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Chapter 8
Tests of Hypotheses Based
on a Single Sample
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.1
Nonstatistical Hypothesis Testing…
A criminal trial is an example of hypothesis testing without
the statistics.
In a trial a jury must decide between two hypotheses. The
null hypothesis is
H0: The defendant is innocent
The alternative hypothesis or research hypothesis is
H1: The defendant is guilty
The jury does not know which hypothesis is true. They must
make a decision on the basis of evidence presented.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.2
Nonstatistical Hypothesis Testing…
In the language of statistics convicting the defendant is
called rejecting the null hypothesis in favor of the alternative
hypothesis. That is, the jury is saying that there is enough
evidence to conclude that the defendant is guilty (i.e., there
is enough evidence to support the alternative hypothesis).
If the jury acquits it is stating that there is not enough
evidence to support the alternative hypothesis. Notice that
the jury is not saying that the defendant is innocent, only that
there is not enough evidence to support the alternative
hypothesis. That is why we never say that we accept the null
hypothesis.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.3
Nonstatistical Hypothesis Testing…
There are two possible errors.
A Type I error occurs when we reject a true null hypothesis.
That is, a Type I error occurs when the jury convicts an
innocent person.
A Type II error occurs when we don’t reject a false null
hypothesis. That occurs when a guilty defendant is acquitted.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.4
Nonstatistical Hypothesis Testing…
The probability of a Type I error is denoted as α (Greek
letter alpha). The probability of a type II error is β (Greek
letter beta).
The two probabilities are inversely related. Decreasing one
increases the other.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.5
Nonstatistical Hypothesis Testing…
In our judicial system Type I errors are regarded as more
serious. We try to avoid convicting innocent people. We are
more willing to acquit guilty people.
We arrange to make α small by requiring the prosecution to
prove its case and instructing the jury to find the defendant
guilty only if there is “evidence beyond a reasonable doubt.”
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.6
Hypothesis Testing…
The critical concepts are these:
1. There are two hypotheses, the null and the alternative
hypotheses.
2. The procedure begins with the assumption that the null
hypothesis is true.
3. The goal is to determine whether there is enough evidence
to reject the truth of the null hypothesis.
4. There are two possible decisions:
Conclude there is enough evidence to reject the null
hypothesis. [Conclude alternative hypothesis is true]
Conclude there is not enough evidence to reject the null
hypothesis. [Conclude null hypothesis is true]
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.7
Nonstatistical Hypothesis Testing…
5. Two possible errors can be made.
Type I error: Reject a true null hypothesis
Type II error: Do not reject a false null hypothesis.
P(Type I error) = α
P(Type II error) = β
In practice, Type II errors are by far the most serious mistake
because you go away believing the null is true and it is not
true. Type I errors would allow you to conduct additional
testing to verify conclusion.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.8
Introduction…
In addition to estimation, hypothesis testing is a procedure
for making inferences about a population.
Population
Sample
Inference
Statistic
Parameter
Hypothesis testing allows us to determine whether enough
statistical evidence exists to conclude that a belief (i.e.
hypothesis) about a parameter is supported by the data.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.9
Statistical Hypotheses Testing…
A statistical hypothesis, or just hypothesis, is a claim either
about the value of a single parameter, or values of several
parameters, or the form of an entire probability.
Example:
Test if population mean µ = 350
Test if population mean µ is between 340 and 360
Test if the samples come from normal distribution
etc……
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.10
Exercise 1…
For each of the following assertion, state whether it is a
legitimate statistical hypothesis and why:
1. H: σ > 100
2. H: ~
x = 45
3. H: s ≤ .20
4. H: σ1/σ2 < 1
5. H: X − Y = 5
6. H: λ ≤ .01, where λ is the parameter of an exponential
distribution used to model component lifetime.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.11
Concepts of Hypothesis Testing …
There are two hypotheses. One is called the null hypothesis
and the other the alternative or research hypothesis. The
usual notation is:
pronounced
H “nought”
H0: — the ‘null’ hypothesis
H1: — the ‘alternative’ or ‘research’ hypothesis
The null hypothesis (H0) will always include “=“ [two tail
test], “<“ [one tail test], and “>” [one tail test] in this course
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.12
Concepts of Hypothesis Testing…
Consider a problem dealing with the mean demand for
computers during assembly lead time. Rather than estimate
the mean demand, our operations manager wants to know
whether the mean is different from 350 units. We can
rephrase this request into a test of the hypothesis:
H0:
= 350
Thus, our research hypothesis becomes:
H1: ≠ 350
This is what we are interested
in determining…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.13
Concepts of Hypothesis Testing …
The testing procedure begins with the assumption that the
null hypothesis is true.
