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final project. It will be on the web site.
The Z statistic
Zx 
Where


x  x
X
X 
x

n
The Z statistic
Zx 
Where


x  x
X
X 
x

n
The Z statistic
Zx 
Where


x  x
X
X 
x

n
The Z statistic
Zx 
Where


x  x
X
X 
x

n
The t Statistic(s)
• Using an estimated  , which we’ll
2
ˆ
call  we can create an estimate of  X
ˆX
which we’ll call 
2

Estimate:
and:

2
(X

X
)

ˆ  i


n 1
2

ˆ
ˆ


ˆX 


n
n
nS 2
n 1

The t Statistic(s)
• Caution: some textbooks and some
calculators use the symbol S2 to
represent this estimated population
variance
The t Statistic(s)
ˆ X instead of  X we get a
• Using, 
statistic that isn’t from a normal (Z)
distribution - it is from a family of
distributions called t


x  x
tn1 
ˆx

The t Statistic(s)
• What’s the difference between t and Z?
The t Statistic(s)
• What’s the difference between t and Z?
• Nothing if n is really large (approaching
infinity)
– because n-1 and n are almost the same
number!
The t Statistic(s)
• With small values of n, the shape of the
t distribution depends on the degrees of
freedom (n-1)
The t Statistic(s)
• With small values of n, the shape of the
t distribution depends on the degrees of
freedom (n-1)
– specifically it is flatter but still symmetric
with small n
The t Statistic(s)
• Since the shape of the t distribution
depends on the d.f., the fraction of t
scores falling within any given range
also depends on d. f.
The t Statistic(s)
• The Z table isn’t useful (unless n is
huge) instead we use a t-table which
gives tcrit for different degrees of
freedom (and both one- and two-tailed
tests)
The t Statistic(s)
• There is a t table on page 142 of your
book
• Look it over - notice how tcrit changes
with the d.f. and the alpha level
The t Statistic(s)
• The logic of using this table to test
alternative hypothesis against null
hypothesis is precisely as with Z scores
- in fact, the values in the bottom row
are given by the Z table and the familiar
+/- 1.96 appears for alpha = .05 (twotailed)
An Example
• You have a theory that drivers in Alberta
are illegally speedy
An Example
• You have a theory that drivers in Alberta
are illegally speedy
– Prediction: the mean speed on highway 2
between Ft. Mac and Calgary is greater
than 110
An Example
• You have a theory that drivers in Alberta
are illegally speedy
– Prediction: the mean speed on highway 2
between Ft. Mac and Calgary is greater
than 110
• Here’s another way to say that: a sample of n
drivers on the highway is not a sample from a
population of drivers with a mean speed of 110
An Example
• Set up the problem:
– null hypothesis: your sample of drivers on highway
2 are representative of a population with an
average speed of 110 km/hr
X  x
and t 
X  110
 t crit in 95% of such samples
ˆx


– alternative hypothesis: sample of drivers is from a
population with a mean speed greater than 110
thus:

X  110 and X  x  t crit
ˆx

An Example
• Here are some (fake) data
X 114.4
Car #
1
2
3
4
5
Speed
105
118
112

121
116


ˆ

2
ˆx

X  110
s2  30.64
ns2 5  30.64


 38.3
n 1
4

ˆ2

38.3


 2.768
n
5
X  X 114.4 110
t4 d . f . 

1.59
ˆX

2.768
An Example
– tcrit for a one-tailed test with 5-1 = 4 d.f. is 2.1318
– Our computed t = 1.59 does not exceed tcrit thus
we cannot reject the null hypothesis
– We conclude there is no evidence to support our
hypothesis that drivers are speeding on highway 2
– Does this mean that drivers are not speeding on
highway 2?
T-test for one sample mean
• We’ve discussed how to create and use
a t statistic when we want to compare a
sample mean to a hypothesized mean
x  x
tn1 
ˆx

