Chapter Eleven
Performing the
One-Sample
t-Test and Testing
Correlation
More Statistical Notation
• Recall the formula for the estimated
population standard deviation.
(X )
X 
N
sX 
N 1
2
2
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Chapter 11 - 2
Using the t-Test
• Use the z-test when  X is known.
• Use the t-test when  X is estimated by
calculating s X .
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Chapter 11 - 3
Performing the
One-Sample t-Test
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Chapter 11 - 4
Setting Up the Statistical Test
1.Set up the statistical hypotheses (H0
and Ha). These are done in precisely
the same fashion as in the z-test.
2.Select alpha
3.Check the assumptions for a t-test
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Chapter 11 - 5
Assumptions for a t-Test
• You have one random sample of
interval or ratio scores
• The raw score population forms a
normal distribution
• The standard deviation of the raw score
population is estimated by computing s X
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Chapter 11 - 6
Computational Formula
for the t-Test
tobt
X 

sX
N
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Chapter 11 - 7
The t-Distribution
The t-distribution is the distribution of all
possible values of t computed for random
sample means selected from the raw
score population described by H0
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Chapter 11 - 8
Comparison of Two t-distributions
Based on Different Sample Ns
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Chapter 11 - 9
Degrees of Freedom
• The quantity N - 1 is called the degrees of
freedom
• Since it is this value that is used to
compute s X , it is the degrees of freedom
(df) that determines how consistently s X
estimates the true  X
• We obtain the appropriate value of tcrit
from the t-tables using both the appropriate
a and df
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Chapter 11 - 10
Two-Tailed t-Distribution
[Insert Figure 11.3 here.]
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Chapter 11 - 11
Estimating the Population Mean by
Computing a Confidence Interval
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Chapter 11 - 12
Estimating 
• There are two ways to estimate the
population mean 
• Point estimation in which we describe
a point on the variable at which the
population mean is expected to fall
• Interval estimation in which we specify
an interval (or range of values) within
which we expect the population
parameter to fall
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Chapter 11 - 13
Confidence Intervals
• We perform interval estimation by
creating a confidence interval
• The confidence interval for a single 
describes an interval containing values
of 
( s X )( tcrit )  X    ( s X )( tcrit )  X
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Chapter 11 - 14
Significance Tests for
Correlation Coefficients
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Chapter 11 - 15
The Pearson
Correlation Coefficient
• The Pearson correlation coefficient (r) is
used to describe the relationship in a
sample
• Ultimately we want to describe the
relationship in the population
• For any correlation coefficient you
compute, you must decide if it is
significant
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Chapter 11 - 16
Hypotheses
• Two-tailed test
– H0: r = 0
– Ha: r ≠ 0
• One-tailed test
– Predicting positive
correlation
H0: r ≤ 0
Ha: r > 0
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Predicting negative
correlation
H0: r ≥ 0
Ha: r < 0
Chapter 11 - 17
Scatterplot of a Population for
Which r = 0
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Chapter 11 - 18
Assumptions for the Pearson r
1.There is a random sample of X and Y
pairs and each variable is an interval or
ratio variable
2.The Y scores and the X scores each
represent a normal distribution. Further,
they represent a bivariate normal
distribution.
3.The null hypothesis is that in the
population there is zero correlation
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Chapter 11 - 19
Sampling Distribution
The sampling distribution of a correlation
coefficient is a frequency distribution
showing all possible values of the
coefficient that can occur when samples
of size N are drawn from a population
where r is zero
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Chapter 11 - 20
Degrees of Freedom
• The degrees of freedom for the
significance test of a Pearson
correlation coefficient are N - 2.
• N is the number of pairs of scores.
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Chapter 11 - 21
Testing the Spearman rs
• Testing the Spearman rs requires a random
sample of pairs of ranked (ordinal) scores.
• Use the critical values of the Spearman
rank-order correlation coefficient for either
a one-tailed or a two-tailed test.
• The critical value is obtained using N, the
number of pairs of scores in the sample.
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Chapter 11 - 22
Maximizing the Power of
a Statistical Test
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Chapter 11 - 23
Maximizing the Power
of the t-Test
1.Larger differences produced by
changing the independent variable
increase power
2.Smaller variability in the raw scores
increases power
3.A larger N increases power
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Chapter 11 - 24
Maximizing the Power of a Correlation
Coefficient
• Avoiding a restricted range increases
power
• Minimizing the variability of the Y scores
at each X increases power
• Increasing N increases power
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Chapter 11 - 25
Example 1
• Use the following data set and conduct
a two-tailed t-test to determine if
 = 12
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
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Chapter 11 - 26
Example 1
• H0:  = 12; Ha:  ≠ 12
• Choose a = 0.05
• Reject H0 if tobt > +2.110 or if
tobt < -2.110
tobt
X   13.67  12 1.67



 4.40
sX
1.61
0.380
N
18
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Chapter 11 - 27
Example 2
• For the following data
set, determine if the
Pearson correlation
coefficient is significant.
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X
Y
1
5
2
2
3
6
4
4
5
3
6
1
Chapter 11 - 28
Example 2
• From chapter 7, we know that r = -0.88
• Using a = 0.05 and a two-tailed test, rcrit
= 0.811. Therefore, we will reject H0 if
robt > 0.811 or if robt < 0.811
• Since robt = -0.88, we reject H0
• We conclude that this correlation
coefficient is significantly different from
0
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Chapter 11 - 29