Transcript Chapter Seven Statistical Intervals Based on a Single Sample

Chapter Seven
Statistical
Intervals Based
on a Single
Sample
Confidence Intervals
For the Population Mean :
100(1- )% Confidence Interval
(L,U) = x  z/2  
n
Common Z Values
99% CI = (.01)/2 = 2.58
95% CI = (.05)/2 = 1.96
90% CI = (.10)/2 = 1.645
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 =
2.80
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CI Example Continued
Is 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?
Sample Size
For Interval Width w:
n = (2 z/2   /
2
w)
For half- width interval B from u; replace 2/w
with 1/B.
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?
Large Sample Confidence Intervals
For the Population mean :
100(1- )% Confidence
Interval (L,U) = x  z/2  s
n
With unknown .
One Sided Confidence Bound
For u:
(L) = x - z 
s
n
(U) = x + z  s
n
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%.
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%?
Student’s t Distribution
f(t) = (v+1)/2
(v/2)v
2
1+t
–(v+1)/2
v
-< t <+
With v degrees of freedom.
Values listed on Table A.5
Student’s t Distribution
For Random Sample
X from a Normal pdf:
T=X–u
s/n
RV T has t distribution with
n – 1 degrees of freedom.
Small Sample Confidence Intervals
For the Population mean :
100(1- )% Confidence
Interval (L,U) = x  t/2,v  s
n
With unknown .
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.
Small Sample CI Example
A manufacture of gunpowder has developed a new
powder which was tested in 8 shells. The resulting
muzzle velocities, in feet per second, were as
follows:
3005 2925 2935 2965
2995 3005 2935 2905
What 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 = 2959
Sample s = 39.4
Confidence Intervals for
2

For the Population Variance 2: 100(1)% Confidence Interval
L = (n-1)  s2
2

/2,
U = (n-1)  s2
2
 1-/2,
Variance Confidence Interval Example
An 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.
Interval Estimation on Variability
X-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
weight:
21.9 23.4 22.1 22.1 24.7 24.6
24.0 24.1 24.2 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 = 27
22.0 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.