Download Ch7 - Statistics

7
Statistical Intervals
Based on a Single
Sample
7.1
Basic Properties of
Confidence Intervals
Basic Properties of Confidence Intervals
The basic concepts and properties of confidence intervals
(CIs) are most easily introduced by first focusing on a
simple and problem situation.
Suppose that the parameter of interest is a population
mean  and that
1. The population distribution is normal
2. The value of the population standard deviation  is
known
3
Basic Properties of Confidence Intervals
Irrespective of the sample size n, the sample mean X is
normally distributed with expected value  and standard
deviation
Standardizing X by first subtracting its expected value
and then dividing by its standard deviation yields the
standard normal variable
(7.1)
4
Basic Properties of Confidence Intervals
Because the area under the standard normal curve
between –1.96 and 1.96 is .95,
(7.2)
The equivalence of each set of inequalities to the original
set implies that
(7.3)
5
Basic Properties of Confidence Intervals
To interpret (7.3), think of a random interval having left
endpoint X – 1.96 
and right endpoint
X + 1.96 
In interval notation, this becomes
(7.4)
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Basic Properties of Confidence Intervals
This CI can be expressed either as
or as
A concise expression for the interval is x  1.96 
,
where – gives the left endpoint (lower limit) and + gives the
right endpoint (upper limit).
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Interpreting a Confidence Interval
With 95% confidence, we can say
that µ should be within roughly

