Download 1 b. Calculate quartiles of life times. c. Prepare a

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SEC/SER
STAT319, Q-112.DOC, ID#
NAME:
A.1 The following data give the life time ( X ) measured in months of 38 randomly selected electronic
components produced by a company during the year 2011:
Stem
1
2
3
4
Leaf
0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4
0, 0, 0, 0, 0, 1, 1, 1, 1, 1
0, 2, 4, 6, 8
3, 6, 9
20
10
5
3
a. Draw a dot plot of the life times of the electronic component. What proportion of life times are less than
or equal to 30 months? Name it in terms of percentile or fractile.
b. Calculate quartiles of life times.
c. Prepare a box plot of the life times.
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A.2 The following data give the life time ( X ) measured in months of 30 randomly selected electronic
components produced by a company during the year 2011:
Stem
2
3
4
5
Leaf
0, 0, 1, 2, 3, 4, 5,, 6, 7, 8, 9, 9
0, 0, 0, 0, 0, 1, 1, 1, 1, 1
0, 2, 4, 6, 8
3, 6, 9
12
10
5
3
a. Prepare a relative frequency distribution of the life time.
b. Draw a relative frequency curve of the life times and comment on the shape of it.
c. What is the amount of variability of life times?
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A.3 The following data give the life time ( X ) measured in months of 30 randomly selected electronic
components produced by a company during the year 2011:
Stem
2
3
4
5
Leaf
0, 0, 1, 2, 3, 4, 5,, 6, 7, 8, 9, 9
0, 0, 0, 0, 0, 1, 1, 1, 1, 1
0, 2, 4, 6, 8
3, 6, 9
a. What is the amount of variability of life times?
b. Calculate Coefficient of Variation, and comment on it.
c. Calculate coefficient of Skewness and comment on it.
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10
5
3
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B.1 Samples of laboratory glass are in small (light packaging) or heavy (large packaging). Suppose that
1% of the sample shipped in small packages break during shipment, and 2% of the sample shipped in
heavy packages break during shipment. It is also known that 60% of the samples are shipped in small
packages.
a. Write out the sample space.
b. What proportion of samples break during shipment?
c. If samples shipped through small and light packaging, what is the probability that samples break during
shipment.
d. If samples break during the shipment, what is the probability that samples are shipped through small
(light packaging).
e. Are the events “small packaging” and “breakage” independent?
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B.2 A batch of 25 injection-molded parts contains 5 that have suffered excessive shrinkage.
a. Write out the sample space if three parts are selected at random.
b. If three parts are selected at random, and without replacement, what is the probability that the third part
selected is one with excessive shrinkage but the first two were ok?
c. If 3 parts are selected, without replacement, what is the probability that only one suffered excessive
shrinkage?
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B.3 ∆ Consider a random experiment involving three boxes ( A, B, C ) , each containing a mixture
of red (R ) and green (G ) balls , with the following contents:
A
B
C
4 R, 5G
6 R, 7G
8 R, 9G
The first ball will be selected from Box A. If the ball is red, the second ball will be selected from Box B;
otherwise the second ball will be selected from Box C. Let Ri (i = 1, 2) be the event that
ball in the i -th draw is red, and Gi (i = 1, 2) be the event that ball in the
determine the following:
i -th draw is green. Then
a. P ( R1 ) b. P (G1 ) c. P ( R2 | R1 ) d. P ( R2 | G1 ) e. P (G2 | G1 ) f. P (G2 | R1 )
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C.1 Pages of data arrive at an input buffer at the rate of 4 pages per second, according to the probability
e −4t ( 4t )
f ( x) =
, x 0, 1, 2,  What is the probability that more than 2 pages will arrive in 3
=
function
x!
x
seconds?
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C.2 Suppose that we have a box containing
N bulbs, of which K (≤ N ) are defective. Three bulbs are
selected in succession without replacement.
a. What is the probability that all 3 bulbs will be defective?
b. If the first two bulbs selected are defective ones, what is the probability that the third ball will not be
defective?
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C.3 A certain type of component is packaged in lots of four. Let X represent the number of properly
functioning components in a randomly chosen lot. Assume that the probability that exactly x components
function is proportional to x; in other words, assume that the probability mass function of X is given by
f ( x) = cx, x = 1, 2,3, 4; f ( x) = 0 elsewhere where c is a normalizing constant.
a. Find the probability that the number of properly functioning components are exactly two.
b. Find the standard deviation of the number of properly functioning components.
c. Find the value of the constant c so that f ( x) is a probability mass function.
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D.1 Chalks manufactured at a factory are known to have length that is normally distributed with mean 100
mm and standard deviation 3 mm. The specifications are that it must have a length between 95 mm and
106 mm.
a. What percentage of chalks would meet the specification?
b. Find the cut of length for the shortest 10% of the chalk produced.
c. A sample of 16 chalks is selected. What is the probability that the sample mean will be less than 102
mm?
