Chapter Seven Statistical Intervals Based on a Single Sample

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Chapter SevenStatistical Intervals Based on a
Single Sample
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Confidence IntervalsFor the Population Mean
?100(1- ?) Confidence Interval (L,U) x ?
z?/2 ? ? ?n
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Common Z Values99 CI (.01)/2 2.5895 CI
(.05)/2 1.9690 CI (.10)/2 1.645
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Confidence Interval (? known)When fission
occurs, many of the nuclear fragments formed have
too many neutrons for stability. Some of these
neutrons are expelled almost instantaneously.
These observations are obtained on RV X, the
number of neutrons released during fission of
plutonium-239 n 40 Average
2.803 2 2 2 2 3 3 33 3 3 3 4 3 2 33 2 3 3 3 3 3
13 3 3 3 3 3 3 33 3 2 3 3 3 3 3
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CI Example ContinuedIs RV X Normally
distributed?Assume ? 0.5. What is the 99
Confidence Interval on ??The reported value of
? is 3.0.Do these data refute this interval?
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Sample SizeFor Interval Width w n (2
z?/2 ? ? / w)2 For half- width interval B
from u replace 2/w with 1/B.
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Sample Size (? known)The helium porosity of coal
samples taken from any particular coal layer
(seam) is Normally distributed with true standard
deviation equal to 0.75. What is the 95 CI for
the true average porosity of a certain coal layer
if the average porosity for 20 specimens from the
seam was 4.85.What sample size is necessary to
estimate true average porosity to within 0.2
with 99 confidence?
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Large Sample Confidence IntervalsFor the
Population mean ? 100(1- ?) Confidence
Interval (L,U) x ? z?/2 ? s
?n With
unknown ?.
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One Sided Confidence BoundFor u (L) x - z?
? s ?n (U) x z? ? s
?n
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Confidence Interval (Large n)A random sample of
64 customers at the UB bookstore found that the
average shopping time was 33 minutes with a
sample variance of 256. Estimate the true average
shopping time per customer ? with a confidence
level of 90.
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Confidence Interval Example (Large n)Using color
infrared photography in identification of normal
Douglas fir trees, a sample of 69 healthy trees
showed a sample mean dye-layer density of 1.028
with a sample standard deviation of 0.163. What
is the 95 (two-sided) CI for the true average
dye-layer density for all healthy trees?Suppose
the investigators made a rough guess of 0.16 for
the value of s before collecting data. What
sample size would be necessary to obtain an
interval width of 0.05 for a confidence level 0f
95?
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Students t Distribution f(t) ?(v1)/2 1t2
(v1)/2 ?(v/2)??v v
-?lt t lt?With v degrees of
freedom.Values listed on Table A.5
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Students t DistributionFor Random SampleX from
a Normal pdf T X u s/?nRV
T has t distribution with n 1 degrees of
freedom.
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Small Sample Confidence IntervalsFor the
Population mean ? 100(1- ?) Confidence
Interval (L,U) x ? t?/2,v ? s
?n With unknown
?.
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Confidence Interval (Small n)A triathlon
consisting of swimming, cycling, and running is a
very strenuous amateur sport. Research on 9 male
triathletes during a swimming performance showed
a sample mean maximum heart rate (beats/minute)
of 188 with a sample standard deviation of 7.2.
Assuming that the heart rate distribution is
approximately Normal, find a 98 CI for the true
mean heart rate of triathletes while swimming.
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Small Sample CI ExampleA manufacture of
gunpowder has developed a new powder which was
tested in 8 shells. The resulting muzzle
velocities, in feet per second, were as
follows3005 2925 2935 29652995 3005 2935
2905What is the 95 Confidence Interval for the
true mean average velocity for shells using this
new type of powder? (Evidence indicates the
velocities to be Normal.)Sample mean
2959Sample s 39.4
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Confidence Intervals for ?2For the Population
Variance ?2 100(1- ?) Confidence Interval
L (n-1) ? s2
?2?/2,? U (n-1) ? s2
?21-?/2,?
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Variance Confidence Interval ExampleAn
experimenter wanted to check the variability of
equipment designed to measure the volume of an
audio source. Three independent measurements
recorded by this equipment for the same sound
were 4.1, 5.2, 10.2. What is an estimate for
the population variance with a confidence level
of 90. Evidence indicates that the measurements
recorded by this equipment is Normally
distributed.
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Interval Estimation on VariabilityX-ray
microanalysis has become an invaluable method of
analysis. With the electron microprobe, both
quantitative and qualitative measures can be
taken and analyzed statistically. One method for
analyzing crystals is called the two-voltage
technique. These measurements are obtained on the
percentage of potassium present in a commercial
product which theoretically contains 26.6 K by
weight21.9 23.4 22.1 22.1 24.7 24.6 24.0 24.1 2
4.2 26.5 23.8 25.3 24.8 24.8 24.5 27.8 24.9 27.2
25.1 25.5 23.7 26.5
n 2722.0 26.7 25.2 23.1 25.4 What is
the 99 Confidence Interval for ?2?What is the
99 Confidence Interval for ?? Suggest a way to
improve the interval estimate for ? based on
these data.