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Discussion section 227
10/19/10
Onesample ttest
•
t tests in general allow us to compare two groups. The onesample ttest is concerned
with comparing one sample to one population and assumes that we don’t know the
value of the population standard deviation (sigma). Therefore we need to estimate it
based on the information from our sample.
•
The test statistic we use here is the tstatistic. It indicates the distance of a sample mean
from a population mean in terms of the estimated standard error
. The formula is very
similar to the Z test but uses the estimated standard error noted as s
m
:
To find the critical values that will define the critical regions we use the t table. Unlike the Z
table, the critical values needed to say a sample mean is in the extreme part of the distribution
will depend on our sample size or degrees of freedom.
What are the degrees of freedom?
Degrees of freedom for one sample are defined as n1
. When we are estimating one population
parameter from a sample statistic, n1 scores are free to vary (can take on any value) in our
sample before the last score is restricted. The notation for degrees of freedom is
df
.
Because we are estimating the standard deviation, there is more chance for error and so the t
distribution does not look as “normal” as the Z distribution (it is wider to reflect more
variability). As sample size increases, it becomes more like a normal curve.
Example 1
A library system lends books for periods of 21 days. This policy is being reevaluated in view of a
possible new loan period that could be either shorter or longer than 21 days. To aid in this
decision, booklending records were consulted to determine the loan periods actually used by
the patrons. A random sample of eight records revealed the following loan periods in days:
21,15,12,24,20,21,13, and 16. Test the null hypothesis with t, using 0.05 level of significance.
M = 17.75
n = 8
s = 4.33
μ = 21
Step 1: state the hypothesis.
H
0
: μ = 21
H
1
: μ ≠ 21
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View Full DocumentStep 2: locate critical regions.
This depends on the degrees of freedom and whether the test is directional or not.
In this case df = n1 = 81 = 7 and a twotailed test. So that’s what we’ll look for in our t
table. The critical value here is 2.365.
Step 3: calculate test statistic.
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 Fall '10
 Fairchild

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