty and causality
 hypothesis testing and probability
differences between groups
types of explanations
 betweensubjects ttest
formula as a concept
actual formula
example
 writing
 relation to H0 and HA
stat hypotheses
research hypotheses
 the t distribution
tails
distribution and df
 types of errors
 return to the conceptual formula and predicting ttests
Probabilty & Causality
I now want to begin to introduce you to probability as it will be used
in hypothesis testing. We’ll start with the very concrete approach.
p = event/number of possibilities
Start with the cards example and start out with the card on the top
of the deck.
What is the probability that the first card is an Ace; a face
card; a heart; the ace of spades
(Convert to decimals: .077; .231; .250; .019)
If I turn over the ace of spades as the top card, why is it there?
*** There are two explanations  chance by shuffling deck or caused
by cheating
*** How do you decide??? (Based on How UNLIKELY)
Now what are the chances that I will turn over another ace?
Sidenote  As long as we are on this topic  let us dismiss
the gambler’s fallacy. What is it? Before I turn over any cards,
what are the chances that I will turn over two aces? 4/52 * 3/51
= .0045; but once I have turned over the first ace, if it happened, what
are the chances that the second card will be an ace? 3/51 = .059.
Before you start the night, the chances are that you will have
some good hands and some bad hands, but once you are into the night, what
is past is past  on each next hand, regardless of previous hands, the
odds of getting a good hand remain the same: low!
Probability and Hypothesis testing
Now what does this have to do with experiments?
Think back to our experiment design:
IV  DV  
Pop >  Sample >  A  Level 1  measure  Compare groups 
B  Level 2  measure 
Differences between groups: There will most likely be some difference when we are done. But the question is to what do we attribute that difference?
Types of explanations: Explanations are chance or caused;
same as with the cards. When were you willing to accuse me of stacking
the deck? When the odds of it occurring by chance are very low.
Caused: due to experimental manipulation
Chance: due to random sampling & measurement
noise
Think of this as shuffling the subjects to the groups.
(Assume no TRUE Difference)
When difference is so large as to be very unlikely, you then claim that it was most likely caused by the experimental manipulation.
This is the purpose of hypothesis testing statistics  to tell
you the probability that the difference you observed occurred by chance.
That allows you, the researcher, to decide if you want to attribute
the outcome to the manipulation or not.
Between subjects ttest
Formula as a concept
Diff
B/T groups
Variability W/I groups
You want to see if the difference between the groups is large compared to what you migh expect based on the chance variability (which is the within group variability).
Actual formula
Diff B/T the means of the two groups = M_{1}
 M_{2}
a combined measure of variability = Combined SEM (Standard
Error of the Mean; S / sqrt n)
When n_{1} = n_{2}
t = (M_{1}  M_{2}) / sqrt (s_{1}^{2}/n_{1} + s_{2}^{2}/n_{2})
When n_{1} does not equal n_{2}
t = (M_{1}  M_{2}) / sqrt [ ( (SS_{1}
+ SS_{2} ) / (n_{1} + n_{2}  2) ) * (1/ n_{1}
+ 1/ n_{2}) ]
Example
Want to know how important having information in a coherent organization
is for memory of the information. Schema Theory suggests that it
is important  information that conforms to prior knowledge is easier
to process and store. New Associations Theory suggests new links
being formed so order doesn’t matter.
Take the story with 80 idea units as basic.
IV: organization
L1: correct order
L2: random order
DV: number of idea units recalled after five minutes
Between Subjects ttest
Presntation Order
i  Correct Order  Random Order 
1  53  43 
2  58  44 
3  60  45 
4  60  47 
5  61  49 
6  63  50 
7  64  51 
8  65  52 
9  65  52 
10  66  53 
11  66  54 
12  66  54 
13  67  54 
14  68  56 
15  68  56 
16  69  57 
17  71  58 
18  72  61 
19  75  62 
20  78  70 
Sum X_{i}  1315  1068 
n  20  20 
M  65.75  53.40 
s  5.84  6.51 
t =
65.75  53.40
sqrt [(5.839)^{2}/20
+ (6.508)^{2}/20]
t = 6.317
df = df1 + df2 = N1  1 + N2  1
= NT  2
= 38
t (38) = 6.317, p < .002
Writing
1. What do we know? (brainstorming)
2. How shall we order things? (outlining)
3. Writing (1st draft)
A betweensubjects ttest revealed that subjects to whom the
story was presented in the correct order recalled more idea units than
subjects to whom the story was presented in random order, t (38) = 6.317,
p < .002.
 noted the stat
 used an active verb: found, revealed, indicated
 stated which level was higher than which other level of IV
 used OpDef of DV
 gave supportive information
 can break this apart
stat in a sentence
that there was a difference in a sentence
and the direction of difference in a sentence
The mean number of idea units recalled by the correct order and random
order groups was 65.75 and 53.40, respectively. The standard deviation
for the correct order and random order groups was 5.84 and 6.51, respectively.
Relation to H0 and HA
Now you probably remember hearing something about a null hypothesis
and an alternate hypothesis and cutoff levels when doing stats.
Let me talk a little bit about these concepts.
Stat hypotheses
Null and Alternative hypotheses are what I’ll refer to as stat
hypotheses. They have to do with things about your ttest.
H_{0}: M_{1} = M_{2}
H_{A}: M_{1} not equal M_{2}
Cutoff levels are for you to know when you reject the null and turn to the alternative. It is what I was talking about: chance vs. caused: is the probability low enough to assume caused. There is not just one cutoff level.
Research hypotheses
Research hypothesis have to do with our theory or theories.
Come straight from our view of science.
Observe > Theory > Hypothesis
> Observe.
Thus they are in the language of the theory (usually English).
For this example:
Organization matters theory predicts that the ordered
group will recall more than the random group.
The learning new association theory predicts that
there will be no difference.
Look at the ttest as disproving 2nd and supporting the 1st, because
the p is so low.
What if it hadn’t have worked? p = .24. Supports
view of no difference, but doesn’t disprove the other.
The t distribution
t scores are normally distributed around zero. Thus numbers
far away from zero occur infrequently (or are low probability). How
far away to be low probability?
Tails
t distributions also happen to have two tails.
Let’s talk about why.
Because either mean could be larger.
If you know which way you expect things to go, then you can do a onetailed
test. Otherwise you need to do a twotailed test.
1tailed puts all of probability (like .05) in one tail
2tailed divides prob between the trails (.025 in each)
1tailed is easier to find something in the direction of interest
2tailed allows you to look both ways
What about our example?
Distribution and df
That depends on the df. The t distribution is actually
a family of distributions. The higher the df, the tighter the distribution
and thus the smaller the numbers are that make something an unlikely t
score.
Types of errors
State of

World  
Outcome of Experiment 




Missed it 

False Alarm p level 

Return to the conceptual formula and predicting ttests
Formula as a concept
Diff
B/T groups
Variability W/I groups
Actual formula
Diff B/T the means of the two groups = M_{1}
 M_{2}
a combined measure of variability = Combined SEM (Standard
Error of the Mean; S / sqrt n)
holding other things constant, what happens to t?
if diff b/t means increases
variability increases
n increases
Rule of thumb for predicting ttests
If the difference between the means is greater than the average of
the standard deviations and if N is ok, then the probability associated
with t will be < .05.
ok N is 1015 per group
large N compensates for diff b/t means being slightly smaller
than sd
small N means diff b/t means has to be much larger than sd to
compensate
Examples












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Prediction?