n
1.96 standard deviations
(1.96/√n) from our sample

mean x.
x

• In 95% of all possible samples
of this size n, µ will indeed fall

in our confidence interval.
• In only 5% of samples would
be farther from µ.
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Example 2
The quantities needed for computation of the 95% CI for
true average preferred height are  = 2.0, n = 31, and
x = 80.0.
The resulting interval is
That is, we can be highly confident, at the 95% confidence
level, that 79.3 <  < 80.7.
This interval is relatively narrow, indicating that  has been
rather precisely estimated.
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Other Levels of Confidence
As Figure 7.4 shows, a probability of 1 –  is achieved by
using z/2 in place of 1.96.
P(–z/2  Z < z/2) = 1 – 
Figure 7.4
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Other Levels of Confidence
Definition
A 100(1 – )% confidence interval for the mean  of a
normal population when the value of  is known is given by
(7.5)
or, equivalently, by
The formula (7.5) for the CI can also be expressed in words
as point estimate of   (z critical value) (standard error of
the mean).
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Example 3
The production process for engine control housing units of
a particular type has recently been modified.
Prior to this modification, historical data had suggested that
the distribution of hole diameters for bushings on the
housings was normal with a standard deviation of .100 mm.
It is believed that the modification has not affected the
shape of the distribution or the standard deviation, but that
the value of the mean diameter may have changed.
A sample of 40 housing units is selected and hole diameter
is determined for each one, resulting in a sample mean
diameter of 5.426 mm.
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Example 3
cont’d
Let’s calculate a confidence interval for true average hole
diameter using a confidence level of 90%.
This requires that 100(1 – ) = 90, from which  = .10 and
z/2 = z.05 = 1.645 (corresponding to a cumulative z-curve
area of .9500). The desired interval is then
With a reasonably high degree of confidence, we can say
that 5.400 <  < 5.452.
This interval is rather narrow because of the small amount
of variability in hole diameter ( = .100).
13
Properties of Confidence Intervals
 User chooses the confidence interval
 We want
High confidence
Small confidence interval
 The confidence interval gets narrower when
z gets smaller
σ is smaller
n is larger
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Confidence Level and Sample Size
A general formula for the sample size n necessary to
ensure an interval width w is obtained from equating w to
2  z/2 
and solving for n.
The sample size necessary for the CI (7.5) to have a width
w is
The smaller the desired width w, the larger n must be. In
addition, n is an increasing function of  (more population
variability necessitates a larger sample size) and of
the confidence level 100(1 – ) (as  decreases, z/2
increases).
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Example 4
Extensive monitoring of a computer time-sharing system
has suggested that response time to a particular editing
command is normally distributed with standard deviation 25
millisec.
A new operating system has been installed, and we wish to
estimate the true average response time  for the new
environment.
Assuming that response times are still normally distributed
with  = 25, what sample size is necessary to ensure that
the resulting 95% CI has a width of (at most) 10?
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Example 4
cont’d
The sample size n must satisfy
Rearranging this equation gives
= 2  (1.96)(25)/10 = 9.80
So
n = (9.80)2 = 96.04
Since n must be an integer, a sample size of 97 is required.
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7.2
Large-Sample Confidence
Intervals for a Population Mean
and Proportion
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A Large-Sample Interval for 
In Ch7.1, we have come across the CI for  which
assumed that
1.The population distribution is normal
2.The value of  is known
In Ch7.2, we now present a large-sample CI whose validity
does not require these assumptions.
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A Large-Sample Interval for 
Let X1, X2, . . . , Xn be a random sample from a population
having a mean  and standard deviation . Provided that n
is large, the Central Limit Theorem (CLT) implies that has
approximately a normal distribution whatever the nature of
the population distribution.
It then follows that
has approximately a
standard normal distribution, so that
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A Large-Sample Interval for 
Proposition
If n is sufficiently large, the standardized variable
has approximately a standard normal distribution. This
implies that
(7.8)
is a large-sample confidence interval for  with
confidence level approximately 100(1 – )%. This formula
is valid regardless of the shape of the population
distribution.
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A Large-Sample Interval for 
Generally speaking, n > 40 will be sufficient to justify the
use of this interval.
This is somewhat more conservative than the rule of thumb
for the CLT because of the additional variability introduced
by using S in place of .
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Example 6
Haven’t you always wanted to own a Porsche? The author
thought maybe he could afford a Boxster, the cheapest
model. So he went to www.cars.com on Nov. 18, 2009, and
found a total of 1113 such cars listed.
Asking prices ranged from $3499 to $130,000 (the latter
price was one of only two exceeding $70,000). The prices
depressed him, so he focused instead on odometer
readings (miles).
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Example 6
cont’d
Here are reported readings for a sample of 50 of these
Boxsters:
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Example 6
cont’d
A boxplot of the data (Figure 7.5) shows that, except for the
two outliers at the upper end, the distribution of values is
reasonably symmetric (in fact, a normal probability plot
exhibits a reasonably linear pattern, though the points
corresponding to the two smallest and two largest
observations are somewhat removed from a line fit through
the remaining points).
A boxplot of the odometer reading data from Example 6
Figure 7.5
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Example 6
cont’d
Summary quantities include n = 50, = 45,679.4,
= 45,013.5, s = 26,641.675, fs = 34,265.
The mean and median are reasonably close (if the two
largest values were each reduced by 30,000, the mean
would fall to 44,479.4, while the median would be
unaffected).
The boxplot and the magnitudes of s and fs relative to the
mean and median both indicate a substantial amount of
variability.
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Example 6
cont’d
A confidence level of about 95% requires z.025 = 1.96, and
the interval is
45,679.4  (1.96)
= 45,679.4  7384.7
= (38,294.7, 53,064.1)
That is, 38,294.7 <  < 53,064.1 with 95% confidence. This
interval is rather wide because a sample size of 50, even
though large by our rule of thumb, is not large enough to
overcome the substantial variability in the sample. We do
not have a very precise estimate of the population mean
odometer reading.
