The one sample t-test
November 14, 2006
From Z to t…
• In a Z test, you compare your sample to a known
population, with a known mean and standard
deviation.
• In real research practice, you often compare two
or more groups of scores to each other, without
any direct information about populations.
– Nothing is known about the populations that the
samples are supposed to come from.
The t Test for a Single Sample
• The single sample t test is used to
compare a single sample to a population
with a known mean but an unknown
variance.
• The formula for the t statistic is similar in
structure to the Z, except that the t statistic
uses estimated standard error.
From Z to t…
X   hyp
t
X   hyp
Z
s

X
X
X 
n
( X   )

N

sX 

2
NX  (X )
N2
2
Note
lowercase
“s”.
s
2
s
n
2
(
X

X
)

n 1
nX 2  (X ) 2
s
n(n  1)
Degrees of Freedom
• The number you divide by (the number of
scores minus 1) to get the estimated
population variance is called the degrees
of freedom.
• The degrees of freedom is the number of
scores in a sample that are “free to vary”.
Degrees of Freedom
• Imagine a very simple situation in which the
individual scores that make up a distribution are
3, 4, 5, 6, and 7.
• If you are asked to tell what the first score is
without having seen it, the best you could do is a
wild guess, because the first score could be any
number.
• If you are told the first score (3) and then asked
to give the second, it too could be any number.
Degrees of Freedom
• The same is true of the third and fourth scores –
each of them has complete “freedom” to vary.
• But if you know those first four scores (3, 4, 5,
and 6) and you know the mean of the
distribution (5), then the last score can only be 7.
• If, instead of the mean and 3, 4, 5, and 6, you
were given the mean and 3, 5, 6, and 7, the
missing score could only be 4.
The t Distribution
• In the Z test, you learned that when the
population distribution follows a normal curve,
the shape of the distribution of means will also
be a normal curve.
• However, this changes when you do hypothesis
testing with an estimated population variance.
– Since our estimate of  is based on our sample…
– And from sample to sample, our estimate of  will
change, or vary…
– There is variation in our estimate of , and more
variation in the t distribution.
The t Distribution
• Just how much the t distribution differs from the normal
curve depends on the degrees of freedom.
• The t distribution differs most from the normal curve
when the degrees of freedom are low (because the
estimate of the population variance is based on a very
small sample).
• Most notably, when degrees of freedom is small,
extremely large t ratios (either positive or negative) make
up a larger-than-normal part of the distribution of
samples.
The t Distribution
• This slight difference in shape affects how extreme a
score you need to reject the null hypothesis.
• As always, to reject the null hypothesis, your sample
mean has to be in an extreme section of the comparison
distribution of means.
The t Distribution
• However, if the distribution has more of its means in the
tails than a normal curve would have, then the point
where the rejection region begins has to be further out
on the comparison distribution.
• Thus, it takes a slightly more extreme sample mean to
get a significant result when using a t distribution than
when using a normal curve.
The t Distribution
• For example, using the normal curve, 1.96 is the cut-off for a twotailed test at the .05 level of significance.
• On a t distribution with 3 degrees of freedom (a sample size of 4),
the cutoff is 3.18 for a two-tailed test at the .05 level of significance.
• If your estimate is based on a larger sample of 7, the cutoff is 2.45, a
critical score closer to that for the normal curve.
The t Distribution
• If your sample size is infinite, the t distribution is
the same as the normal curve.
•
•
•
Since it takes into
account the changing
shape of the
distribution as n
increases, there is a
separate curve for
each sample size (or
degrees of freedom).
However, there is not
enough space in the
table to put all of the
different probabilities
corresponding to
each possible t score.
The t table lists
commonly used
critical regions (at
popular alpha levels).
•
•
If your study has
degrees of freedom
that do not appear
on the table, use
the next smallest
number of degrees
of freedom.
Just as in the
normal curve table,
the table makes no
distinction between
negative and
positive values of t
because the area
falling above a
given positive
value of t is the
same as the area
falling below the
same negative
value.
The t Test for a Single Sample:
Example
You are a chicken farmer… if only you had paid more
attention in school. Anyhow, you think that a new type of
organic feed may lead to plumper chickens. As every
chicken farmer knows, a fat chicken sells for more than a
thin chicken, so you are excited. You know that a
chicken on standard feed weighs, on average, 3 pounds.
You feed a sample of 25 chickens the organic feed for
several weeks. The average weight of a chicken on the
new feed is 3.49 pounds with a standard deviation of
0.90 pounds. Should you switch to the organic feed?
Use the .05 level of significance.
Hypothesis Testing
1.
2.
3.
4.
5.
6.
State the research question.
State the statistical hypothesis.
Set decision rule.
Calculate the test statistic.
Decide if result is significant.
Interpret result as it relates to your
research question.
The t Test for a Single Sample:
Example
• State the research question.
– Does organic feed lead to plumper chickens?
• State the statistical hypothesis.
HO :   3
HA :   3
•
Set decision rule.
  .05
df  25  1  24
t crit  1.711
The t Test for a Single Sample:
Example
• Calculate the test statistic.
t
X   hyp
sX
sX 
t
s
n