Thus, until we have further statistical evidence, we will
assume:
H0:
= 350 (assumed to be TRUE)
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11.14
Concepts of Hypothesis Testing …
The goal of the process is to determine whether there is
enough evidence to infer that the null hypothesis is most
likely not true and consequently the alternative hypothesis
then must be true.
That is, is there sufficient statistical information to determine
if this statement:
H1: ≠ 350, is true?
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.15
Concepts of Hypothesis Testing …
There are two possible decisions that can be made:
Conclude that there is enough evidence to reject the null
hypothesis
Conclude that there is not enough evidence to reject the
null hypothesis
NOTE: we do not say that we accept the null hypothesis,
although most people in industry will say this.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.16
Concepts of Hypothesis Testing…
Once the null and alternative hypotheses are stated, the next
step is to randomly sample the population and calculate a test
statistic (in this example, the sample mean).
If the test statistic’s value is inconsistent with the null
hypothesis we reject the null hypothesis and infer that the
alternative hypothesis is true.
For example, if we’re trying to decide whether the mean is
not equal to 350, a large value of (say, 600) would provide
enough evidence. If is close to 350 (say, 355) we could
not say that this provides a great deal of evidence to infer
that the population mean is different than 350. Is 355 really close??
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.17
Concepts of Hypothesis Testing …
Two possible errors can be made in any test:
A Type I error occurs when we reject a true null hypothesis
and
A Type II error occurs when we don’t reject a false null
hypothesis.
There are probabilities associated with each type of error:
P(Type I error) =
P(Type II error ) =
Maximum value of α is called the significance level.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.18
Types of Errors…
A Type I error occurs when we reject a true null hypothesis
(i.e. Reject H0 when it is TRUE)
H0
T
Reject
I
Reject
F
II
A Type II error occurs when we don’t reject a false null
hypothesis (i.e. Do NOT reject H0 when it is FALSE)
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.19
Types of Errors…
Back to our example, we would commit a Type I error if:
Reject H0 when it is TRUE
We reject H0 ( = 350) in favor of H1 ( ≠ 350) when in
fact the real value of is 350.
We would commit a Type II error in the case where:
Do NOT reject H0 when it is FALSE
We believe H0 is correct ( = 350), when in fact the real
value of is something other than 350.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.20
Exercise 2…
For the following pairs of assertions, indicate which do not
comply with our rules for setting up hypotheses and why
(the subscripts 1 and 2 differentiate between quantities for
two different populations or samples)
1. H0: µ = 100, Ha: µ > 100
2. H0: σ = 20, Ha: σ ≤ 20
3. H0: p ≠ .25, Ha: p = .25
4. H0: µ1 – µ2 = 25, Ha: µ1 – µ2 > 100
5. H0:S12 = S 22 , Ha: S12 ≠ S 22
6. H0: µ = 120, Ha: µ = 150
7. H0: σ1/σ2 = 1, Ha: σ1/σ2 ≠ 1
8. H0: p1 – p2 = –.1, Ha: p1 – p2 < –.1
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.21
Exercise 3…
To determine whether the pipe welds in a nuclear power
plant meet specifications, a random sample of welds is
selected, and tests are conducted on each weld in the sample.
Weld strength is measured as the force required to break the
weld. Suppose the specifications state that the mean strength
of welds should exceed 100 lb/in2; the inspection team
decides to test H0: µ = 100 versus Ha: µ > 100. Explain why
it might be preferable to use this Ha rather than µ < 100.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.22
Exercise 4…
Let µ denote the true average radioactivity level (picocuries
per litter). The value 5pCi/L is considered the dividing line
between safe and unsafe water. Would you recommend
testing H0: µ = 5 vs. Ha: µ > 5 or H0: µ = 5 vs. Ha: µ < 5 ?
Explain your reasoning. [Hint: Think about the consequnces
of a type I and type II error for each possibility.]
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.23
Recap I…
1) Two hypotheses: H0 & H1
2) ASSUME H0 is TRUE
3) GOAL: determine if there is enough evidence to infer that
H1 is TRUE
4) Two possible decisions:
Reject H0 in favor of H1
NOT Reject H0 in favor of H1
5) Two possible types of errors:
Type I: reject a true H0 [P(Type I)= ]
Type II: not reject a false H0 [P(Type II)= ]
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.24
Recap II…
The null hypothesis must specify a single value of the
parameter (e.g. =___)
Assume the null hypothesis is TRUE.