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One-Sided Confidence Intervals (Confidence Bounds)
Starting with P(–1.645 < Z)  .95 and manipulating the
inequality results in the upper confidence bound. A similar
argument gives a one-sided bound associated with any
other confidence level.
Proposition
A large-sample upper confidence bound for  is
and a large-sample lower confidence bound for  is
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7.3
Intervals Based on a Normal
Population Distribution
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Intervals Based on a Normal Population Distribution
The CI for  presented in 7.2 is valid when n is large.
The resulting interval can be used whatever the nature of
the population distribution. The CLT cannot be invoked,
however, when n is small.
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Intervals Based on a Normal Population Distribution
The result on which inferences are based introduces a new
family of probability distributions called t distributions.
Theorem
When is the mean of a random sample of size n from a
normal distribution with mean , the rv
(7.13)
has a probability distribution called a t distribution with n – 1
degrees of freedom (df).
31
Properties of t Distributions
Properties of t Distributions
Let tn denote the t distribution with n df.
1. Each tn curve is bell-shaped and centered at 0.
2. Each tn curve is more spread out than the standard
normal (z) curve.
3. As n increases, the spread of the corresponding tn curve
decreases.
4. As n , the sequence of tn curves approaches the
standard normal curve (so the z curve is often called the
t curve with df = ).
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Properties of t Distributions
Figure 7.7 illustrates several of these properties for
selected values of n.
tn and z curves
Figure 7.7
33
Properties of t Distributions
Notation
Let t,n = the number on the measurement axis for which
the area under the t curve with n df to the right of t,n is ;
t,n is called a t critical value.
For example, t.05,6 is the t critical value that captures an
upper-tail area of .05 under the t curve with 6 df. The
general notation is illustrated in Figure 7.8.
Illustration of a t critical value
Figure 7.8
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The One-Sample t Confidence Interval
The standardized variable T has a t distribution with n – 1
df, and the area under the corresponding t density curve
between –t/2,n – 1 and t/2,n – 1 is 1 –  (area /2 lies in each
tail), so
P(–t/2,n – 1 < T < t/2,n – 1) = 1 – 
(7.14)
Expression (7.14) differs from expressions in previous
sections in that T and t/2,n – 1 are used in place of Z and
but it can be manipulated in the same manner to
obtain a confidence interval for .
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The One-Sample t Confidence Interval
Proposition
Let and s be the sample mean and sample standard
deviation computed from the results of a random sample
from a normal population with mean . Then a
100(1 – )% confidence interval for  is
(7.15)
or, more compactly
36
The One-Sample t Confidence Interval
An upper confidence bound for  is
and replacing + by – in this latter expression gives a lower
confidence bound for , both with confidence level
100(1 – )%.
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Example 11
Even as traditional markets for sweetgum lumber have
declined, large section solid timbers traditionally used for
construction bridges and mats have become increasingly
scarce.
The article “Development of Novel Industrial Laminated
Planks from Sweetgum Lumber” (J. of Bridge Engr., 2008:
64–66) described the manufacturing and testing of
composite beams designed to add value to low-grade
sweetgum lumber.
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Example 11
cont’d
Here is data on the modulus of rupture (psi; the article
contained summary data expressed in MPa):
6807.99
6981.46
6906.04
7295.54
7422.69
7637.06
7569.75
6617.17
6702.76
7886.87
6663.28
7437.88
6984.12
7440.17
6316.67
6165.03
6872.39
7093.71
8053.26
7713.65
6991.41
7663.18
7659.50
8284.75
7503.33
6992.23
6032.28
7378.61
7347.95
7674.99
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Example 11
cont’d
Figure 7.9 shows a normal probability plot from the R
software.
A normal probability plot of the modulus of rupture data
Figure 7.9
40
Example 11
cont’d
The straightness of the pattern in the plot provides strong
support for assuming that the population distribution of
MOR is at least approximately normal.
The sample mean and sample standard deviation are
7203.191 and 543.5400, respectively (for anyone bent on
doing hand calculation, the computational burden is eased
a bit by subtracting 6000 from each x value to obtain
yi = xi – 6000; then
from which = 1203.191 and sy = sx as given).
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Example 11
cont’d
Let’s now calculate a confidence interval for true average
MOR using a confidence level of 95%. The CI is based on
n – 1 = 29 degrees of freedom, so the necessary t critical
value is t.025,29 = 2.045. The interval estimate is now
We estimate 7000.253 <  < 7406.129 that with 95%
confidence.
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Example 11
cont’d
If we use the same formula on sample after sample, in the
long run 95% of the calculated intervals will contain .
Since the value of  is not available, we don’t know
whether the calculated interval is one of the “good” 95% or
the “bad” 5%.
Even with the moderately large sample size, our interval is
rather wide. This is a consequence of the substantial
amount of sample variability in MOR values.
A lower 95% confidence bound would result from retaining
only the lower confidence limit (the one with –) and
replacing 2.045 with t.05,29 = 1.699.
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Intervals Based on Nonnormal Population Distributions
The one-sample t CI for  is robust to small or even
moderate departures from normality unless n is quite small.
By this we mean that if a critical value for 95% confidence,
for example, is used in calculating the interval, the actual
confidence level will be reasonably close to the nominal
95% level.
If, however, n is small and the population distribution is
highly nonnormal, then the actual confidence level may be
considerably different from the one you think you are using
when you obtain a particular critical value from the t table.
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Intervals Based on Nonnormal Population Distributions
It would certainly be distressing to believe that your
confidence level is about 95% when in fact it was really
more like 88%!
The bootstrap technique, has been found to be quite
successful at estimating parameters in a wide variety of
nonnormal situations.
In contrast to the confidence interval, the validity of the
prediction and tolerance intervals described in this section
is closely tied to the normality assumption.
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Intervals Based on Nonnormal Population Distributions
These latter intervals should not be used in the absence of
compelling evidence for normality.
The excellent reference Statistical Intervals, cited in the
bibliography at the end of this chapter, discusses
alternative procedures of this sort for various other
situations.
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