T- test purpose
1- Sample t- test
04/05/2010
Hajieh Jabbari
T formula
T obtained = (ybar - y)
Sy/sqrt N
Procedure:
Compute t obtained from sample data
Determine cutoff point (not a z, but now a critical
t) based on
Reject the null hypothesis if your observed t value
falls in critical region (|t observed| > |t critical|)
Z test requires that you know from
pop
Use a t- test when you don t know
the population standard deviation.
One sample t- test:
Compare a sample mean to a
population with a known mean but an
unknown variance
Use Sy (sample std dev) to estimate
(pop std dev)
T distribution
Can t use unit normal table to find
critical value must use t table to
find critical t
Based on degrees of freedom (df):
# scores free to vary in t obtained
Start w/ sample size N, but lose 1 df
due to having to estimate pop std dev
Df = N- 1
Find t critical based on df and alpha
level you choose
(cont.)
To use the t table, decide what alpha level
to use & whether you have a 1- or 2- tailed
test
gives column
Then find your row using df.
For = .05, 2 tailed, df= 40, t critical =
2.021
Means there is only a 5% chance of finding a
t > = 2.021 if null hyp is true, so we should reject
Ho if t obtained > 2.021
Note
Note that only positive values given in t
table, so
If 1- tailed test,
Use + t critical value for upper- tail test (1.813)
Use t critical value for lower- tail test (you
have to remember to switch the sign, - 1.813)
If 2- tailed test,
Use + and signs to get 2 t critical values, one
for each tail (1.813 and 1.813)
1
Example
Is EMT response time under the new
system (ybar = 28 min) less than old
system ( = 30 min)? Sy = 3.78 and
N= 10
Ha: new < old ( < 30)
Ho: no difference ( = 30)
Use .05 signif., 1- tailed test (see Ha)
T obtained = (28- 30) / (3.78 / sqrt10) =
(28- 30) / 1.20 = - 1.67
(cont.)
Cutoff score for .05, 1- tail, 9 df = 1.833
Remember, we re interested in lower tail (less
response time), so critical t is 1.833
T obtained is not in critical region (not >
| - 1.833 |), so fail to reject null
No difference in response time now
compared to old system
1- sample t test in SPSS
Use menus for:
Analyze
Compare Means
One sample t
Gives pop- up menu need 2 things:
select variable to be tested/compared to
population mean
Notice test value window at bottom. Enter the
population/comparison mean here (use given to
you)
Hit OK, get output and find sample mean,
observed t, df, sig value (p value)
Won t get t critical, but SPSS does the comparison
for you (if sig value < , reject null)
Paired- Samples t Test
Learning Objectives
Learn purpose of paired- samples t test
Discover particulars of the test
Paired- Samples t Test
Purpose of Paired- Samples t Test
Purpose of Test
To determine if means for two paired
(matched) scores differs significantly
from each other
Understand definitions related to test
Discover how to conduct a t test
To determine significance of the difference b/w
means of two paired samples (i.e. dependent
means)
Reminder: Mean is the average (as in
mathematical average) & must be used
with interval/ratio variables
2
Purpose of Paired- Samples t Test cont.
Purpose of Paired- Samples t Test cont.
Purpose of Test cont.
Or in other words
Purpose of Test cont.
t- statistic tests null hypothesis
Test is used when score underlying one
mean has been paired w/ a score underlying
another mean
Reminder:
Null hypothesis
·
A test using t - statistic that establishes
whether two means collected from same
sample (or related observations) differ
significantly
Assertion hoped to be disproven by data
Hypothesis (alternative hypothesis)
·
·
An assertion conjecture about the distribution
of one or more random variables
What you think may be true a prediction
(Field, 2005)
Purpose of Paired- Samples t Test cont.
Purpose of Test cont.
Example:
Pre- test scores & Post- test scores from
same sample
·
·
·
·
Obtain mean from pre- test
Obtain mean from post- test
Same sample took both tests (they match!)
