SPSS Problem and slides
Is this quarter fair?
• How could you determine this?
• You assume that flipping the coin a large number of times would result in heads half the time (i.e., it has a .50 probability)
Is this quarter fair?
• Say you flip it 100 times
• 52 times it is a head
• Not exactly 50, but its close– probably due to random error
Is this quarter fair?
• What if you got 65 heads?
• 70?
• 95?
• At what point is the discrepancy from the expected becoming too great to attribute to chance?
Basic logic of research
Start with two equivalent groups of subjects
D ep en d en t V ariab leIf p e rson lives
E xp erim en ta l G rou pG ive m ed ica tion
S u b jec ts
D ep en d en t V ariab leIf p e rson lives
C on tro l G rou pD o n o t g ive m ed ica tion
S u b jec ts
Treat them alike except for one thing
D ep en d en t V ariab leIf p e rson lives
E xp erim en ta l G rou pG ive m ed ica tion
S u b jec ts
D ep en d en t V ariab leIf p e rson lives
C on tro l G rou pD o n o t g ive m ed ica tion
S u b jec ts
See if both groups are different at the end
D ep en d en t V ariab leIf p e rson lives
E xp erim en ta l G rou pG ive m ed ica tion
S u b jec ts
D ep en d en t V ariab leIf p e rson lives
C on tro l G rou pD o n o t g ive m ed ica tion
S u b jec ts
Or – Single Group
Subjects
Give Treatment -- Prozac
Dependent VariableHappiness
Do something
Subjects
Give Treatment -- Prozac
Dependent VariableHappiness
Measure DV
Subjects
Give Treatment -- Prozac
Dependent VariableHappiness
Compare Group to Population
Subjects
Give Treatment -- Prozac
Dependent VariableHappiness
Population Happiness
Score
Example• You randomly select 100 college students
living in a dorm
• They complete a happiness measure– (1 = unhappy; 4 = neutral; 7 = happy)
• You wonder if the mean score of students living in a dorm is different than the population happiness score (M = 4)
The Theory of Hypothesis Testing
• Data are ambiguous
• Is a difference due to chance?– Sampling error
Population
• You are interested in the average self-esteem in a population of 40 people
• Self-esteem test scores range from 1 to 10.
Population Scores
• 1,1,1,1• 2,2,2,2• 3,3,3,3• 4,4,4,4• 5,5,5,5
• 6,6,6,6• 7,7,7,7• 8,8,8,8• 9,9,9,9• 10,10,10,10
Histogram
012345
678910
1 2 3 4 5 6 7 8 9 10
What is the average self-esteem score of this population?
• Population mean = 5.5
• Population SD = 2.87
• What if you wanted to estimate this population mean from a sample?
What if. . . .
• Randomly select 5 people and find the average score
Group Activity
• Why isn’t the average score the same as the population score?
• When you use a sample there is always some degree of uncertainty!
• We can measure this uncertainty with a sampling distribution of the mean
EXCEL
INTERNET EXAMPLE
• http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
Sampling Distribution of the Mean
• Notice: The sampling distribution is centered around the population mean!
• Notice: The sampling distribution of the mean looks like a normal curve!– This is true even though the distribution of
scores was NOT a normal distribution
Central Limit Theorem
For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution a N (sample size) get larger. Furthermore, the sampling distribution of the mean will have a mean equal to and a standard deviation equal to / N
Sampling Distribution
• Tells you the probability of a particular sample mean occurring for a specific population
Sampling Distribution
• You are interested in if your new Self-esteem training course worked.
• The 5 people in your course had a mean self-esteem score of 5.5
Sampling Distribution
• Did it work?– How many times would we expect a sample
mean to be 5.5 or greater?• Theoretical vs. empirical
– 5,000 random samples yielded 2,501 with means of 5.5 or greater
– Thus p = .5002 of this happening
Sampling Distribution
5.5
2,499 2,501
P = .4998 P =.5002
Sampling Distribution
• You are interested in if your new Self-esteem training course worked.
• The 5 people in your course had a mean self-esteem score of 5.8
Sampling Distribution
• Did it work?– How many times would we expect a sample
mean to be 5.8 or greater?
– 5,000 random samples yielded 2,050 with means of 5.8 or greater
– Thus p = .41 of this happening
Sampling Distribution
5.8
2,700 2,300
P = .59 P =.41
Sampling Distribution
• The 5 people in your course had a mean self-esteem score of 9.8.
• Did it work?
– 5,000 random samples yielded 4 with means of 9.8 or greater
– Thus p = .0008 of this happening
Sampling Distribution
9.8
4,996 4
P = .9992 P =.0008
Logic
• 1) Research hypothesis– H1
– Training increased self-esteem– The sample mean is greater than general population mean
• 2) Collect data
• 3) Set up the null hypothesis– H0
– Training did not increase self-esteem– The sample is no different than general population mean
Logic
• 4) Obtain a sampling distribution of the mean under the assumption that H0 is true
• 5) Given the distribution obtain a probability of a mean at least as large as our actual sample mean
• 6) Make a decision– Either reject H0 or fail to reject H0
Hypothesis Test – Single Subject
• You think your IQ is “freakishly” high that you do not come from the population of normal IQ adults.
• Population IQ = 100 ; SD = 15
• Your IQ = 125
Step 1 and 3
• H1: 125 > μ
• Ho: 125 < or = μ
Step 4: Appendix Z shows distribution of Z scores under null
-3 -2 -1 1 2 3
Step 5: Obtain probability
-3 -2 -1 1 2 3
125
Step 5: Obtain probability
-3 -2 -1 1 2 3
125
(125 - 100) / 15 = 1.66
Step 5: Obtain probability
-3 -2 -1 1 2 3
125
Z = 1.66
.0485
Step 6: Decision
• Probability that this score is from the same population as normal IQ adults is .0485
• In psychology– Most common cut-off point is p < .05– Thus, your IQ is significantly HIGHER than
the average IQ
One vs. Two Tailed Tests
• Previously wanted to see if your IQ was HIGHER than population mean– Called a “one-tailed” test– Only looking at one side of the distribution
• What if we want to simply determine if it is different?
One-Tailed
-3 -2 -1 1 2 3
p = .05
Did you score HIGHER than population mean?
Want to see if score falls in top .05
μ
H1: IQ > μ
Ho: IQ < or = μ
Two-Tailed
-3 -2 -1 1 2 3
p = .05
Did you score DIFFERNTLY than population mean?
μ
p = .05
H1: IQ = μ
Ho: IQ = μ
Two-Tailed
-3 -2 -1 1 2 3
p = .05
Did you score DIFFERNTLY than population mean?
PROBLEM: Above you have a p = .10, but you want to test at a p = .05
μ
p = .05
H1: IQ = μ
Ho: IQ = μ
Two-Tailed
-3 -2 -1 1 2 3
p = .025
Did you score DIFFERNTLY than population mean?
μ
p = .025
H1: IQ = μ
Ho: IQ = μ
Step 6: Decision
• Probability that this score is from the same population as normal IQ adults is .0485
• In psychology– Most common cut-off point is p < .05– Note that on the 2-tailed test the point of significance
is .025 (not .05)– Thus, your IQ is not significantly DIFFERENT than the
average IQ
Problems
• Problems with Null hypothesis testing
• Logic is backwards:
• Most think we are testing the probability of the hypothesis given the data
• Really testing the probability of the data given the null hypothesis!