1
STAT 319, TERM 091, HW 091
Chapter 1
1. The following data are a sample for the bursting strengths in pounds per squares inch for 30
soft-drink bottles:
210
215
220
230
235
240
250
250
230
222
250
250
250
251
250
251
251
255
255
251
255
259
260
270
255
270
270
271
280
285
a. Construct a relative frequency distribution and related histogram for these data. Comment on
the shape of the data.
b. Calculate the sample mean, sample median and sample mode.
c. Calculate deviations of observations from the sample mean. Do they add to zero?
d. Calculate squared deviations and hence find
variance and standard deviation.
CSS (Centered Sum of Squares), sample
e. Find the sample variance by the formula s 
2
2
1 
1
x 2   x   .


n 1 
n

f. Find the sample standard deviation by using the standard deviation function available in the
calculator.
g. Construct the intervals: sample mean  2( sample standard deviation). What proportion of
sample observations are in this interval?
h. Calculate quartiles, interquartile range of the sample.
i. Determine outliers in the sample by the formula
(Q1  3I ,Q 3  3I ) where I  Q 3  Q1 is the
inter-quartile range.
j. Construct a box plot of the sample and explain.
k. Calculate standardized scores of the sample observations, prepare a relative frequency
distribution, relative frequency curve and hence comment on the shape of the distribution of the
sample.
j. Construct a cumulative relative frequency curve, and use it to find the 80 th percentile.
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STAT 319, TERM 091, HW 091
1.2 The following frequency distribution represent the temperature (in
February 2001 in a city.
Temperature
 8 to  4
 4 to 0
0 to 4
4 to 8
8 to 12
0
C ) for the month of
Frequency
2
4
15
4
3
a. Plot the frequency curve and comment on the shape of the distribution.
b. Find the mean and median of the temperature of the month of February. Use Frequency set-up
of your calculator.
c. Find variance of the temperature of the month of February. While using frequency set-up of
your calculator you may replace the temperature in every class by the mid-values of every class
by the lower limits, or mid-values or upper limits. What is the standard deviation of the
temperature of the month of February.
d. Calculate standard deviation of the following sample
Temperature
1
2
3
4
5
Frequency
2
4
15
4
3
and check that the standard deviation in part (c ) is 4 times the standard deviation obtained in
here.
e. What is the coefficient of variation of the temperature of the month of February. Explain it.
c. Calculate coefficient of skewness. Does it provide different interpretation from what you have in
part (a)? Explain.
d. Interpolate 80th percentile from the Cumulative Relative Frequency Curve.
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STAT 319, TERM 091, HW 091
Chapter 2
2.1 Two drivers will drive through 2 traffic signals. It is equally likely that a driver gets caught by
red signal at any of the 2 traffic signals. What is the probability that all the drivers get caught by
the red light at the (a) first (b) second (c) same traffic signal?
2.2 The primary causes of failure of a space power cell are solar radiation with probability 0.10,
launch vibration 0.35, material weakness 0.19, collision 0.36. For any particular failure, these
causes may be considered as mutually exclusive events. Since there can be only one primary
reason for failure. The launch vibration and collision are mechanical accidents. What is the
probability that the primary causes of failure of a space power cell are mechanical?
2.3 A shipment of 50 personal computers has 40 intact, eight damaged but operative, and two
inoperative. If three computers are randomly selected without replacement from the shipment
a.
b.
c.
d.
what is the probability that at least two are intact?
what is the probability that one computer is damaged but operative?
what is the probability that the third computer is damaged?
what would be your answer in part (b) if you sampled with replacement?
2.4 A DC-10 airplane has three engines-a central engine, and an engine on each wing. The plane
will crash only if the central engine fails and at least one of the two wing engines fails. The
probability of failure during any given flight is 0.005 for the central engine and 0.008 for each of
the wing engines. Assuming that the three engines operate independently, what is the probability
that the plane will crash during a flight?
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STAT 319, TERM 091, HW 091
Chapter 3 /4
1. A contractor is going to bid a project, and the days, X, required to completion follows the
probability distribution (probability mass function) given as follows:
x
f (x )
10
0.1
11
0.3
12
0.4
13
0.1
14
0.1
a. The contractor’s profit is Y  2000(12  X ). Find the contractors expected profit.
b. Find the expected the days required to complete the project.
c. Also find the variability of the number of days required to complete the project.
d. Find 90th percentile.
2. Suppose that it has been empirically determined that the probability mass function for the
random variable counting the number of I/O for a particular system in a particular time frame is
given by
e 1
f (x ) 
,
x!
if x  0,1, 2,
a. What is the probability of observing two or more errors?
b. Find mean and variance of the number of I/O for the system.
(Dougherty, 1990, 131)
3. The percentage (Y ) of antiknock additive in a particular gasoline has density function
where