Sample from the population, and build a statistic related to
the parameter hypothesized (e.g. the sample mean, )
Compare the statistic with the value specified in the first step
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.25
Example …
A department store manager determines that a new billing
system will be cost-effective only if the mean monthly
account is more than $170. [One tail test]
A random sample of 400 [n] monthly accounts is drawn, for
which the sample mean is $178 [X-bar]. The accounts are
approximately normally distributed with a standard deviation
of $65 [Sigma].
Can we conclude that the new system will be cost-effective?
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.26
Example …
The system will be cost effective if the mean account
balance for all customers is greater than $170.
We express this belief as a our research hypothesis, that is:
H1:
> 170 (this is what we want to determine)
Thus, our null hypothesis becomes:
H0: < 170 (we use the “=“ part to conduct the test)
Some people will leave the less than part off of the null hypothesis.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.27
Example …
What we want to show:
H1: > 170
H0: < 170 (we’ll assume the = is true)
We know:
n = 400,
= 178, and
= 65
What to do next?! Determine sampling distribution of X-bar.
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11.28
Example …
To test our hypotheses, we can use two different approaches:
The rejection region approach based on sampling
distribution of X-bar (typically used when computing
statistics manually), and
The p-value approach (which is generally used with a
computer and statistical software and also based on the
sampling distribution of X-bar).
We will explore both …
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.29
Example … Rejection Region…[Based on sampling distribution
of X-bar]
The rejection region is a range of values such that if the test
statistic falls into that range, we decide to reject the null
hypothesis in favor of the alternative hypothesis.
is the critical value of
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
to reject H0.
11.30
Example …
It seems reasonable to reject the null hypothesis in favor of
the alternative if the value of the sample mean is large
relative to 170, that is if
>
. Normally stated as “if Xbar is greater than the critical value”.
= P( > )
is also…
= P(rejecting H0 given that H0 is true)
= P(Type I error)
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11.31
Example …
All that’s left to do is calculate
and compare it to 170.
we can calculate this based on any level of
significance ( ) we want…
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11.32
Example …
At a 5% significance level (i.e.
=0.05), we get
Solving we compute
=175.34
Since our sample mean (178) is greater than the critical value we
calculated (175.34), we reject the null hypothesis in favor of H1, i.e.
that:
> 170 and that it is cost effective to install the new billing
system
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11.33
Example: The Big Picture…
H1:
H0:
> 170
= 170
=175.34
=178
Reject H0 in favor of
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11.34
Standardized Test Statistic…
An easier method is to use the standardized test statistic:
and compare its result to
: (rejection region: z > )
Since z = 2.46 > 1.645 (z.05), we reject H0 in favor of H1…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.35
Conclusions of a Test of Hypothesis…
If we reject the null hypothesis, we conclude that there is
enough evidence to infer that the alternative hypothesis is
true.
If we do not reject the null hypothesis, we conclude that
there is not enough statistical evidence to infer that the
alternative hypothesis is true.
Remember: The alternative hypothesis is the more
important one. It represents what we are investigating.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.36
One– and Two–Tail Testing…
The department store example was a one tail test, because
the rejection region is located in only one tail of the
sampling distribution:
More correctly, this was an example of a right tail test.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.37
Right-Tail Testing…
Calculate the critical value of the mean ( ) and compare
against the observed value of the sample mean ( )…
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11.38
Left-Tail Testing…
Calculate the critical value of the mean ( ) and compare
against the observed value of the sample mean ( )…
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11.39
Two–Tail Testing…
Two tail testing is used when we want to test a research
hypothesis that a parameter is not equal (≠) to some value
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11.40
Example …
AT&T’s argues that its rates are such that customers won’t see a
difference in their phone bills between them and their competitors.
They calculate the mean and standard deviation for all their customers
at $17.09 and $3.87 (respectively).
These are assumed to be the values of the population parameters [mean
and standard deviation]
You don’t believe them, so you sample 100 customers at random and
calculate a monthly phone bill based on competitor’s rates.
What we want to show is whether or not:
H1: ≠ 17.09. We do this by assuming that:
H0: = 17.09
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11.41
Example …
At a 5% significance level (i.e. = .05), we have
/2 = .025. Thus, z.025 = 1.96 and our rejection region is:
z < –1.96
-z.025
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-or-
0
z > 1.96
+z.025
z
11.42
Example …
From the data, we calculate
= 17.55
Using our standardized test statistic:
We find that:
Since z = 1.19 is not greater than 1.96, nor less than –1.96
we cannot reject the null hypothesis in favor of H1. That is
“there is insufficient evidence to infer that there is a
difference between the bills of AT&T and the competitor.”
