Download Introduction to the Practice of Statistics

David S. Moore • George P. McCabe
Introduction to the
Practice of Statistics
Fifth Edition
Chapter 6:
Introduction to Inference
Copyright © 2005 by W. H. Freeman and Company
Introduction to Inference
•
•
•
•
6.1 – Estimating with Confidence
6.2 – Tests of Significance
6.3 – Use and Abuse of Tests
6.4 – Power and Inference as a Decision (optional)
We will explore these ideas in the context of
estimating and/or testing hypotheses about a
population mean, .
Section 6.1 Estimating with
Confidence
Example 6.2 (page 384):
• Goal: Estimate the mean SATM score, , for all
California seniors.
• Take SRS of 500 California seniors.
• Find the sample mean, x  461 .
• How well does x estimate  ?
Question: What is the sampling distribution of x
(suppose we know   100)??? Why??

100
N ( X   , X 
)  N ( X   , X 
)  N (  X   ,  X  4.5)
n
500
Values of x are distributed according to this density curve. How
likely is it that your value of x will be within 2 X  9 of the
(unknown) value of  ?
??
N (  X   ,  X  4.5)
Sampling Distribution of
x
95% of the time,
2 X  9 of
x
5% of the time
2 X  9 of
x will be within
!
x
x will not be within

x
x
If
is within

of

is within 2 X  9 of
, then
!
x
Thus, we are 95% confident that
x
x 9
x 9
x
x2
x9
x9
x 9
x

n
   x2
Or in our case (
x9

n
x  461 )
461  9    461  9
452    470
There is a 5% chance that this interval does not capture
the population mean!
that
Confidence Intervals
• We have just computed a
95% confidence interval for 
• The endpoints of the interval (452, 470) depended on three
values:
– The point estimate for  , x
– The level of confidence which determined how many standard
deviations to use (95%: we used approximately two standard
deviations)

100
– The standard deviation of the point estimate,  X 

 4.5
n
500
which also depends on the sample size, n.
• The interval also depended on the fact that we knew that the
sampling distribution of the statistic, x is normal (or
approximately normal, since n is large)!
We are 95% confident that the
true population mean is
contained in the confidence
interval. WHY? Because
according to the Central Limit
Theorem, there is a 95%
chance that the estimate of the
population mean (i.e. the
sample mean), will be within
2 standard deviations (of the
sampling distribution!) of the
true population mean.
Page 386
Let’s try playing with the
“Confidence Interval” applet
on your CD….
For finding a level C confidence interval for a population mean, how does the
confidence level determine the number of standard deviations to use to compute
the interval?
For a 95% confidence interval for  we used approximately 2 standard
deviations to compute the endpoints of our interval:
How do we
find z* ?
x  2 x    x  2 x
x2
x  z*

n

n
   x2
   x  z*

n

n
What happens
to z* when the
C-level
changes?
*
Let’s find z for various C-levels for a C-level confidence
interval for a population mean,  . (see page 387)
Standard Normal
Density Curve
CLevel
99%
Value of
95%
1.960
??
90%
1.645
??
85%
1.440
??
Use the invNorm key on your TI-83!
z
*
2.576
??
Let’s use our TI-83 calculators to find some C-level confidence
intervals and explore their behavior!
Example 6.3 (page 388): Find a 95% confidence interval for .
(What is  ??)
You are given what values?
What is the margin of error for this confidence interval?
What is the confidence interval?
Write a sentence that explains the meaning of this interval.
Let’s also use the Stat|Tests|ZInterval key on your TI-83!
What if we change the C-level to 99%? How will the interval
change (see picture page 391)? What if we change the C-level to
90%
What if the sample size in Example 6.3 (page 388) was actually
320 instead of 1280? How will this effect the margin of error, the
confidence interval?
New margin of error?
New confidence interval?
See Figure 6-5 on page 390 (and below):
• Goals for Estimating Population Parameters:
– High Confidence
– Low Margin of Error
• How to Reduce the Margin of Error
mz
*
– Change C-Level?

n
Lower C-Level Results in Smaller Value of z*.
– Change Sample Size?
Larger n will reduce m since you will divide by a larger value.
– Change Population Standard Deviation?
Smaller  will reduce m. This may not be possible to do.
Page 390
Suppose we want a certain margin of error for our confidence
interval for a population mean. What sample size will be needed to
get the desired result? (Assume we know the population standard
deviation.)
mz
*

Let’s solve this equation for n!!
n
Let’s work Problem 6.28 on page 399!
Section 6.2 Tests of Significance
Example 6.14 (page 410):
• Goal: Test a claim about a population parameter (in this
case, the mean systolic b.p., , for all middle aged male
executives).
• Is the mean significantly different than the mean for the
general
of is
males
to 44
(which equals
What is population
the claim, what
beingaged
tested35
about
the parameter?
128)?
• Take SRS of size 72 from the middle aged male
executives (i.e. from
the the
population
Collect
evidence. for ).
• Find the sample mean, x  126.07 . What is this value?
• Does the data from the sample support the claim or not?
I.e. based on the data ( x ), do you believe the mean
systolic
b.p.appropriate
for middle
aged male
executives
( ) is
Draw the
conclusions
based
on the evidence.
significantly different than mean for the general
population of males (i.e. 128 mmHG)?
Formulating Hypotheses
For us, the null hypothesis will contain an equality
involving the parameter!
The alternative hypothesis is what we hope or suspect is true about the parameter
(instead of the equality in the null hypothesis). This will depend on the problem
statement.
For our example:
H 0 :   128
H a :   128
Says the mean systolic b.p. of middle-aged male
executives is not different (i.e. not statistically
significantly different) than 128
Says the mean systolic b.p. of middle-aged male
executives is different (i.e. statistically
significantly) than 128
The Logic of Hypothesis Testing
• Assume the null hypothesis is TRUE!
• Determine how likely it is to get data as extreme as what
you got IF this is the case (i.e. IF the null hypothesis is
TRUE).
• If it is very unlikely to get data as extreme as what you
got, then you doubt the assumption that the null
hypothesis is true (i.e. you will reject the null hypothesis).
• If it is not very unlikely to get data as extreme as what
you got, then you have no reason to doubt the null
hypothesis (i.e. you will fail to reject the null hypothesis).
• The direction “as extreme” is determine by the
alternative hypothesis.
Following the Logic for Our
Example (6.14 page 410)
H 0 :   128
H a :   128
IF the null hypothesis is TRUE (and assume the
population of middle-aged executives has the same
standard deviation as the males aged 35 to 44, i.e.
assume   15 ), then what is the (approximate?)
sampling distribution for x ? Why?

