Chapter 11 Introduction to Hypothesis Testing 11.1

Transcription

Chapter 11
Introduction to Hypothesis
Testing
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.
11.2
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.
11.3
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.
11.4
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]
11.5
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.
11.6
Concepts of Hypothesis Testing (1)…
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]
11.7
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…
11.8
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)
11.9
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?
11.10
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.
11.11
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??
11.12
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)
11.13
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.
11.14
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)= ]
11.15
Example 11.1…
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?
11.16
Example 11.1…
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.
11.17
Example 11.1…
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.
11.18
Example 11.1…
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 …
11.19
Example 11.1… 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
to reject H0.
11.20
Example 11.1…
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)
11.21
Example 11.1…
All that’s left to do is calculate
and compare it to 170.
Solve for the upper critical value. [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
11.22
Example 11.1… The Big Picture…
H1:
H0:
> 170
= 170
=175.34
=178
Reject H0 in favor of
11.23
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…
11.24
p-Value
The p-value of a test is the probability of observing a test
statistic at least as extreme as the one computed given that
the null hypothesis is true.
In the case of our department store example, what is the
probability of observing a sample mean at least as extreme
as the one already observed (i.e. = 178), given that the null
hypothesis (H0: = 170) is true?
p-value
11.25
Interpreting the p-value…
The smaller the p-value, the more statistical evidence exists
to support the alternative hypothesis.
•If the p-value is less than 1%, there is overwhelming
evidence that supports the alternative hypothesis.
•If the p-value is between 1% and 5%, there is a strong
•If the p-value is between 5% and 10% there is a weak
•If the p-value exceeds 10%, there is no evidence that
supports the alternative hypothesis.
We observe a p-value of .0069, hence there is
overwhelming evidence to support H1: > 170.
11.26
Interpreting the p-value…
Compare the p-value with the selected value of the
significance level:
If the p-value is less than , we judge the p-value to be
small enough to reject the null hypothesis.
If the p-value is greater than , we do not reject the null
hypothesis.
Since p-value = .0069 <
= .05, we reject H0 in favor of H1
11.27
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.
11.28
Chapter-Opening Example…
The objective of the study is to draw a conclusion about the
mean payment period. Thus, the parameter to be tested is the
population mean. We want to know whether there is enough
statistical evidence to show that the population mean is less
than 22 days. Thus, the alternative hypothesis is
H1:μ < 22
The null hypothesis is
H0:μ > 22
11.29
The [standardized]test statistic is
x 
z
/ n
We wish to reject the null hypothesis in favor of the
alternative only if the sample mean and hence the value of
the test statistic is small enough. As a result we locate the
rejection region in the left tail of the sampling distribution.
We set the significance level at alpha = .10 or 10%.
11.30
z   z    z.10  1.28
Rejection region:
From the data in SSA we compute
and
x

x
4,759

 21 .63
220
220
z
x 
/ n
i

21 .63  22
 .91
6 / 220
p-value = P(Z < -.91) = .5 - .3186 = .1814
11.31
Conclusion: Since -.91 > -1.28, there is not enough evidence
to infer that the mean is less than 22 at a 10% level of
significance. [reason, conclusion, level of significance]
There is not enough evidence to infer that the plan will be
profitable.
11.32
One– and Two–Tail Testing…
11.33
Example 11.2…
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
11.34
Example 11.2…
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
-or-
0
z > 1.96
+z.025
z
11.35
Example 11.2…
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”.
11.36
Summary of One- and Two-Tail Tests…
One-Tail Test
(left tail)
Two-Tail Test
One-Tail Test
(right tail)
11.37
Example 11.1 - Calculating P(Type II errors)…
Our original hypothesis…
our new assumption…
11.38
Effects on
of Changing
Decreasing the significance level , increases the value of
and vice versa.
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)
11.39
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.
) is judged
11.40
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.
11.41

Chapter 11 Introduction to Hypothesis Testing 11.1

Transcription

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