Tuesday October 1
Point Estimation;
Using SPSS and Excel to conduct ttests
Last Time
• We practiced doing an independent samples ttest by hand
• We learned how to do related samples t-tests
– We learned when to do this type of test (one group,
two scores per person, or two groups with pairs of
participants who are matched or related in some way
– We learned that this test is essentially the same as a
one-sample t-test on difference scores, with H0: μD=0
– We learned how to do a related samples t-test in
SPSS
Estimation
• In inferential statistics we use sample statistics to
make inferences about population parameters
• One way we can do this is to estimate population
parameters based on sample statistics
• Estimate can be
– Point estimate
– Interval estimate
– Confidence interval
Point Estimation
– Single value (e.g. M is the point estimate of μ, or s
is the estimate of σ)
– Single value that represents a difference (e.g., MD
is the point estimate of μD, MA-MB is the point
estimate of μA-μB)
– Point estimates we are interested in are often in
the numerators of our test statistics.
Interval estimates
• We can specify a range of scores rather than a
single value when we estimate population
parameters
• We can use our formulae for our t statistics to help
us do this:
t=
(sample mean or mean difference) - (population mean or mean difference)
estimated standard error
• Using algebra, this formula converts to:
Population mean
(or mean diff)
=
sample mean +/- t * (estimated standard error)
(or mean diff)
Interval estimates
Type of test
Formula for t
statistic
One-sample ttest:
M -m
t=
sM
Independentsamples t-test:
Pairedsamples t-test:
t=
( M A - M B ) - (mA - mB )
sM A -M B
M D - mD
t=
sM D
Formula for
confidence
interval
m = M ± t ( SM )
( m A - m B ) = ( M A - M B ) ± t ( sM
A
( )
m D = M D ± t sM
D
- MB
)
Confidence Intervals
• Interval estimate accompanied by specific
level of confidence or probability (e.g., p=.95)
Confidence Intervals
m = M ± t ( SM )
• Question: M and SM are calculated from your sample
data, but what is t?
• (Before when we calculated t, we had a known value for
μ or assumed it was equal to 0 - now we are trying to
estimate μ based on sample data)
• Answer: t is the critical value of t (from the t-table) based
on the confidence level or p-value we desire for our
confidence interval!
•5% chance of being wrong = 95% chance of being correct
Confidence Intervals
Example: A sample of 25 3rd graders read on average
22 words correctly per minute, with a sample
standard deviation of 5. Use this information to
estimate the average number of words the population
of 3rd graders can read correctly per minute, with p =
.95 confidence.
M=22, SM=5/5 = 1,
M
tcrit=2.064
m = M ± t (S
)
m = 22 ± 2.064(1) = (19.936,24.066)
Confidence Intervals
Example II: A sample of 20 3rd graders read on
average 21 words correctly per minute prior to a
reading intervention. Following the intervention, the
average increased to 23 words correctly per minute. If
the standard error for the mean difference scores is .8,
estimate the average improvement you would expect
to find in the population following this intervention.
find the 95% confidence intervals
( )
m D = M D ± t sM
D
MD = 2, SM D = 0.8,
tcrit=2.093
mD = 2 ± 2.093(.8 ) = 2 ± 1.6744 = ( 0.3256, 3.6744 )
Statistical Tests Summary
Design
One sample, σ
known
One sample, σ
unknown
Two independent
samples, σ
unknown
Two related samples
(or one sample with
repeated measures),
σ unknown
Statistical test
zM =
M - mM
sM
M - mM
t=
sM
M A - M B ) - (mA - mB )
(
t=
sM A -M B
M D - mD
t=
sM D
(Estimated) Standard error
sM =
s
n
s
sM =
n
sM A - M B
sM D
sP2 sP2
=
+
nA nB
sD
=
nD
Using spss to conduct t-tests
• One-sample t-test: Analyze =>Compare Means =>One
sample t-test. Select the variable you want to analyze, and
type in the expected mean based on your null hypothesis.
• Independent samples t-test: Analyze =>Compare Means
=>Independent samples t-test. Specify test variable and
grouping variable, and click on define groups to specify how
grouping variable will identify groups.
• Paired or related samples t-test: Analyze =>Compare Means
=>Paired samples t-test. Select the variables you want to
compare and drag them into the “pair 1” boxes labeled
“variable 1” and “variable 2”
Using excel to compute t-tests
• =ttest(array1,array2,tails,type)
• Select the arrays that you want to
compare, specify number of tails (1 or 2)
and type of t-test (1=dependent,
2=independent w/equal variance assumed,
3=independent w/unequal variance
assumed).
• Returns the p-value associated with the ttest.
Next Homework (due Thursday, October 3)
See Handout