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366_7
T-distribution
• T-test vs. Z-test
• Z assumes we know, or can calculate the
standard error of the distribution of
something in a population
• We never really do
T-distribution
• Reality: We estimate standard error of a
mean we observe from (duh) what we
observe
• Insert formula for estimate of standard dev. of
sample here (p. 138) & std error of mean (p.
139)
T-ratio
• t-ratio used to test if/how an observation from
a sample reflects the population mean
• very similar to Z-scores (at infinity)
• Slightly different distribution
T-ratio
• Suppose observing
children 10 cooperating
• how many cooperative
acts?
• how confident it reflects
population
1
5
2
3
4
1
2
2
4
3
1) mean = 2.7
T-ratio
• Suppose observing
children 10 cooperating
2) Calculate std. deviation
sum of sq distances
from mean
sum X2=89
sd=SQRT (89/10) – 2.7
=1.27
T-ratio
• Suppose observing
children 10 cooperating
3) Translate std dev. into std.
error
s.e= s.d / SQRT(N-1)
= .42
4) establish degrees of
freedom & alpha (.05)
df = N-1
=9
5) Check Table C
T-ratio
6) t= 2.26
• how confident?
• 95% confident the
population mean is
between 1.74 to 3.66
acts.
7) Confidence intervals
95%= 2.7 +/-(2.62*.42)
= 2.7 +/- .96
= 1.74 +/- 3.66
Hypothesis Testing with t
Research Hypothesis (H1):
Something is going on.
There is a difference between groups, Men have higher score.
H1: Xm > Xf
Null Hypothesis (H0):
There is no difference
Mean for group 1 = the mean for group 2
H0: X1 = X2
Hypothesis Testing with t
Observe difference between means:
Magnitude of difference
Variance in measure of X1 and X2
Number of observations
What is the likelihood that such a difference
would occur by chance?
T-test
• Assume
– Random samples, independent of each other
– Variable being compared is interval or ratio
– Distributions are normal
– Roughly equal variance of each group
T-test
• Decide criteria, or critical t
– Alpha to reject (chance of a Type 1 error)
• t= 1.65 for alpha = .10 (at infinity. Critical t larger if
sample small)
• t= 1.96 for alpha - .05 (larger if sample small)
– Directional test?
t test
• Calculate t
t=
(x1 – x2)
________________
s x1-x2
----------std. error of the
difference between 2 means
this part is messy, but
includes info about sample sizes
t-test
• result is one value (t-statistic) we can use to
check if difference between groups is
significant
• Example:
– Corruption, south vs. non south
• What hypothesis?
Projects
• Identify testable hypotheses
–
–
–
–
x causes y
x explains differences in y
differences in x explain y
x and y go together in some interesting way
• Identify how to measure what you want to test
– What actual questions
• We’ll worry about the test statistic later
Mean of group 1 significantly different
than mean of group 2?
Non south, x= .33; s.e. .03
South, x= .43; s.e .05
t-test
• Southern states,
zoomed in....
Results
ttest percap_convic, by(var82)
Two-sample t test with equal variances
-----------------------------------------------------------------------------Group |
Obs
Mean
Std. Err.
Std. Dev.
[95% Conf. Interval]
---------+-------------------------------------------------------------------0 |
39
.3358051
.0313071
.1955129
.2724272
.3991831
1 |
11
.4324917
.0577252
.1914528
.303872
.5611115
---------+-------------------------------------------------------------------combined |
50
.3570762
.0278429
.1968793
.3011237
.4130287
---------+-------------------------------------------------------------------diff |
-.0966866
.0664606
-.2303146
.0369414
-----------------------------------------------------------------------------diff = mean(0) - mean(1)
t = -1.4548
Ho: diff = 0
degrees of freedom =
48
Ha: diff < 0
Pr(T < t) = 0.0761
.
Ha: diff != 0
Pr(|T| > |t|) = 0.1522
Ha: diff > 0
Pr(T > t) = 0.9239
Results
•
•
•
•
•
Note each mean is given
variation around mean is given
confidence intervals
difference between means is given (-.096)
std. error of differences btwn means given
• AND t values
Results
• Note different t values are given
• Each is for a specific hypothesis
– Difference is greater than 0, positive (one tail)
– Difference is greater than 0, negative (one tail)
– “Absolute difference” (two tail)
t test results
• Do we accept of reject null hypothesis?