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T-distribution & comparison of means
Z as test statistic
Use a Z-statistic only if you know the population standard
deviation (σ).
Z-statistic converts a sample mean into a z-score from the
null distribution.
X  H
X  H
Z test 
0
SE

0

n
p-value is the probability of getting a Ztest as extreme as
yours under the null distribution
T-distribution & comparison of means
t as a test statistic
t-test: uses sample data to evaluate a hypothesis about a
population mean when population stdev () is unknown
We use the sample stdev (s) to estimate the standard error
standard x
error
z
=

n
X  H0
X
estimated
standard sx
error
t
=
s
n
X  H0
sX
T-distribution & comparison of means
t distribution
You can use s to approximate σ, but then the sampling
distribution is a t distribution instead of a normal
distribution
Why are Z-scores normally distributed, but t-scores are
not?
Random (normal)
variable
constant
ztest 
X  H0
X
constant
Random (normal)
variable
constant
ttest 
X  H0
sX
Random (non-normal)
variable
T-distribution & comparison of means
t distribution
Very large sample - estimated standard error almost = the
true standard error (t almost exactly the same as Z)
t distribution is a family of curves. As n gets bigger, t
becomes more normal.
For smaller n, the t distribution is platykurtic (narrower peak,
fatter tails)
Use “degrees of freedom” to decide which t curve to use.
Basic t-test, df = n-1
T-distribution & comparison of means
t distribution
Level of significance for a one-tailed test
df
.05
1
2
3
4
6.314
2.920
2.353
2.132
.025
.01
.005
12.706 32.821 63.657
4.303 6.965 9.925
3.182 4.541 5.841
2.776 3.747 4.604
tcrit
T-distribution & comparison of means
Practice with Table B.3
With a sample of size 6, what are the degrees of
freedom? For a one-tailed test, what is the critical
value of t for an alpha of .05?
And for an alpha of .01?
Df = 5, tcrit = 2.015; tcrit = 3.365
T-distribution & comparison of means
Practice with Table B.3
For a sample of size 25, doing a two-tailed test,
what are the degrees of freedom and the critical
value of t for an alpha of .05 and for an alpha of
.01?
Df = 24, tcrit = 2.064; tcrit = 2.797
T-distribution & comparison of means
Practice with Table B.3
You have a sample of size 13 and you are doing a
one-tailed test. Your tcalc = 2. What do you
approximate the p-value to be?
p-value between .025 and .05
What if you had the same data, but were doing a
two-tailed test?
p-value between .05 and .10
T-distribution & comparison of means
Two-sample testing
All the inferential statistics we have considered involve
using one sample as the basis for drawing conclusion
about one population.
Most research studies aim to compare of two (or more)
sets of data in order to make inferences about the
differences between two (or more) populations.
What do we do when our research question concerns a
mean difference between two sets of data?
T-distribution & comparison of means
Two-sample testing
Steps for Calculating a Test Statistic:
Ho: 1 - 2 = 0
HA: 1 - 2  0
To test the null hypothesis – compute a t statistic and
look up in Table B.3
T-distribution & comparison of means
Two-sample testing
Steps for Calculating a Test Statistic:
General t formula
t = sample statistic - hypothesized population parameter
estimated standard error
T-distribution & comparison of means
Two-sample testing
Steps for Calculating a Test Statistic:
One Sample t
ttest 
X  H0
Remember?
sX
For two-sample test
( X 1  X 2 )  ( 1   2 )
t
s X1  X 2
T-distribution & comparison of means
Two-sample testing
Steps for Calculating a Test Statistic:
Standard Error for a Difference in Means
s X1  X 2
The one-sample standard error ( sx ) measures how much
error expected between X and .
The two sample standard error (sx1-x2) measures how
much error is expected when you are using a sample
mean difference (X1 – X2) to represent a population mean
difference.
T-distribution & comparison of means
Two-sample testing
Steps for Calculating a Test Statistic:
Standard Error for a Difference in Means
s X1  X 2
s12 s22


