Transcript Grunge Template - Professor Fell

Chapter 10
The Hypothesis of Difference
Sampling Distribution of Differences
•Use a Sampling Distribution of Differences when we want to examine a hypothesis of difference
•We want to compare sets of scores from two samples
• EX: We want to find the average IQ of all students at Omega College (pop=6000)
-put all 6000 student names in a fishbowl
-randomly select 30 students with our left hand & 30 students with our right hand
-give them each an IQ test
-find a mean of 118 for students selected with our left hand (M1=118)
-find a mean of 115 for students selected with our right hand (M2=115)
-then we calculate the difference between these two means (118-115= 3)
-then, we randomly select another 30 students with the left hand and 30 with the right hand,
give them the IQ test & calculate the difference between the means….so on and so forth.
Mean of the Distribution of Differences: μM1-M2
-theoretically, if you add the differences between the means together and divide by the number of
differences, you could calculate the mean of the distribution of differences
-if we did that we should get a mean of approximately zero since all of the samples come from a
single population
**WHY? Because there should be the same number of positive differences as there are
negative differences so they will cancel each other out
Standard Error of Difference
Standard Error of Difference: allows us to predict the value of the standard deviation of the entire
distribution of differences between the means of successively drawn pairs of random samples.
-ie. the estimated standard deviation of the sampling distribution of differences
-based on information contained in just two samples
*Symbolized as SED
Formula:
Step 1: Calculate the means for M1 and M2
Step 2: Calculate the standard deviations for SD1 and SD2
Step 3: Calculate the standard error of the mean for SEM1 and SEM2
Step 4: Calculate the standard error of the difference
Standard Error of Difference Worksheet
Standard Error of Difference Homework due Next Class
Independent t-test
• Two-sample t-test for Independent Samples
-allows us to make a probability statement regarding whether two independently selected
samples represent a single population.
-“independent” means the samples are not dependent on each other
*ie: we can’t use this test if we are doing a repeated measure or matched subjects design
-Research Question: Are these two samples from the same population?
Ho: μ1 = μ2 The samples are from the same population
Ha: μ1 ≠ μ2 The samples are not from the same population (they are significantly different)
• Formula:
Evaluation of t
• In order to Reject Ho the t-value must fall within the .05 or .01 critical areas
*critical areas=rejection region
• We use the Levels of Significance (critical areas) to reduce the probability of
committing an alpha error
*the smaller the critical area, the stronger the conclusion to Reject Ho is
•After we calculate t, we need to determine the t-values at the .05 and .01 levels so
we can make a comparison
*ie. Does our observed t-value fall within the critical areas?
•We must calculate the degrees of freedom:
•In order to Reject Ho, the calculated t-value must be GREATER than the the tabled
t-value (Table C & D)
Evaluation of t
Evaluation of t
One Tail vs. Two-Tail
Advantage of the one-tail test: do not have to obtain as high of a
calculated t in order to reject the null, as we do with a two-tailed
Disadvantage of the one-tail test: the sign of the t-value matters
(whether it is positive or negative) thus limiting the ability to reject the
null
-if you predict a positive t-value (saying M1 is greater than M2)
then you must calculate a positive t-value to be able to reject the null
-if you predict a negative t-value (saying M1 is less than M2)
then you must calculate a negative t-value to be able to reject the null
Independent t-test Worksheet
Independent t-test Homework due Next Class
Type I & Type II Errors
Real
Life
Example:
Real
Life
Example:
ThePolly
producer
of Survivor
andshe
Pimp
Mybe
Ride,
Bruceand
BeresfordPregnantface
thinks
may
pregnant
takes a home
Redman,
is being
pregnancy
test.accused of murdering his wife in Mexico.
Ηo: Bruce is innocent
o: Polly is NOT pregnant
ΗaΗ
: Bruce
is guilty
Ηa: Polly is pregnant
Evidence strongly suggests that Bruce is innocent α
The pregnancy
test
givesinaNYC
negative
result.
-There
is video of
Bruce
on day
of murder
thinks
sheaisguilty
not pregnant
ThePolly
Court
passes
verdict but Polly is, in fact, pregnant.
β
Power
Power is our ability to Reject Ho
-we want power!!
-the more power we have, the more likely we are to reject Ho
-power is 1-β (β=probability that you won’t reject Ho)
Q: if β=.15 then what is the power of the test?
*conceptually, we want to pull this curve to the left
*researchers generally want power to get to .80
How can we increase power?
-increase our alpha levels
**Problem is, while you will reject more nulls this includes true nulls—increase Type 1 errors
-control for extraneous variables
**if we can get rid of variability that we don’t want, it can’t interfere with the variability we want
-reduce measurement error
**ie. Be precise in your measurement
-increase your sample size
**the more people you have, the more likely to find what you are looking for
**However, if it’s too large then you are likely to detect variability that isn’t meaningful