Transcript Test Statistic

Lecture Slides
Elementary Statistics
Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Section 8.2-1
Chapter 8
Hypothesis Testing
8-1 Review and Preview
8-2 Basics of Hypothesis Testing
8-3 Testing a Claim about a Proportion
8-4 Testing a Claim About a Mean
8-5 Testing a Claim About a Standard Deviation or
Variance
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Section 8.2-2
Definitions
A hypothesis is a claim or statement about a property of a
population.
A hypothesis test is a procedure for testing a claim about a
property of a population.
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Section 8.2-3
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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Section 8.2-4
Null Hypothesis
• The null hypothesis (denoted by H0) 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 in the sense
that we assume it is true and reach a conclusion
to either reject H0 or fail to reject H0.
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Section 8.2-5
Alternative Hypothesis
•
The alternative hypothesis (denoted by H1 or
HA) 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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Section 8.2-6
Examples
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 0.62oF.
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Section 8.2-7
Examples
d) The majority of college students have credit
cards.
e) The standard deviation of duration times (in
seconds) 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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Section 8.2-8
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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Section 8.2-9
Test Statistic - Formulas
z
Test statistic for
proportion
pˆ  p
pq
n
Test statistic
for mean
z
x 

or t 
x 
s
n
Test statistic for
standard deviation
n
 
2
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 n  1 s

2
2
Section 8.2-10
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 figures.
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Section 8.2-11
Significance Level
The significance level (denoted by α) is the
probability that the test statistic will fall in the
critical region when the null hypothesis is
actually true (making the mistake of rejecting the
null hypothesis when it is true).
This is the same α introduced in Section 7-2.
Common choices for α are 0.05, 0.01, and 0.10.
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Section 8.2-12
Find the Value of the Test Statistic, Then
Find Either the P-Value or the Critical
Value(s)
First transform the relevant sample statistic to a
standardized score called the test statistic.
Then find the P-Value or the critical value(s).
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Section 8.2-13
P-Value
The 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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Section 8.2-14
Left-tailed Test
H 0 :
H 1 :
All α in the left tail
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Section 8.2-15
Right-tailed Test
H 0 :
H 1 :
All α in the right tail
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Section 8.2-16
Two-tailed Test
H 0 :
H 1 :
α is divided equally between the two
tails of the critical region
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Section 8.2-17
Decision Criterion: P-Value
Method
The null hypothesis is rejected if the Pvalue is very small.
Using the significance level :
If P-value , then Reject H0
If P-value > , then Fail to reject H0
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Section 8.2-18
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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Section 8.2-19
Example
For each test, find the P-value and determine the
decision about the null hypothesis if the test is
conducted at a 0.05 level of significance.
a) If H1:  < 95, and the test statistic has been
found to be z = –1.89.
b) If H1: p 0.6, and the test statistic has been
found to be z = 1.62.
c) If H1:  > 41 and the test statistic has been
found to be z = 2.05.
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Section 8.2-20
Restate the Decision Using Simple
and Nontechnical Terms
State a final conclusion that addresses
the original claim with wording that can
be understood by those without
knowledge of statistical procedures.
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Section 8.2-21
Wording of Final Conclusion
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Section 8.2-22
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.
b) Original claim: The percentage of Americans who know
their credit score is equal to 20%.
Decision: Fail to reject the null hypothesis.
c) Original claim: The percentage of on-time U.S. airline
flights is less than 75%.
Decision: Reject the null hypothesis.
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Section 8.2-23
Type I Error
• A Type I error is the mistake of
rejecting the null hypothesis when it is
actually true.
• The symbol α is used to represent the
probability of a type I error.
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Section 8.2-24
Type II Error
• A Type II error is the mistake of failing to
reject the null hypothesis when it is
actually false.
• The symbol β (beta) is used to represent
the probability of a type II error.
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Section 8.2-25
Type I and Type II Errors
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Section 8.2-26
Controlling Type I and
Type II Errors
• For any fixed α, an increase in the sample size
n will cause a decrease in β
• For any fixed sample size n, a decrease in α will
cause an increase in β. Conversely, an
increase in α will cause a decrease in β.
• To decrease both α and β, increase the sample
size.
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Section 8.2-27
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.
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Section 8.2-28