Run procedure to determine if any significant
difference b/w two means
Definitions for Paired- Samples t Test
Definitions of Test
Sample mean is what researcher will
find
The value (score) by using statistical
analysis (mean)
Paired sample = paired scores
Paired = matched (they go together)
Definitions for Paired- Samples t Test cont.
Definitions of Test cont.
Reminder: Significantly different
Researchers use statistical procedures to
determine significance of difference
Discrimination b/ w 2 statistical
hypotheses
·
·
Null hypothesis
Alternative hypothesis
Example of Paired- Samples t Test
Ex: An experiment with paired scores
A researcher drew a random sample from a
population and administered a depression scale to
the sample. Administration of the scale yielded
pre- test scores, for which the researcher computed
a mean. Then, the researcher administered a new
antidepressant drug to the sample. Next, the
researcher administered the depression scale
again, which yielded posttest scores. As a result,
for each pretest score earned by an individual,
there is an associated posttest score for the same
individual. These sets of scores are paired scores.
3
Example of Paired- Samples t Test
Interval/ratio data
Conducting a Paired- Samples t Test
Instrument
Ex. cont.
1st set of scores
Mean for 1st set of scores
A researcher drew a random sample from a
population and administered a depression scale to
the sample. Administration of the scale yielded
pre- test scores, for which the researcher computed
a mean. Then, the researcher administered a new
antidepressant drug to the sample. Next, the
researcher administered the depression scale
again, which yielded posttest scores. As a result,
for each pretest score earned by an individual,
there is an associated posttest score for the same
individual. These sets of scores2nd
aresetpaired
scores.
of
Treatment
Instrument
Paired
scores
scores
Conducting Paired- Samples t Test cont.
SPSS Procedures - Paired- Samples t
test Open SPSS data file or
Create new data file
In Variable View
Click Analyze
Click Compare Means
Click Paired- Samples T Test
Interpreting SPSS Output for t Test
Interpreting Output
SPSS Procedures cont.
In Paired- Samples T Test box
Ch. 12 Holcomb Paired-Samples t Test Output
Click on 1 st variable want to use for test
·
First set of scores (ex: pre- test scores)
Pair
1
Second set of scores (ex: post- test scores)
Mean
11.5556
9.6667
Pretest
Posttest
Click on 2 nd variable want to use for test
·
N
9
9
Std. Deviation
2.29734
2.23607
Std. Error
Mean
.76578
.74536
Paired Samples Correlations
N
Pair 1
Pretest & Posttest
9
Click on arrowhead
·
Unformatted
Paired Samples Statistics
Correlation
.527
Sig.
.145
Paired Samples Test
Paired Differences
Moves chosen variables to Paired Variables
box
Pair 1
Pretest - Posttest
Std. Error
Mean
.73493
Mean
Std. Deviation
1.88889
2.20479
95% Confidence
Interval of the
Difference
Lower
Upper
.19414
3.58364
t
2.570
df
8
Sig. (2-tailed)
.033
Click Ok
Interpreting Output for t Test cont.
Mean for each set of
scores
2 variables
Ch. 12 Holcomb Paired-Samples t Test Output
Sample size for each set
of scores
Standard
Deviation for
each set of
scores
Unformatted
Paired Samples Statistics
Pair
1
Reminder:
SD is the
# of score
points out Pair 1
from the
Mean of a
normal
distribution
Pair 1
Mean
11.5556
9.6667
Pretest
Posttest
N
9
9
Std. Deviation
2.29734
2.23607
Std. Error
Mean
.76578
.74536
Standard
Error of the
Mean for
each set of
scores
Paired Samples Correlations
N
Pretest & Posttest
Pretest - Posttest
9
Correlation
.527
Sig.
.145
Paired Samples Test
Paired Differences
Mean
Std. Deviation
1.88889
2.20479
Aka SD of sampling
distribution
Std. Error
Mean
.73493
95% Confidence
Interval of the
Difference
Lower
Upper
.19414
3.58364
t
2.570
df
Sig. (2-tailed)
8
.033
Interpreting SPSS Output for t Test
Sample size for both variables
Pair of
(should be exactly the same!)
variables
being Ch. 12 Holcomb Paired-Samples t Test Output Unformatted
tested
Paired Samples Statistics
Pair
1
Mean
11.5556
9.6667
Pretest
Posttest
N
9
9
Std. Deviation
2.29734
2.23607
2nd box in
output
describes
Correlations
Std. Error
Mean
.76578
.74536
Relationship
one thing is
to another
Paired Samples Correlations
N
Pair 1
Pretest & Posttest
9
Correlation
.527
Sig.
