HYPOTHESIS TESTING:
ONE SAMPLE Z AND T TESTS
If the sample size is small and σ is not known, then the tstatistic is used instead of Z.
How do I know what test to use?
Use Zstatistic if:
Use tstatistic if:
If the population standard
deviation
σ
is known or the sample
is more than 30.
If the population standard
deviation (
σ
)is unknown and the
sample is less than 30.
Comparison of Zstatistic and tstatistic formulas
Zstatistic
Tstatistic
Use the tstatistic for testing a sample mean against a population mean if σ is unknown and n<30.
When using the tstatistic, you need to find the critical value that corresponds to n1 degrees of freedom.
Distributions of the tstatistic symmetrical and bellshaped but are flatter and more spread out (greater
variability).
Tdistribution is the same as normal distribution when sample size is infinite
The critical value for the tstatistic uses the tDistribution Table.
1.
Decide whether you’re doing a one or twotailed test. (directional vs. nondirectional)
2.
Select your alpha level.
3.
Find the row that corresponds to
n1
degrees of freedom.
Table C:
Percentile points of
t
Distribution (p.621)
Example:
95% confidence, twotailed test, n=14
Critical value is 2.160
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What are degrees of freedom?
The number in a sample that are free to vary.
The sample mean places a restriction on the value of one
score in the sample.
Therefore, there are
n1
values that are free to vary.
We have
n1
degrees of freedom with the
t statistic.
Review Questions
1.
As the value for
df
gets smaller, the
t
distribution resembles a normal distribution more and more.
(true or false?)
2.
As the value for
df
gets larger,
s
provides a better estimate of σ.
(true or false?)
3.
For
df =
10, what
t
values are associated with:
a.
The top 1% of the t distribution
b.
The bottom 5% of the t distribution
c.
A twotailed distribution for 99% confidence.
Example 1
The average writing score on a test in the population is 368.
A random sample of 25 students took the test
with a sample mean of 372.5 with a sample standard deviation of 15.
Do the students have a different
mean than the population? Test at the 0.05 level.
1. State the null and alternative hypotheses.
2. Select a level of significance (i.e. 0.10, 0.05, 0.01)
3. Identify the test statistic
4. Formulate a decision rule
5.
Take a sample and arrive at decision:
a)
Calculate test statistic
b)
Compare it to our critical value(s).
c)
Make a decision.
Example 3:
An insurance company reports the mean cost to process a claim is $60. An industry comparison showed this
amount to be larger than most other insurance companies, so the company instituted costcutting measures. To
evaluate the effect of the costcutting measures, the Supervisor of the Claims Department selected a random sample
of 26 claims processed last month.
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 '10
 MURPHY,PRICILLAK
 Normal Distribution, Variance, Statistical hypothesis testing, critical value, DI, Student's ttest

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