Statistics for the Social Sciences
Psychology 340Fall 2006
Hypothesis testing
Statistics for the Social Sciences
Outline (for week)
• Review of: – Basic probability– Normal distribution– Hypothesis testing framework
• Stating hypotheses• General test statistic and test statistic distributions
• When to reject or fail to reject
Statistics for the Social Sciences
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.
Statistics for the Social Sciences
Hypothesis testing
• Example: Testing the effectiveness of a new memory treatment for patients with memory problems
Memory treatment
No Memorytreatment
Memory patients
MemoryTest
MemoryTest
55 errors60 errors
5 error diff
•Is the 5 error difference: – A “real” difference due to the effect of the treatment
– Or is it just sampling error?
Statistics for the Social Sciences
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
Statistics for the 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
Statistics for the 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
Statistics for the Social Sciences
Flipping a coin example
What are the odds of getting two “heads”?
Number of heads
2
1
1
0
One 2 “heads” outcomeFour total outcomes
=0.25
This situation is known as the binomial# of outcomes = 2n
n = 2
Statistics for the Social Sciences
Flipping a coin example
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
Statistics for the Social Sciences
Flipping a coin example
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
Statistics for the Social Sciences
Flipping a coin example
Number of heads3
2
1
0
2
2
1
1
X f p
3 1 .125
2 3 .375
1 3 .375
0 1 .125
Number of heads0 1 2 3
.1
.2
.3
.4
probability
.125 .125.375.375
Distribution of possible outcomes(n = 3 flips)
Statistics for the Social Sciences
Flipping a coin example
Number of heads0 1 2 3
.1
.2
.3
.4
probability
What’s the probability of flipping three heads in a row?
.125 .125.375.375 p = 0.125
Distribution of possible outcomes(n = 3 flips)
Can make predictions about likelihood of outcomes based on this distribution.
Statistics for the Social Sciences
Flipping a coin example
Number of heads0 1 2 3
.1
.2
.3
.4
probability
What’s the probability of flipping at least two heads in three tosses?
.125 .125.375.375 p = 0.375 + 0.125 = 0.50
Can make predictions about likelihood of outcomes based on this distribution.
Distribution of possible outcomes(n = 3 flips)
Statistics for the Social Sciences
Flipping a coin example
Number of heads0 1 2 3
.1
.2
.3
.4
probability
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
Can make predictions about likelihood of outcomes based on this distribution.
Distribution of possible outcomes(n = 3 flips)
Statistics for the Social Sciences
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
Statistics for the Social Sciences
The Normal Distribution
• The distribution of days before and after due date (bin width = 4 days).
0 14-14Days before and after due date
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The Normal Distribution
• Normal distribution
Statistics for the Social Sciences
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:
€
1
2πσ 2e−(X −μ )2 / 2σ 2
1 2-1-2 0
Statistics for the Social Sciences
The Unit Normal Table
z .00 .01
-3.4-3.3::0::
1.0::
3.33.4
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 normal distribution is often transformed into z-scores.
Statistics for the Social Sciences
Using the Unit Normal Table
z .00 .01
-3.4-3.3::0::
1.0::
3.33.4
0.0003
0.0005::
0.5000::
0.8413::
0.9995
0.9997
0.0003
0.0005::
0.5040::
0.8438::
0.9995
0.9997
15.87% (13.59% and 2.28%) of the scores are to the right of the score100%-15.87% = 84.13% to the left
At z = +1:
13.59%2.28%
34.13%
50%-34%-14% rule
1 2-1-2 0
Similar to the 68%-95%-99% rule
Statistics for the Social Sciences
Using the Unit Normal Table
z .00 .01
-3.4-3.3::0::
1.0::
3.33.4
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 area’s percentage (using 50%-34%-14% rule)
• Steps for figuring the percentage above of below a particular raw or Z score:
Statistics for the Social Sciences
Using the Unit Normal Table
z .00 .01
-3.4-3.3::0::
1.0::
3.33.4
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 table5. If needed, add or subtract 50% from this percentage6. Check the exact percentage is within the range of the estimate from Step 3
• Steps for figuring the percentage above of below a particular raw or Z score:
Statistics for the 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: = 500, = 100, and it is Normally distributed
€
z =X − μ
σ=
630 − 500
100=1.3 From the
table:
z(1.3) =.0968
-1-2 1 2
That’s 9.68% above this score
So 90.32% got your score or worse
Statistics for the Social Sciences
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 starts3. Find the exact Z score using the unit normal table4. Check that your Z score is similar to the rough estimate from Step 25. If you want to find a raw score, change it from the Z score
Statistics for the Social Sciences
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
• 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
– A five step program
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– Step 1: State your hypotheses: as a research hypothesis and a null hypothesis about the populations• Null hypothesis (H0)
• Research hypothesis (HA)
Hypothesis testing
• There are no differences between conditions (no effect of treatment)
• Generally, not all groups are equal
This is the one that you test
• Hypothesis testing: a five step program
– 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!)
Statistics for the Social Sciences
In our memory example experiment:
Testing Hypotheses
Treatment > No Treatment
Treatment < No
Treatment
H0
:HA:
– Our theory is that the treatment should improve memory (fewer errors).
– Step 1: State your hypotheses
• Hypothesis testing: a five step program
One -tailed
Statistics for the Social Sciences
In our memory example experiment:
Testing Hypotheses
Treatment > No Treatment
Treatment < No
Treatment
H0
:HA:
– Our theory is that the treatment should improve memory (fewer errors).
– Step 1: State your hypotheses
• Hypothesis testing: a five step program
Treatment = No Treatment
Treatment ≠ No
Treatment
H0
:HA:
– Our theory is that the treatment has an effect on memory.
