Transcript Chapter 8 - The Citadel

Section 8-2
Basics of Hypothesis
Testing
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8.1 - 1
Rare Event Rule for
Inferential Statistics
If, under a given assumption, the
probability of a particular observed event
is exceptionally small, we conclude that
the assumption is probably not correct.
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Components of a
Formal Hypothesis
Test
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Null Hypothesis:
H0
• The null hypothesis (denoted by H 0 ) is
a statement that the value of a
population parameter (such as
proportion, mean, or standard
deviation) is equal to some claimed
value.
•
We test the null hypothesis directly.
•
Either reject H 0 or fail to reject H 0 .
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Alternative Hypothesis:
H1
• The alternative hypothesis (denoted
by H1 or H a or H A ) is the statement that
the parameter has a value that
somehow differs from the null
hypothesis.
• The symbolic form of the alternative
hypothesis must use one of these
symbols: , ,  .
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Note about Forming Your
Own Claims (Hypotheses)
If you are conducting a study and want
to use a hypothesis test to support
your claim, the claim must be worded
so that it becomes the alternative
hypothesis.
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Note about Identifying
H 0 and H1
Figure 8-2
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Example:
a) The mean annual income of employees who took
a statistics course is greater than $60,000.
b) The proportion of people aged 18 to 25 who
currently use illicit drugs is equal to 0.20.
c) The standard deviation of human body
temperatures is equal to 0.62 degrees F.
d) The majority of college students have credit
cards.
e) The standard deviaton of duration times of the
Old Faithful geyser is less than 40 sec.
f) The mean weight of airline passengers (including
carry-on bags) is at most 195 lbs.
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Test Statistic
The test statistic is a value used in making
a decision about the null hypothesis, and is
found by converting the sample statistic to
a score with the assumption that the null
hypothesis is true.
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Test Statistic - Formulas
pˆ  p
z
pq
n
Test statistic for
proportion
Test statistic
for mean
z
x 

n
Test statistic for
standard deviation
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x 
or t 
s
n
n  1 s

 
2
2
2

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Critical Region
The critical region (or rejection region) is the
set of all values of the test statistic that
cause us to reject the null hypothesis. For
example, see the red-shaded region in the
previous figure.
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Significance Level
The significance level (denoted by  ) is the
probability that the test statistic will fall in the
critical region when the null hypothesis is
actually true. This is the same  introduced
in Section 7-2. Common choices for  are
0.05, 0.01, and 0.10.
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P-Value
The P-value (or p-value or probability value)
is the probability of getting a value of the test
statistic that is at least as extreme as the one
representing the sample data, assuming that
the null hypothesis is true.
Critical region
in the left tail:
P-value = area to the left of
the test statistic
Critical region
in the right tail:
P-value = area to the right of
the test statistic
Critical region
in two tails:
P-value = twice the area in the
tail beyond the test statistic
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Caution
Don’t confuse a P-value with a proportion p.
Know this distinction:
P-value = probability of getting a test
statistic at least as extreme as
the one representing sample
data
p = population proportion
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Decision Criterion
P-value method:
Using the significance level  :
If P-value   , reject H 0 .
If P-value   , fail to reject H 0 .
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Decision Criterion
Find the P-value and determine the decision about the
null hypothesis if the test is conducted at the 0.05
level of significance.
a) H1:  < 95 and z = -1.89
b) H1: p is not equal to 0.6 and z = 1.62
c) H1:  > 41 and z = 2.05
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Wording of Final Conclusion
Figure 8-7
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Caution
Never conclude a hypothesis test with a
statement of “reject the null hypothesis”
or “fail to reject the null hypothesis.”
Always make sense of the conclusion
with a statement that uses simple
nontechnical wording that addresses the
original claim.
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Example
State the final conclusion.
a) Original claim: The percentage of blue M&Ms is
greater than 5%.
Decision: Fail to reject the null hypothesis.
a) Original claim: The percentage of Americans who
know their credit score is equal to 20%.
Decision: Fail to reject the null hypothesis.
a) Original claim: The percentage of on-time U.S.
airline flights is less than 75%.
Decision: Reject the null hypothesis.
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Type I and Type II Errors
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Controlling Type I and
Type II Errors
• For any fixed , an increase in the sample
size n will cause a decrease in  
• For any fixed sample size n, a decrease in
 will cause an increase in .
Conversely, an increase in  will cause a
decrease in  .
• To decrease both  and  , increase the
sample size.
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Hypothesis Test –P-Value Method
• State the null and alternative hypotheses in symbolic
form.
• Select the level of significance  based on the
seriousness of a type 1 error. The values of 0.05 and 0.01
are most common.
• Identify the statistic that is relevant to this test and
determine its sampling distribution (such as normal z, t, or
chi-square.)
• Find the test statistic and find the P-value. Draw a graph
and show the test statistic and P-value. Technology may
be used for this step.
• Reject H0 if the P-value is less than or equal to the level of
significance, , or Fail to reject H0 if the P-value is greater
than .
• Restate the previous decision in simple non-technical
terms, and address the original claim.
8.1 - 22
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