X   hyp
sX
0.90
25
 .18
3.49  3

 2.72
.18
The t Test for a Single Sample:
Example
• Decide if result is significant.
– Reject H0, 2.72 > 1.711
• Interpret result as it relates to your
research question.
– The chickens on the organic feed weigh
significantly more than the chickens on the
standard feed.
The t Test for a Single Sample:
Try in pairs
Odometers measure automobile mileage. How close to
the truth is the number that is registered? Suppose 12
cars travel exactly 10 miles (measured beforehand) and
the following mileage figures were recorded by the
odometers:
9.8, 10.1, 10.3, 10.2, 9.9, 10.4, 10.0, 9.9, 10.3, 10.0, 10.1, 10.2
Using the .01 level of significance, determine if you can
trust your odometer.
s = .19
Mean = 10.1
Hypothesis Testing
1.
2.
3.
4.
5.
6.
State the research question.
State the statistical hypothesis.
Set decision rule.
Calculate the test statistic.
Decide if result is significant.
Interpret result as it relates to your
research question.
The t Test for a Single Sample:
Example
• State the research question.
– Are odometers accurate?
• State the statistical hypotheses.
H O :   10
H A :   10
The t Test for a Single Sample:
Example
• Set the decision rule.
  .01
df  n  1  12  1  11
t crit  3.106
The t Test for a Single Sample:
Example
Calculate the
test statistic.
X
X2
9.8
96.04
10.1
102.01
10.3
106.09
10.2
104.04
9.9
98.01
10.4
108.16
10.0
100.00
9.9
98.01
10.3
106.09
10.0
100.00
10.1
102.01
10.2
104.04
121.20 1224.50
X 
121.20
 10.1
12
s
nX 2  ( X ) 2
n(n  1)
s
(12)1224.50  (121.20) 2
12(11)
s
14694  14689.44
132
4.56
132
s  .19
s
.19
sX 

 .06
n
12
s
t
X   hyp
sX

10.1  10.0
 1.67
.06
The t Test for a Single Sample:
Example
• Decide if result is significant.
– Fail to reject H0, 1.67<3.106
• Interpret result as it relates to your
research question.
– The mileage your odometer records is not
significantly different from the actual mileage
your car travels.
Confidence Intervals
• You can estimate a population mean based on
confidence intervals rather than statistical
hypothesis tests.
– A confidence interval is an interval of a certain width,
which we feel “confident” will contain the population
mean.
– You are not determining whether the sample mean
differs significantly from the population mean.
– Instead, you are estimating the population mean
based on knowing the sample mean.
When to Use Confidence Intervals
• If the primary concern is whether an effect
is present, use a hypothesis test.
• You should consider using a confidence
interval whenever a hypothesis test leads
you to reject the null hypothesis, in order
to determine the possible size of the
effect.
The t Test for a Single Sample:
Example
You are a chicken farmer… if only you had paid more
attention in school. Anyhow, you think that a new type of
organic feed may lead to plumper chickens. As every
chicken farmer knows, a fat chicken sells for more than a
thin chicken, so you are excited. You know that a
chicken on standard feed weighs, on average, 3 pounds.
You feed a sample of 25 chickens the organic feed for
several weeks. The average weight of a chicken on the
new feed is 3.49 pounds with a standard deviation of
0.90 pounds. Should you switch to the organic feed?
Construct a 95 percent confidence interval for the
population mean, based on the sample mean.
The t Test for a
Single Sample:
Example
Construct a 95
percent
confidence
interval.
X  (t conf )( s X )
3.49  (2.064)(
0.9
25
3.49  (2.064)(. 18)
3.49  .37
3.86

3.12
)
The t Test for a
Single Sample:
Example
Construct a 99
percent
confidence
interval.
X  (t conf )( s X )
3.49  (2.797)(
0 .9
25
3.49  (2.797)(. 18)
3.49  .50
3.99

2.99
)
Confidence Intervals
• Notice that the 99 percent confidence
interval is wider than the corresponding 95
percent confidence interval.
• The larger the sample size, the smaller the
standard error, and the narrower (more
precise) the confidence interval will be.
Confidence Intervals
•It’s tempting to claim that
once a particular 95 percent
confidence interval has been
constructed, it includes the
unknown population mean
with a 95 percent
probability.
•However, any one particular
confidence interval either
does contain the population
mean, or it does not.
•If a series of confidence
intervals is constructed to
estimate the same
population mean,
approximately 95 percent of
these intervals should
include the population mean.
Next Week
• Finish Chps. 12 & 13
• You are now ready to ready to do the
tutorial and the first problem set of
homework #4