n1 n2
T-distribution & comparison of means
Two-sample testing
Steps for Calculating a Test Statistic:
Each of the two sample means - there is some error.
Calculate the total amount of error involved in using
two sample means - find the error from each sample
separately and then add the two errors together.
T-distribution & comparison of means
Two-sample testing
Steps for Calculating a Test Statistic:
Standard Error for a Difference in Means
s X1  X 2
BUT:
s12 s22


n1 n2
Only works when n1 = n2
T-distribution & comparison of means
Two-sample testing
Steps for Calculating a Test Statistic:
Change the formula slightly - use the pooled sample variance
instead of the individual sample variances.
Pooled variance = weighted estimate of the variance derived
from the two samples.
SS1  SS2
s 
df1  df 2
2
p
T-distribution & comparison of means
One-Sample T
1. Calculate sample mean
2. Calculate standard error
3. Calculate T and d.f.
4. Use Table B.3
T-distribution & comparison of means
SS1  SS2
s 
df1  df 2
Two-Sample T
2
p
1. Calculate X1-X2
2. Calculate pooled variance
s2p
3. Calculate standard error
n1
4. Calculate T and d.f.
5. Use Table B.3
t
(X1  X 2 )  (1  2 )
sx1  x2

s2p
n2
d.f. = (n1 - 1) + (n2 - 1)
T-distribution & comparison of means
EXAMPLE
A developmental psychologist would like to examine the
difference in verbal skills for 8-year-old boys versus 8year-old girls. A sample of 10 boys and 10 girls is
obtained, and each child is given a standardized verbal
abilities test. The data for this experiment are as follows:
Girls
Boys
n1 = 10
n = 10
X2 2
= 37
= 31
SS1 = 150 SS2 = 210
X1
T-distribution & comparison of means
EXAMPLE
Girls
Boys
n1 = 10
n2 = 10
X1 = 37
X 2 = 31
SS1 = 150 SS2 = 210
STEP 1: get mean difference
X1  X 2  6
T-distribution & comparison of means
EXAMPLE
Girls
Boys
n1 = 10
n2 = 10
X1 = 37
X 2 = 31
SS1 = 150 SS2 = 210
STEP 2: Compute Pooled Variance
SS1  SS2
150  210
360
s 


 20
df1  df 2 (10  1)  (10 1) 18
2
p
T-distribution & comparison of means
EXAMPLE
Girls
Boys
n1 = 10
n2 = 10
X1 = 37
X 2 = 31
SS1 = 150 SS2 = 210
STEP 3: Compute Standard Error
2
SE 
sp
2
sp
20 20



 4 2
n1 n2
10 10
T-distribution & comparison of means
EXAMPLE
Girls
Boys
n1 = 10
n2 = 10
X1 = 37
X 2 = 31
SS1 = 150 SS2 = 210
STEP 4: Compute T statistic and df
t
(X1  X 2 )  (1  2 ) (37  31)  0

3
sx1  x2
2
d.f. = (n1 - 1) + (n2 - 1) = (10-1) + (10-1) = 18
T-distribution & comparison of means
EXAMPLE
Girls
Boys
n1 = 10
n2 = 10
X1 = 37
X 2 = 31
SS1 = 150 SS2 = 210
STEP 5: Use table B.3
T = 3 with 18 degrees of freedom
For alpha = .01, critical value of t is 2.878
Our T is more extreme, so we reject the null
There is a significant difference between boys and girls
T-distribution & comparison of means
T-Test
for Independent
Samples
SUMMARY
OF EQUATIONS
Sample
Data
One-sample
t-statistic
Two-sample
t-statistic
X
X1  X 2
Hypothesized
Population
Parameter

1  2
Sample
Variance
2
s 
2
sp 
SS
df
SS1  SS2
df1  df 2
Estimated
Standard
Error
s2
n
s2p
n1
t-statistic
t

X
sx
s2p
n2
t
(X1  X 2 )  (1  2 )
sx1  x2