Transcript Chapter 16

CHAPTER 16:
Inference in Practice
Chapter 16 Concepts
2

Conditions for Inference in Practice

Cautions About Confidence Intervals

Cautions About Significance Tests

Planning Studies: Sample Size for Confidence
Intervals

Planning Studies: The Power of the Statistical
Test
Chapter 16 Objectives
3






Describe the conditions necessary for inference
Describe cautions about confidence intervals
Describe cautions about significance tests
Calculate the sample size for a desired margin of
error in a confidence interval
Define Type I and Type II errors
Calculate the power of a significance test
z Procedures
4
So far, we have met two procedures for statistical inference. When the “simple
conditions” are true: the data are an SRS, the population has a Normal
distribution and we know the standard deviation s of the population, a confidence
interval for the mean m is:
x ±z*
s
n
To test a hypothesis H0: m = m0 we use the one-sample z statistic:
z=
x - µ0
s
n
These are called z procedures because they both involve a one-sample z
statistic and use the standard Normal distribution.
Conditions for Inference in Practice
5
Any confidence interval or significance test can be trusted only under specific
conditions.
Where did the data come from?
When you use statistical inference, you are acting as if your data are a
random sample or come from a randomized comparative experiment.
• If your data don’t come from a random sample or randomized
comparative experiment, your conclusions may be challenged.
• Practical problems such as nonresponse or dropouts from an
experiment can hinder inference.
• Different methods are needed for different designs.
• There is no cure for fundamental flaws like voluntary response.
What is the shape of the population distribution?
Many of the basic methods of inference are designed for Normal populations.
• Any inference procedure based on sample statistics like the sample
mean that are not resistant to outliers can be strongly influenced by a
few extreme observations.
Cautions About Confidence Intervals
6
A sampling distribution shows how a statistic varies in repeated random
sampling.
This variation causes random sampling error because the statistic misses
the true parameter by a random amount.
No other source of variation or bias in the sample data influences the
sampling distribution.
The margin of error in a confidence interval covers only random
sampling errors. Practical difficulties such as undercoverage and
nonresponse are often more serious than random sampling error. The
margin of error does not take such difficulties into account.
Cautions About Significance Tests
7
Significance tests are widely used in most areas of statistical work. Some
points to keep in mind when you use or interpret significance tests are:
How small a P is convincing?
The purpose of a test of significance is to describe the degree of
evidence provided by the sample against the null hypothesis. How
small a P-value is convincing evidence against the null hypothesis
depends mainly on two circumstances:
• If H0 represents an assumption that has been believed for
years, strong evidence (a small P) will be needed.
• If rejecting H0 means making a costly changeover, you need
strong evidence.
Cautions About Significance Tests
8
Significance tests are widely used in most areas of statistical work. Some
points to keep in mind when you use or interpret significance tests are:
Significance Depends on the Alternative Hypothesis
The P-value for a one-sided test is one-half the P-value for the twosided test of the same null hypothesis based on the same data.
• The evidence against the null hypothesis is stronger when the
alternative is one-sided because it is based on the data plus
information about the direction of possible deviations from the
null.
• If you lack this added information, always use a two-sided
alternative hypothesis.
Cautions About Significance Tests
9
Significance tests are widely used in most areas of statistical work. Some
points to keep in mind when you use or interpret significance tests are:
Sample Size Affects Statistical Significance
Because large random samples have small chance variation, very
small population effects can be highly significant if the sample is
large.
Because small random samples have a lot of chance variation, even
large population effects can fail to be significant if the sample is small.
Statistical significance does not tell us whether an effect is large
enough to be important. Statistical significance is not the same as
practical significance.
Beware of Multiple Analyses
The reasoning of statistical significance works well if you decide what
effect you are seeking, design a study to search for it, and use a test
of significance to weigh the evidence you get.
Sample Size for Confidence Intervals
10