t Tests for Two Sample Means
• We’re often interested in a more
sophisticated and powerful experimental
design…
t Tests for Two Sample Means
• We’re often interested in a more
sophisticated and powerful experimental
design…
• Usually we perform some experimental
manipulation and look for a change on
some score or variable
– e.g. before and after taking a drug
t Tests for Two Sample Means
• We manipulate a variable (eg. drug
dose) and we want to know whether
some other variable (e.g. fever)
depends on our manipulation
t Tests for Two Sample Means
• We manipulate a variable (eg. drug
dose) and we want to know whether
some other variable (e.g. fever)
depends on our manipulation
• Let’s introduce some formal terms:
– independent variable: the variable that you
control
– dependent variable: the variable that
depends on the experimental manipulation
(the one you measure)
t Tests for Two Sample Means
• Example: Let’s ask whether or not
Tylenol reduces fever - there are two
ways you could do this…
t Tests for Two Sample Means
• Example: Let’s ask whether or not
Tylenol reduces fever - there are two
ways you could do this…
1. Get a bunch of people with fevers, give half
of them Tylenol and half of them a placebo
and then measure their temperatures
t Tests for Two Sample Means
• Example: Let’s ask whether or not
Tylenol reduces fever - there are two
ways you could do this…
1. Get a bunch of people with fevers, give half
of them Tylenol and half of them a placebo
and then measure their temperatures
2. Get a bunch of people with fevers, measure
their temperatures, then give them Tylenol
and measure them again
t Tests for Two Sample Means
• Repeated Measures - an experiment in
which the same subject (or object) is
measured in two (or more!) conditions
t Tests for Two Sample Means
• Repeated Measures - an experiment in
which the same subject (or object) is
measured in two (or more!) conditions
• The two samples are actually pairs of
scores and those pairs are correlated or
dependent
t Tests for Two Sample Means
• Repeated Measures - an experiment in
which the same subject (or object) is
measured in two (or more!) conditions
• The two samples are actually pairs of
scores and those pairs are correlated or
dependent
• This type of t test is called a test for two
dependent sample means (sometimes
called a paired t-test)
t Tests for Two Dependent
Sample Means
• When comparing two paired samples we’re
often not interested in the absolute scores but
we are interested in the differences between
scores
Sample 1 Sample 2
X21
X11
X22
X12
.
.
.
.
.
.
X2n
X1n
Difference
X11 - X21
X12 - X22
.
.
.
X1n - X2n
•This is a sample
of differences
taken from a
population of
differences
•it has a mean and
standard deviation
t Tests for Two Dependent
Sample Means
• If we’re wondering whether an
independent variable has some effect on
the dependent variable then our null
hypothesis is that there is no difference
between the two paired measurements in
our sample
t Tests for Two Dependent
Sample Means
• If we’re wondering whether an
independent variable has some effect on
the dependent variable then our null
hypothesis is that there is no difference
between the two paired measurements in
our sample
• Some differences would be positive,
some would be negative, on average the
difference would be zero
t Tests for Two Dependent
Sample Means
• We can use a t-test to test if the sample of differences
has a mean that is significantly different from zero
D  D
tn1 
ˆD