One -tailed Two -tailedno directionspecifieddirection
specified
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One-Tailed and Two-Tailed Hypothesis Tests
• Directional hypotheses– One-tailed test
• Nondirectional hypotheses– Two-tailed test
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Testing Hypotheses
– Step 1: State your hypotheses– Step 2: Set your decision criteria
• Hypothesis testing: a five step program
• 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
Statistics for the Social Sciences
Testing Hypotheses
– Step 1: State your hypotheses– Step 2: Set your decision criteria– Step 3: Collect your data
• Hypothesis testing: a five step program
Statistics for the Social Sciences
Testing Hypotheses
– Step 1: State your hypotheses– Step 2: Set your decision criteria– Step 3: Collect your data – Step 4: Compute your test statistics
• Hypothesis testing: a five step program
• Descriptive statistics (means, standard deviations, etc.)• Inferential statistics (z-test, t-tests, ANOVAs, etc.)
Statistics for the Social Sciences
Testing Hypotheses
– 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: a five step program
• 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
Statistics for the Social Sciences
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-)
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Error types
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
There really isn’t an effect
There really isan effect
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Error types
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
I conclude that there is an effect
I can’t detect an effect
Statistics for the Social Sciences
Error types
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error Type
II error
α
β
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Performing your statistical test
H0: is true (no treatment effect) H0: is false (is a treatment effect)
Two populations
One population
• What are we doing when we test the hypotheses?
Real world (‘truth’)
XA
they aren’t the same as those in the population of memory patients
XA
the memory treatment sample are the same as those in the population of memory patients.
Statistics for the Social Sciences
Performing your statistical test
• What are we doing when we test the hypotheses?– Computing a test statistic: Generic test
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test statistic =observed difference
difference expected by chance
Could be difference between a sample and a population, or between different samples
Based on standard error or an estimate
of the standard error
Statistics for the Social Sciences
“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 criterionDistribution of the test statistic
-level determines where these boundaries go
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“Generic” statistical test
If test statistic is here Reject H0
If test statistic is here Fail to reject H0
Distribution of the test statistic
• 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
Statistics for the Social Sciences
“Generic” statistical test
Reject H0
Fail to reject H0
• The alpha level gives us the decision criterion
One -tailedTwo -tailedReject H0
Fail to reject H0
Reject H0
Fail to reject H0
= 0.05
0.025
0.025split up into the two tails
Statistics for the Social Sciences
“Generic” statistical test
Reject H0
Fail to reject H0
• The alpha level gives us the decision criterion
One -tailedTwo -tailedReject H0
Fail to reject H0
Reject H0
Fail to reject H0
= 0.050.05
all of it in one tail
Statistics for the Social Sciences
“Generic” statistical test
Reject H0
Fail to reject H0
• The alpha level gives us the decision criterion
One -tailedTwo -tailedReject H0
Fail to reject H0
Reject H0
Fail to reject H0
= 0.05
0.05
all of it in one tail
Statistics for the Social Sciences
“Generic” statistical test
An example: One sample z-test
Memory example experiment:• We give a n = 16 memory patients a memory improvement treatment.
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60, = 8?
• After the treatment they have an average score of = 55 memory errors.
€
X
• Step 1: State your hypotheses
H0
:the memory treatment sample are the same as those in the population of memory patients.HA: they aren’t the same as those in the population of memory patients
Treatment > pop > 60
Treatment < pop < 60
Statistics for the Social Sciences
“Generic” statistical test
An example: One sample z-test
Memory example experiment:• We give a n = 16 memory patients a memory improvement treatment.
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60, = 8?
• After the treatment they have an average score of = 55 memory errors.
€
X
• Step 2: Set your decision criteria
= 0.05One -tailed
H0: Treatment > pop > 60 HA: Treatment < pop < 60
Statistics for the Social Sciences
“Generic” statistical test
An example: One sample z-test
Memory example experiment:• We give a n = 16 memory patients a memory improvement treatment.
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60, = 8?
• After the treatment they have an average score of = 55 memory errors.
€
X
= 0.05One -tailed
• Step 3: Collect your data
H0: Treatment > pop > 60 HA: Treatment < pop < 60
Statistics for the Social Sciences
“Generic” statistical test
An example: One sample z-test
Memory example experiment:• We give a n = 16 memory patients a memory improvement treatment.
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60, = 8?
• After the treatment they have an average score of = 55 memory errors.
€
X
= 0.05One -tailed• Step 4: Compute your
test statistics
€
zX
=X − μ
X
σX
€
=55 − 60
816
⎛ ⎝ ⎜
⎞ ⎠ ⎟
= -2.5
H0: Treatment > pop > 60 HA: Treatment < pop < 60
Statistics for the Social Sciences
“Generic” statistical test
An example: One sample z-test
Memory example experiment:• We give a n = 16 memory patients a memory improvement treatment.
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60, = 8?
• After the treatment they have an average score of = 55 memory errors.
€
X
= 0.05One -tailed
€
zX
= −2.5
• Step 5: Make a decision about your null hypothesis
-1-2 1 2
5%
Reject H0
H0: Treatment > pop > 60 HA: Treatment < pop < 60
Statistics for the Social Sciences
“Generic” statistical test
An example: One sample z-test
Memory example experiment:• We give a n = 16 memory patients a memory improvement treatment.
• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, = 60, = 8?
• After the treatment they have an average score of = 55 memory errors.
€
X
= 0.05One -tailed
€
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
H0: Treatment > pop > 60 HA: Treatment < pop < 60