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CHAPTER 13
STUDENT’S t TEST FOR SINGLE SAMPLES
CHAPTER OUTLINE
I.
Introduction
A. Use of the t test. Use the t test when
1. the mean of the Null Hypothesis Population can be specified
2. the standard deviation is unknown (which also means the z test cannot be
used)
B. Applications covered in this chapter.
1. Analysis of data in experiments with a single sample
2. Determining the significance of Pearson r.
II.
Comparison of z and t Tests
A. Formula.
zobt 
tobt 
X obt  μ
sX
X obt  μ
σ
; where σ X 
σX
N
where s X = estimated standard error of the mean =
s
N
B. z and t equation differences. Difference between equations is that in the t
equation, s and s X are used instead of  and σ X respectively.
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III. Sampling Distribution of t
A. Definition. the sampling distribution of t is a probability distribution of the t
values which would occur if all possible different samples of a fixed size N were
drawn from the Null Hypothesis Population. It gives (1) all the possible t values
for samples of size N and (2) the probability of getting each value if sampling is
random from the Null Hypothesis Population.
B. Characteristics.
1. Family of curves; many curves with a different curve for each sample size
N.
2. Shape. Shaped similarly to z distribution if
a. sample size  30 or
b. H0 population is normally distributed.
C. Degrees of freedom (df). The number of scores that are free to vary.
df = N 1
D. Comparison of z and t distributions.
1.
2.
3.
4.
t and z are both symmetrical about 0
As df increases t becomes more similar to z
As df approaches , t becomes identical to z
At any value of df < , the t distribution has more extreme values than z
(i.e., tails of the t distribution are more elevated than in the z distribution)
5. For a given alpha level, tcrit > zcrit
IV. Calculations and Use of t
A. Calculation of t from raw scores.
t obt 
X obt  μ
SS
N (N  1)
B. Requirements. Appropriate use of t requires that sampling distribution of X is
normal. This can result if N  30 or population of raw scores is normal.
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V. Size of Effect Using Cohen’s d
A. Rationale. The statistic we are using to measure size of effect is symbolized by
“d.” It is a standardized statistic that relies on the relationship between the size
of effect and X obt  μ . As the size of effect gets greater, so does X obt  μ ,
regardless of the direction of the effect. The statistic d uses the absolute value
of X obt  μ since we are interested in the size of real effect, and are not
concerned about direction. This allows d to have a positive value that
increases with the size of the difference between X obt and μ regardless of the
direction of the real effect. X obt  μ is divided by σ to create a standardized
value, much as was done with z scores.
B. Formula for Cohen’s d.
d
X obt  μ
σ
Conceptual equation for size of effect, single sample t test
Since σ is unknown, we estimate it using s, the sample standard deviation.
Substituting s for σ, we arrive at the computational equation for size of effect.
Since s is an estimate, d̂ is used instead of d.
X obt  μ
dˆ 
s
Computational equation for size of effect, single sample t test
C. Interpreting the Value of d̂ . To interpret the value of d̂ , we are using the
criteria that Cohen has provided. These criteria are given in the following
table.
Value of d̂
0.00 – 0.20
0.21 – 0.79
≥0.80
Interpretation of d̂
Small effect
Medium effect
Large effect
VI. Confidence Intervals for the Population Mean
A. Definition. A confidence interval is a range of values which probably contains
the population mean. Confidence limits are the values that bound the
confidence interval. Example: The 95% confidence interval is an interval such
that the probability is 0.95 that the interval contains the population value.
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B. Formula for confidence interval.
μlower  X obt  sX tcrit
μupper  X obt  sX tcrit
VII. Testing Significance of Pearson r.
A. Rho (). this is the Greek letter to symbolize the population correlation
coefficient.
B. Nondirectional H0. Asserts   0.
C. Directional H0 Asserts that  is positive or negative depending on the predicted
direction of the relationship.
D. Sampling distribution of r. Generated by taking all samples of size N from a
population in which  = 0 and calculating r for each sample. By systematically
varying the population scores and N, the sampling distribution of r is generated.
E. Using t test to evaluate significance of r:
tobt 
robt  ρ
1  robt 2
; where sr 
sr
N 2
with df = N  2 and N equals the number of pairs of X, Y scores.
F. Using rcrit to evaluate the significance of r:
if | robt |  | rcrit |, reject H0
The values of rcrit can be calculated directly. Values of rcrit are shown in Table
E.