• This is done by simply treating your column of
differences as a one-sample t-test with a null
 that u = 0
hypothesis
t Tests for Two Dependent
Sample Means
• Some curiosities that make your life easier with regard
to paired t-tests
– Note that:
D  X1  X 2
– And that n1 always equals n2
with the z-test, the t distribution is symmetric so you treat
– As
negative differences as if they were positive for comparing to
tcrit
– Also as with the z-test, one- or two-tailed tests are
possible…simply use the appropriate column from the t table
t Test for Two Independent
Sample Means
• Often we have a situation in which
repeated measures is inappropriate or
impossible (e.g. any time measuring the
dependant variable once alters
subsequent measurements)
t Test for Two Independent
Sample Means
• Often we have a situation in which
repeated measures is inappropriate or
impossible (e.g. any time measuring the
dependant variable once alters
subsequent measurements)
• In this situation we must use a betweensubjects design
t Test for Two Independent
Sample Means
• The data are laid out like the repeated measures
case except they aren’t pairs of scores, the two
columns are measurements of different subjects
(objects, etc.)
• We thus usually only refer to a single measurement
with respect to the mean of that sample
Sample 1 Sample 2
X1
X2
Difference
D  X1  X 2
t Test for Two Independent
Sample Means
• The null hypothesis states that these
two independent samples are random
samples from the same population
t Test for Two Independent
Sample Means
• The null hypothesis states that these
two independent samples are random
samples from the same population
– so you would expect the difference to be
zero on average
t Test for Two Independent
Sample Means
• The null hypothesis states that these
two independent samples are random
samples from the same population
– so you would expect the difference to be
zero on average
– therefore the numerator of the t statistic in
this situation works just like the dependent
samples case
D  D
where
D  0
t Test for Two Independent
Sample Means
• The denominator is different because…
t Test for Two Independent
Sample Means
• The denominator is different because…
• How many degrees of freedom are there?
t Test for Two Independent
Sample Means
• The denominator is different because…
• How many degrees of freedom are there?
– The mean difference is based on two different
samples, each with their own degrees of freedom
t Test for Two Independent
Sample Means
• The denominator is different because…
• How many degrees of freedom are there?
– The mean difference is based on two different
samples, each with their own degrees of freedom
– So there are n1-1+n2-1 = n1+n2-2 d.f.
t Test for Two Independent
Sample Means
• The denominator is different because…
• How many degrees of freedom are there?
– The mean difference is based on two different
samples, each with their own degrees of freedom
– So there are n1-1+n2-1 = n1+n2-2 d.f.
– The best estimate of the population standard
deviation will incorporate both samples so that it
has more degrees of freedom
t Test for Two Independent
Sample Means
• We can pool the sums of squares
(which weights the variances according
to the number in each sample)
n1
n2
i1
i1
SS pooled  (X1i  X1) 2  (X 2i  X 2 ) 2 n1S12  n2 S22
t Test for Two Independent
Sample Means
• We can pool the sums of squares
(which weights the variances according
to the number in each sample)
n1
n2
i1
i1
SS pooled  (X1i  X1) 2  (X 2i  X 2 ) 2 n1S12  n2 S22
• Then divide by the pooled degrees of
2

freedom to estimate
ˆ 

2
SS pooled
n1  n 2  2
t Test for Two Independent
Sample Means
• Estimate  D : Both samples contribute
to the standard error of the mean
differences so
2
2
ˆ
ˆ


pooled
pooled
2
ˆ
D 


n1
n2
and…

ˆD 

2
ˆ
pooled
n1

2
ˆ
pooled
n2
t Test for Two Independent
Sample Means
• Now we can construct a t statistic
X1  X 2 D
t n1 n 2 2 

ˆ X X
ˆD


1

2
t Test for Two Independent
Sample Means
• Notice that this t statistic has more
degrees of freedom than its dependent
samples counterpart
• Why does a repeated measures design
still tend to have more power?
t Test for Two Independent
Sample Means
• Consider an example:
– Are northbound drivers slower than southbound
drivers on highway 2 ?
– Null hypothesis: samples of n speeds taken from
northbound and southbound traffic are from the
same population
– Alternative hypothesis: samples of southbound
drivers are from a population with a mean greater
than that of northbound drivers
t Test for Two Independent
Sample Means
Northbound Car #
1
2
3
4
5
ˆ

2
pooled
ˆX

1 X 2
Speed
105
118
112
121
116
Southbound Car #
1
2
3
4
5
Speed
121
119
127
124
115
X1  114.4
S 21  30.64
X 2  121.2
S 2 2  16.96
n1S12  n 2 S22
5  30.64  5 16.96


 29.75


(n1 1)  (n 2 1)
(5 1)  (5 1)

t8d . f . 
ˆ2

n1

ˆ2

n2

29.75 29.75

 3.45
5
5
X1  X 2 114.4 121.2

 1.97
ˆ
X X
3.45
1
2
For a one-tailed test at  = .05, with 8 d.f. , tcrit = 1.86. We can therefore reject
the null hypothesis and conclude that southbound drivers are faster.
t Test for Two Independent
Sample Means
• Some caveats and disclaimers about
independent-sample t-tests:
– There is an assumption of equal variance in the
two underlying populations
• If this assumption is violated, your Type I error rate is
greater than the indicated alpha!
• However, for samples of equal n, the t-test is quite robust
to violations of this assumption (so you usually don’t
have to worry about it)
– Note that n need not be equal! (but it’s better if
possible)
Next Time:
• Too many t tests spoils the statistics