A wise user of statistics never plans a sample or an experiment without also
planning the inference. The number of observations is a critical part of planning
the study.
The margin of error ME of the confidence interval for the population mean µ is
z *×
s
n
Choosing Sample Size for a Desired Margin of Error When Estimating µ
To determine the sample size n that will yield a level C confidence interval
for a population mean with a specified margin of error ME:
• Get a reasonable value for the population standard deviation σ from an
earlier or pilot study.
• Find the critical value z* from a standard Normal curve for confidence
level C.
• Set the expression for the margin of error to be less than or equal to ME
and solve for n:
s
z*
n
£ ME
Sample Size for Confidence Intervals
11
Researchers would like to estimate the mean cholesterol level µ of a particular
variety of monkey that is often used in laboratory experiments. They would like
their estimate to be within 1 milligram per deciliter (mg/dl) of the true value of µ
at a 95% confidence level. A previous study involving this variety of monkey
suggests that the standard deviation of cholesterol level is about 5 mg/dl.
 The critical value for 95% confidence is z* = 1.96.
 We will use σ = 5 as our best guess for the standard deviation.
5
1.96
£1
n
Multiply both sides by
square root n and divide
both sides by 1.
Square both sides.
(1.96)(5)
1
£ n
(1.96× 5) 2 £ n
96.04 £ n
We round up to 97
monkeys to ensure the
margin of error is no
more than 1 mg/dl at
95% confidence.
The Power of a Statistical Test
12
When we draw a conclusion from a significance test, we hope our conclusion will
be correct. But sometimes it will be wrong. There are two types of mistakes we
can make.
If we reject H0 when H0 is true, we have committed a Type I error.
If we fail to reject H0 when H0 is false, we have committed a Type II
error.
Truth about the population
Conclusion
based on
sample
H0 true
H0 false
(Ha true)
Reject H0
Type I error
Correct
conclusion
Fail to reject
H0
Correct
conclusion
Type II error
The Power of a Statistical Test
13
The probability of a Type I error is the probability of rejecting H0 when it is really
true. This is exactly the significance level of the test.
Significance and Type I Error
The significance level α of any fixed level test is the probability of a Type I
error. That is, α is the probability that the test will reject the null hypothesis
H0 when H0 is in fact true. Consider the consequences of a Type I error
before choosing a significance level.
A significance test makes a Type II error when it fails to reject a null hypothesis
that really is false. There are many values of the parameter that satisfy the
alternative hypothesis, so we concentrate on one value. We can calculate the
probability that a test does reject H0 when an alternative is true. This probability
is called the power of the test against that specific alternative.
The power of a test against a specific alternative is the probability that
the test will reject H0 at a chosen significance level α when the
specified alternative value of the parameter is true.
The Power of a Statistical Test
14
A potato-chip producer wonders whether the significance test of H0: p = 0.08 versus Ha: p >
0.08 based on a random sample of 500 potatoes has enough power to detect a shipment
with, say, 11% blemished potatoes.
What if p = 0.11?
We would reject
H0 at α = 0.05 if
our sample
yielded a
sample
proportion to the
right of the
green line.
The power of a test against any alternative
is 1 minus the probability of a Type II error
for that alternative; that is, power = 1 – β.
Since we reject H0 at α = 0.05 if
our sample yields a proportion >
0.0999, we’d correctly reject the
shipment about 75% of the time.
The Power of a Statistical Test
15
How large a sample should we take when we plan to carry out a significance
test? The answer depends on what alternative values of the parameter are
important to detect.
Summary of influences on the question “How many
observations do I need?”
•If you insist on a smaller significance level (such as 1% rather
than 5%), you have to take a larger sample. A smaller
significance level requires stronger evidence to reject the null
hypothesis.
• If you insist on higher power (such as 99% rather than 90%),
you will need a larger sample. Higher power gives a better
chance of detecting a difference when it is really there.
• At any significance level and desired power, detecting a small
difference requires a larger sample than detecting a large
difference.
Chapter 16 Objectives Review
16






Describe the conditions necessary for
inference
Describe cautions about confidence intervals
Describe cautions about significance tests
Calculate the sample size for a desired margin
of error in a confidence interval
Define Type I and Type II errors
Calculate the power of a significance test