Does not mean that you have proven that “the true mean is $17.09”.
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11.43
Probability of a Type II Error –
It is important that that we understand the relationship
between Type I and Type II errors; that is, how the
probability of a Type II error is calculated and its
interpretation.
Recall example about department store …
H0: = 170
H1: > 170
At a significance level of 5% we rejected H0 in favor of H1
since our sample mean (178) was greater than the critical
value of
(175.34)
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11.44
Probability of a Type II Error –
A Type II error occurs when a false null hypothesis is not
rejected.
In example, this means that if is less than 175.34 (our
critical value) we will not reject our null hypothesis, which
means that we will not install the new billing system.
Thus, we can see that:
= P(
< 175.34 given that the null hypothesis is false)
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11.45
Example - Calculating P(Type II errors)…
Our original hypothesis…
our new assumption…
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11.46
Effects on
of Changing
Decreasing the significance level
and vice versa.
, increases the value of
Consider this diagram again. Shifting the critical value line
to the right (to decrease ) will mean a larger area under the
lower curve for … (and vice versa)
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11.47
Another example…
A certain type of automobile is known to sustain no visible
damage 25% of time in 10-mph crash tests. A modified
bumper design has been proposed in an effort to increase this
percentage. The experiment involves n = 20 independent
crashes with prototypes of the new design.
Let p denote the proportion of all 10-mph crashes with new
bumper that result in no visible damage.
Hypothesis:
H0: p = .25
Ha: p > .25
No Improvement
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11.48
Another example…
Intuitively, H0 should be rejected if a substantial number of
crashes show no damage.
How large is
enough ?
Test statistic: X = the number of crashes with no visible
damages.
Rejection region: R8 = {8, 9, 10, … , 19, 20}
i.e, we will reject H0 if x ≥ 8.
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11.49
Calculating type I error…
When H0 is true, X has a binomial probability distribution
with n = 20 and p = .25
α = P(type I error ) = P( H 0 is rejected when it is true)
= P( X ≥ 8 when X ~ Bin(20,.25)) = 1 − B(7;20,.25)
= 1 − .898 = .102
So, when H0 is actually true, roughly 10% of all experiment
consisting of 20 crashes would result in H0 being incorrectly
rejected.
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11.50
Calculating type II error…
In contrast to α, there is no single β. Instead, there is a
different β for each different p that exceeds .25.
Let’s pick p =.3 (i.e, we assume that the true p = .3)
β (.3) = P(type II error when p = .3)
= P ( H 0 is not rejected when it is false because p = .3)
= P ( X ≤ 7 when X ~ Bin(20,.3)) = B (7;20,.3) = .772
When p is actually .3 rather than .25 (a “small” departure
from H0), roughly 77% of all experiments of this type would
result in H0 being incorrectly not rejected!
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.51
Calculating type II error…
Picking different values of p results different type II error
p
β(p)
.3
.772
.4
.416
.5
.132
.6
.021
.7
.001
.8
.000
Clearly, β decreases as the value of p moves farther to the
right of the null value .25
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11.52
Example…
What if we want a smaller α ?
Reduce the rejection region to R9={9,10,…,19,20}
α = P(type I error ) = P( H 0 is rejected when it is true)
= P( X ≥ 9 when X ~ Bin(20,.25)) = 1 − B(8;20,.25)
= .041
β (.3) = P( H 0 is not rejected when it is false because p = .3)
= P ( X ≤ 8 when X ~ Bin(20,.3)) = B(8;20,.3) = .887
β (.5) = B(8;20,.5) = .252
Compare these with the previous β’s for R8 .772 and
.132 respectively
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.53
Judging the Test…
A statistical test of hypothesis is effectively defined by the
significance level (
) and the sample size (n), both of
which are selected by the statistics practitioner.
Therefore, if the probability of a Type II error (
to be too large, we can reduce it by
increasing , and/or
increasing the sample size, n.
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) is judged
11.54
Judging the Test…
The power of a test is defined as 1– .
It represents the probability of rejecting the null hypothesis
when it is false [the true mean is some value other than that
specified by the null hypothesis].
I.e. when more than one test can be performed in a given
situation, its is preferable to use the test that is correct more
often. If one test has a higher power than a second test, the
first test is said to be more powerful and the preferred test.
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11.55