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381
Hypothesis Testing
(Introduction-II)
QSCI 381 – Lecture 26
(Larson and Farber, Sect 7.1)
Overview
381

To test a claim using data, we:


Develop two statistical hypotheses
(the null and alternative hypotheses).
Select a level of significance (which
determines the level of type I error – the
probability of (unintentionally) rejecting the
null hypothesis when it is true).
Statistics Tests -I
381

We now need to summarize the data in the
form of a
. Common examples
of test statistics and their associated sampling
distributions are:
Population
Parameter
Test
Statistic
Sampling
Distribution
Standardized
test statistic

x
Normal (n30)
Student t
p
p̂
Normal
z
t
z
2
s2
Chi-square
2
Statistics Tests -II
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
Therefore, given a claim related to , p
or 2:

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We select the appropriate test statistic
from the previous table.
We choose the appropriate sampling
distribution.
We standardize the test statistic.
Examples-I
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We wish to test the claim that 30% of
the diet of Pacific cod is walleye pollock.

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Test statistic =
Sampling distribution =
Standardized test statistic =
Examples-II
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Identify the null and alternative hypotheses,
the test statistic, the sampling distribution,
and comment on the appropriate levels of
type I and type II error:

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The probability of a building by a given contractor
collapsing is less than 1%.
The density of a fish species based on a survey
consisting of 15 trawls is 15 kg / ha.
The standard deviation of the survey is 5 kg / ha.
p-values
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
Assuming that the null hypothesis is
true, the
(or probability value)
of a hypothesis test is the probability of
obtaining a sample statistic with a value
as extreme or more extreme than the
one determined from the sample data.
p-values and Tests-I
381
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
The nature of a hypothesis test depends on whether
it is left-, right- or two-tailed. This in turn depends on
the nature of the alternative hypothesis (one- or twosided alternatives).
There are three cases:
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The alternative hypothesis contains the symbol “<“
(i.e. the null hypothesis involves the symbol “”).
The alternative hypothesis contains the symbol “>”
(i.e. the null hypothesis involves the symbol “”).
The alternative hypothesis contains the symbol “” (i.e.
the null hypothesis involves the symbol “=”).
p-values and Tests-II
381
Left tailed test:
Null hypothesis involves a
population
parameter  something
-2
0
2
Left-tailed test
-2
0
Two-tailed test
-2
0
2
Right-tailed test
2
The p-value is the total shaded area
and measures the probability of
getting a test statistic as extreme or
more extreme than the observed
value.
Making Decisions Based on p-values-I
381
1.
2.
3.
State the claim mathematically and verbally.
Identify the null and alternative hypotheses.
H0 = ?;
Ha = ?
Specify the level of significance.
=?
Determine the standardized sampling
distribution if the hypothesis is true (sketch it)
-2
0
2
Making Decisions Based on p-values-II
381
4.
5.
6.
Calculate the test statistic and its standardized value
(z in this case). (add it your sketch).
Find the p-value
Apply the decision rule:
0
z
Is the p-value less than or equal to ?
Yes
Reject H0
7.
Interpret the results
No
Fail to reject H0
Making Decisions Based on p-values-III
381
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Note that rejection of the null hypothesis is
not proof that the null hypothesis is false,
just that it is (very) unlikely.
Rejection of the null hypothesis is also not
proof that the alternative hypothesis is true.
The following lectures cover various
situations in which the algorithm outlined
above is used to make decisions regarding
hypotheses.
Caveat
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The inability to reject the null
hypothesis can arise because:
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The null hypothesis is true (is a null
hypothesis ever true?)
The sample size is too small to show that
the null hypothesis is false.
The null hypothesis may be rejected
even if it is not substantially false.