15
N ( X   , X 
)  N (  X  128,  X 
)
n
72
How “far out” is our data? I.e. how many standard deviations from
the mean is x  126.07 for our sampling distribution?
z
x  0
X

x  0 126.07  128

 1.09
/ n
15 / 72
This is called the z test statistic for the data, note it is just the z-score for the data!.
The z-test statistic for our data tells us that our data differs from the hypothesized
mean by z = -1.09 standard deviations (assuming the null hypothesis is true).
What other values of the z-test statistic are as extreme as our data’s test statistic
(i.e. differ by at least as many standard deviations from the hypothesized mean as
our data)? Note that our alternative hypothesis is H a :   128 .
z < -1.09 OR z > 1.09 (This is a two-tailed test!)
How will this z test statistic be distributed (Hint: We are standardizing a variable
that is approximately normal: x !)???
This z test statistic has an
approximately standard normal
distribution
(mean=0, standard deviation=1)
How likely is it to get a result
“as extreme” as our data?
P-value
Let’s compute the P-value for
our test two ways:
1. Using the invNorm key:
2. Using the Stat|Tests|Z-Test
key:
Our data is not too “rare,” thus it will not
cause us reject the null hypothesis in
favor of the alternative hypothesis.
P-value
Typical  levels in the “real world” are
  0.10
  0.05
  0.01
Most statistical software will report the P-value – but will not tell
you what conclusion to reach! That is the hardest part!
Steps in Hypothesis Testing (page 407) OR
What Have We Done (in Example 6.14)?
• State the null and alternative hypothesis (clearly identify the
population parameter!):
Where  is the mean
H 0 :   128
H a :   128
systolic b.p. of middleaged male executives.
• Calculate the value of the test statistic on which the test will be based:
x  0 x  0 126.07  128
z


 1.09
X
/ n
15 / 72
• Find the P-value for the observed data:
P=0.2758
• State a conclusion:
Since P is large, we will fail to reject the null hypothesis in favor
of the alternative (i.e. it is not very unlikely to get a result as
extreme as ours). Thus the data do not provide strong evidence
that the mean systolic b.p. of middle-aged male executives
differs significantly from the mean of other men (i.e. 128).
We Have Just Conducted a (two-tailed) z-test
for a Population Mean,
!

Assumptions (see page
393):
• Data from a SRS of the
population.
•Population standard
deviation is known (i.e. 
known).
•Data does not have
extreme outliers (since x
not resistant to outliers)
•The smaller n is, the more
concerned we need to be
about the population
distribution and outliers.
One-tailed
Test
One-tailed
Test
Two-tailed
Test
Page 410
Let’s Work Another Example: Example 6.15 page 412
• State the null and alternative hypothesis (clearly identify the
population parameter!):
• Calculate the value of the test statistic on which the test will be based:
• Find the P-value for the observed data:
• State a conclusion:
Relationship Between Two-Sided Significance Tests and
Confidence Intervals
Let’s Revisit Example 6.15 page 412:
•What if our alternative hypothesis was two sided? What would be
the P-value of the test?
•Would you reject the null hypothesis at any of the standard alpha
levels?
•Find the 1   -level confidence intervals for  using the standard
alpha levels.
•Which confidence intervals computed above contain the value of  0
stated in the null hypothesis of the test?
Section 6.3 Use and Abuse of Tests
• Read on Your Own!
• You will have a hand-in homework
assignment from this section.
• You will be asked questions about this
material on future tests and quizzes.
Section 6.4 Power and Inference as a
Decision*
•
•
•
•
•
Type I and Type II Errors
The Power of a Statistical Hypothesis Test
Increasing the Power of a Test
Significance and Type I Errors
Power and Type II Errors
*Optional Section
Possible Errors in Hypothesis Testing
Type I and Type II Errors
Page 436
An Example
Page 435
H 0 : Lot of bearings meets standards
H a : The lot does not meet standards
More on Type I and Type II Error – Example 6.28
H0 :   0
Ha :   0
What is the likelihood of a
Type I Error?
0.05
Test at   0.05 level of significance.
Reject null if z-test statistic is greater than what?
Reject if get a sample mean greater than what?
H 0 TRUE
If it is true that   1 , then
what is the likelihood of a
Type II Error?
0.20
If it is true that   1 , then
what is the likelihood of
correctly rejecting the null
hypothesis?
0.80
  1 TRUE
Power, Significance, and Type I and Type II Errors