One Sample t-test
Hypothesis Testing:
• Unknown Parameters Requires t-test
• Comparison of One Sample Mean to
a Specific Value
M  0 M  0
t

sM
s/ n
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Outline
Differences between t and z
1.
2.
3.
4.
Unknown parameters
Degrees Freedom
Using the tables
Hypothesis Testing
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What Is a t-test?
• In most research situations, the parameters  and
 are unknown because the test is novel
• Estimates, based on sample statistics must be used
in place of the parameters
• Use of estimation reduces the certainty of the tests
by a quantifiable probability which depends upon
the size of the sample
• For very large samples, the t-test and z-test are
identical
• t-tests are just like z-tests, except that they
compensate for the increasing uncertainty of small
sample sizes
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The larger the df is, the more closely the t
distribution approximates a normal distribution.
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Use of t-test vs. z-test
1. z-test
a)  is known:
b) Mx is computed
c) M is known ( is known)
2. t-test
a)
 is hypothesized or predicted
(not computed and generally not known):
b) Mx is computed
c) M is unknown ( is unknown) :
sM is computed
d) Degrees freedom (d.f.) is computed as the one less than the
sample size (the denominator of the standard deviation):
df = n - 1
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A Note On The Influence of
Sample Size (slide 1 of 2)
•
For the z-test, sample size influences the
shape of the sampling distribution; the
larger the sample size, the more
leptokurtic the sampling distribution
because larger n means smaller M
Platykurtic
Leptokurtic
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A Note On The Influence of
Sample Size (Slide 2 of 2)
For the t-test, sample size has two effects:
1. Makes the sampling distribution more leptokurtic because larger
n means smaller sM
2. t-distribution is more platykurtic than comparable z-distribution
• The smaller the d.f. the more extreme the effect must be to be
detected (rejecting the null hypothesis)
• For df larger than thirty, the
t-distribution is a very close
approximation to
the z-distribution
(see Table B.1)
Platykurtic
Leptokurtic
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The t-statistic
M  0 M  0
t

sM
s/ n
• This value is used just like a z-statistic: if the
value of t exceeds some threshold or critical
valued, t , then an effect is detected (i.e., the
hypothesis of no difference is rejected)
• Critical values t are found in Table B.2
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Finding Critical Values
A portion of the t distribution table
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Finding Critical Values
The t-distribution for df = 3, 2-tailed α = 0.10
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Finding Critical Values
The t-distribution for df =15, 2-tailed α = 0.05
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Finding Critical Values
The t-distribution for df =15, one-tailed 2-tailed α = 0.05
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The one-sample t-test for a
population mean (Slide 1 of 3)
Step 1 The null hypothesis is H0:  = 0 (the real mean
equals some proposed theoretical constant 0); the alternative
hypothesis is one of the following:
Ha:   0
Ha:  < 0
Ha:  > 0
(Two Tailed) (Left Tailed) (Right Tailed)
Step 2 Decide on the significance level, 
Step 3 The critical values are
±t/2
-t
+t
(Two Tailed) (Left Tailed) (Right Tailed)
with df = n - 1.
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The one-sample t-test for a
population mean (Slide 2 of 3)
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The one-sample t-test for a
population mean (Slide 3 of 3)
Step 4 Compute the value of the test statistic
M  0 M  0
t

sM
s/ n
Step 5 If the value of the test statistic falls in the
rejection region, reject H0, otherwise do not
reject H0.
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Criterion for deciding whether or
not to reject the null hypothesis
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Summary of hypothesis-testing
The null hypothesis H0:  = 0
Type
z-test
t-test
Conditions
Test Statistic
M  0
M  0

/ n
μ0 is known
σ is known
z
μ0 is hypothesized
σ is unknown
M  0 M  0
t

sM
s/ n
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M
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Sample Problem
• Given a multiple choice test where each question
has 4 choices, I want to know if a sample of 24
students did better than chance
•
M = 37, s = 14
• What is µ0?
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Sample Problem
• Given a multiple choice test where each question
has 4 choices, I want to know if a sample of 24
students did better than chance
•
M = 37, s = 14
M  0
t
sM
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Sample Problem
• Given a multiple choice test where each question
has 4 choices, I want to know if a sample of 24
students did better than chance
•
M = 37, s = 14
M  0
12
t

 4.2
14
sM
24
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Second Sample Problem
You are conducting an experiment to see if
a given therapy works to reduce test
anxiety. A standard measure of test anxiety
is known to produce a µ = 20. In the
sample you draw of 81 the mean M = 18
with s = 9.
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Second Sample Problem
You are conducting an experiment to see if a given therapy
works to reduce test anxiety. A standard measure of test
anxiety is will known to produce a µ = 20. In the sample
you draw of 81 the mean M = 18 with s = 9.
9
sM 
1
81
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Second Sample Problem
You are conducting an experiment to see if a given therapy
works to reduce test anxiety. A standard measure of test
anxiety is will known to produce a µ = 20. In the sample
you draw of 81 the mean M = 18 with s = 22.
9
sM 
1
81
18  20
t
 2
1
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Third Sample Problem
You are conducting an experiment on ESP.
People who claim to be high in ESP are
asked to guess which of four cards an
experimenter draws from a deck.
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Third Sample Problem
µ = 0.25
tcrit = 1.83
X
X2
0.220
0.048
0.220
0.048
0.220
0.048
0.230
0.053
0.240
0.058
0.240
0.058
SS
n 1
0.270
0.073
0.260
0.068
s
n
0.220
0.048
0.240
0.058
SS  X
s
sM 
t
2
2

X 

n
M 
sM
Σ
2.360
M=
0.236
Σ
SS=
0.003
s=
0.018
sM=
0.006
t=
-2.492
0.560
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