.145
Paired Samples Test
Paired Differences
Pair 1
Pretest - Posttest
Mean
Std. Deviation
1.88889
2.20479
Std. Error
Mean
.73493
95% Confidence
Interval of the
Difference
Lower
Upper
.19414
3.58364
t
2.570
df
8
Sig. (2-tailed)
.033
Amount of variability across
sample from same population
4
Interpreting SPSS Output for t Test
Standard Error of
Mean
3rd box in Output is results of
Paired-Samples t test
Ch. 12 Holcomb Paired-Samples t Test Output
Unformatted
Pair
1
Mean
11.5556
9.6667
Pretest
Posttest
N
9
9
Std. Deviation
2.29734
2.23607
Further Explanation of Terms cont.
t (Reminder)
Paired Samples Statistics
SD
Interpreting Output for t Test
This tests
hypothesis
Std. Error
Mean
.76578
.74536
A negative value of t indicates
·
Paired Samples Correlations
N
Difference
in mean
b/w set of
scores
Pair 1
Pretest & Posttest
9
Correlation
.527
Significance aka
Probability value (p)
Sig.
.145
t score
Paired Differences
Pair 1
Pretest - Posttest
Mean
Std. Deviation
1.88889
2.20479
95% certain true difference b/w
means in this range
Std. Error
Mean
.73493
95% Confidence
Interval of the
Difference
Lower
Upper
.19414
3.58364
t
2.570
df
8
Sig. (2-tailed)
.033
·
·
Further Explanation of Terms cont.
Sig. (2- tailed) (Reminder)
SPSS defaults to a 2 tailed t test
Most common type of t test
Sig. = significance
Sig. also called the probability value
·
p value
Sig. must be .05 or less to be significant
Sig. tests the hypothesis
Interpreting Output for t Test cont.
Further Explanation of Terms cont.
Probability Values cont.
If p value is equal or less than .05 but
greater than .01
·
·
p = < .05
Declare the difference to be statistically
significant at the .05 level
If p value is greater than .05
·
·
P > .05
Declare the difference to be not statistically
significant at the .05 level
If positive then t is on positive side of
distribution
A t score can also help indicated whether
p value is significant or not
Degrees of freedom N-1
Interpreting Output for t Test cont.
If negative then t is on negative side of
distribution
A positive value of t indicates
·
Paired Samples Test
t
Larger the t value = smaller the p value
Size of t score also dependent on sample size
Interpreting Output for t Test cont.
Further Explanation of Terms cont.
Probability Values (p Values) (Reminder)
If p value is equal to or less than .001
·
·
p = < .001
Declare the difference to be statistically
significant at the .001 level
If p value is equal to or less than .01
·
·
p = < .01
Declare the difference to be statistically
significant at the .01 level
Hypotheses & Significance
Hypothesis & Sig (Reminder)
If p value is significant (p < .05)
Reject the null
Null hypothesis is rejected
If p value is not significant (p > .05)
Failure to reject the null
Null hypothesis is not rejected
df = 8). Thus, we fail to reject the null.
5
Summary
Researchers use a Paired- Samples t test for
a determine if means b/w two paired
(matched) set of scores from the same
sample differ significantly from each other
Means from each set of scores are
compared in procedure
Only interval/ratio (scale) variables are used
Summary cont.
Results of the t test help researchers determine
statistical significance
Gives Sig. (p value)
Gives t value
Gives 95% CI of Difference range
Can either reject the null hypothesis
Or fail to reject the null
Results of t test presented in APA report
Include mean, SD, t , whether statistically
significant or not, df, & if reject null hypothesis
or fail to reject the null hypothesis
6
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