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E.1 The lifetime in thousand hours of a certain kind of radio tube has a probability function f ( x) = 0,
x ≤ 0, and f ( x) = 0.5 / e 0.5 x , x > 0.
a. Determine the 95th percentile of the life time of a radio tube?
c. What is the (true) mean lifetime of a radio tube?
c. What is the standard deviation (σ ) of the lifetime of a radio tube?
d. If 36 radio tubes are selected what is the probability that the average lifetime will exceed its mean ( µ )
lifetime by more than the double of its standard deviation (σ / 6) ?
e. What is the 95th percentile of the average lifetime of a radio tube?
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F.1 Measurements of Rockwell hardness of the head of shearing pins are normally distributed with a
standard deviation 1.21. A sample of 25 pins has yielded an average value of 48.50 with a sample
standard deviation of 1.5. Construct a 90% confidence interval for the mean Rockwell hardness.
F.2 Measurements of Rockwell hardness of the head of shearing pins are normally distributed with a
standard deviation 1.21. A sample of 30 pins has yielded an average value of 48.50 with a sample
standard deviation of 1.5. Construct a 90% confidence interval for the mean Rockwell hardness.
F.3 Measurements of Rockwell hardness of the head of shearing pins has standard deviation 1.21. A random sample of 20 shearing
pins is taken in a study of the Rockwell hardness of the head on the pin. Measurements on the Rockwell hardness were made for each
of the sample pins, yielding an average value of 48.50. How would you estimate 90% confidence interval for the mean Rockwell
hardness?
F.4 Measurements of Rockwell hardness of the head of shearing pins are not normally distributed but
they have a standard deviation 1.21. A sample of 30 pins has yielded an average value of 48.50 and a
sample standard deviation of 1.5. Construct a 90% confidence interval for the mean Rockwell hardness.
F.5 Measurements on the Rockwell hardness were made for each of the 30 sample pins, yielding an
average value of 48.50 with a sample standard deviation of 1.5. If the measurements are not normally
distributed, how would you estimate 90% confidence interval for the mean Rockwell hardness?
F.6 Measurements of Rockwell hardness of the head of shearing pins are normally distributed. A sample
of 20 sample pins has yielded an average value of 48.50 and standard deviation 1.5. How would you
estimate 90% confidence interval for the mean Rockwell hardness?
F.7 Measurements of Rockwell hardness of the head of shearing pins are normally distributed. A sample
of 30 sample pins has yielded an average value of 48.50 and standard deviation 1.5. How would you
estimate 90% confidence interval for the mean Rockwell hardness?
F.8 Measurements on the Rockwell hardness were made for each of 20 sample pins, yielding an average value of 48.50 and standard
deviation 1.5. How would you estimate 90% confidence interval for the mean Rockwell hardness?
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G.1 Measurements of Rockwell hardness of the head of shearing pins are normally distributed with a
standard deviation 1.21. A sample of 25 pins has yielded an average value of 48.50 with a sample
standard deviation of 1.5. Test the hypothesis that Rockwell hardness is less than 49.
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G.2 Measurements of Rockwell hardness of the head of shearing pins are normally distributed with a
standard deviation 1.21. A sample of 30 pins has yielded an average value of 48.50 with a sample
standard deviation of 1.5. Test the hypothesis that Rockwell hardness is less than 49.
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G.3 Measurements of Rockwell hardness of the head of shearing pins has standard deviation 1.21. A random sample of 20 shearing
pins is taken in a study of the Rockwell hardness of the head on the pin. Measurements on the Rockwell hardness were made for each
of the sample pins, yielding an average value of 48.50. Test the hypothesis that Rockwell hardness is different from 49.
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G.4 Measurements of Rockwell hardness of the head of shearing pins are not normally distributed but
they have a standard deviation 1.21. A sample of 30 pins has yielded an average value of 48.50 and a
sample standard deviation of 1.5. Test the hypothesis that Rockwell hardness is different from 49.
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G.5 Measurements on the Rockwell hardness were made for each of the 30 sample pins, yielding an
average value of 48.50 with a sample standard deviation of 1.5. If the measurements are not normally
distributed, how would you test the hypothesis that Rockwell hardness is different from 49?
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G.6 Measurements of Rockwell hardness of the head of shearing pins are normally distributed. A sample
of 20 sample pins has yielded an average value of 48.50 and standard deviation 1.5. Test the hypothesis
that Rockwell hardness is less than 49.
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G.7 Measurements of Rockwell hardness of the head of shearing pins are normally distributed. A sample
of 30 sample pins has yielded an average value of 48.50 and standard deviation 1.5. Test the hypothesis
that Rockwell hardness is different from 49.
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G.8 Measurements on the Rockwell hardness were made for each of 20 sample pins, yielding an average value of 48.50 and standard
deviation 1.5. How would you estimate 90% confidence interval for the mean Rockwell hardness? Test the hypothesis that Rockwell
hardness is different from 49.