 (1  y) y 2 , 0  y  1
f ( y)  
0 elsewhere
is a normalizing constant.
a. Find the normalizing constant.
b. Evaluate P (Y  1 / 2)
c. Evaluate P [{Y  0.75}|{Y  0.25}]
d. Find 95th , 96th, 97th , 98th, 99th percentile of the above distribution.
e. Evaluate the expected value and standard deviation of the above distribution.
f. What proportion of the percentage of antiknock additive lie within 2 standard deviation of the
average.
4. The life lengths ( y ) of a battery, in 100 hours, follows
0.5 e 0.5 y ,
(0  y  )
f (y )  
0, (  y  0)
a. Derive the following percentiles corresponding to tail probabilities:
0.05
0.04
0.03
0.02
0.01
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STAT 319, TERM 091, HW 091
5. The life lengths ( y ) of a battery, in 100 hours, is given the following density function:
0.5 e 0.5 y ,
(0  y  )
f (y )  
0, (  y  0)
a. What is the expected life length of the battery?
b. Find 95th percentile of the life lengths of the battery.
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STAT 319, TERM 091, HW 091
Chapter 5
5.1 In shooting a rifle the probability that John hits the target is 0.95, the probability that he hits
bull's eye is 0.20.
a. What is the probability that he fails to hit the target at least 3 times before the first success ?
b. What is the probability that he hits bull's eye at least two times if he tried 4 times?
5.2 Suppose that there are 18 defective glass bricks that include10 that have cracks but no
discoloration, five that have discoloration but no cracks, and three that have cracks and
discoloration.
a. What is the probability that among six of the bricks (chosen at random for further checks) three
will have discoloration but no cracks?
b. What is the probability that the first one and every alternate one of a sample of 6 bricks will
have discoloration but no cracks? Solve this part without resorting to part (c).
N n 


 K  x  where the symbols have usual meanings.
c. Solve part (b) by the formula
N 
 
K 
c. Use ( c) to get probability in part (a).
c. Use the formula of the text book to get the probability in part (a).
5.3 In shooting a rifle the probability that John hits the target is p .
a. If John tries 3 times, derive the expected number of tries to hit the target by the formula
E (X )   xf (x )  0f (0)  1f (1)  2f (2)  3f (3) .
b. If John tries n times, what will be the expected number of tries to hit the target?
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STAT 319, TERM 091, HW 091
Chapter 6
Police records that the speed of cars on the Kwinana Freeway is normally distributed with mean
speed   98 km/hr with a standard deviation of 10 km/hr.
a. Suppose that 1.39% of the drivers are caught speeding. What is the minimum speed to get
caught speeding? (cf. Smith, Peter1997, Springer Verlag, p317)
b. What is the probability that 2 out of 5 drivers are caught speeding?
c. If cars of two friends are checked, what is the probability that the total speed of the two cars
will exceed 196 km?
d. Find 95th percentile of the distribution.
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STAT 319, TERM 091, HW 091
Chapter 8
Police records that the speed of cars on the Kwinana Freeway is normally distributed with mean
speed   98 km/hr with a standard deviation of 10 km/hr.
a. If the drivers of the most highly speeding 5% cars are jailed, what is the minimum speed
with which a driver is jailed?
b. If cars of two friends are checked, what is the probability that the total speed of the two
cars will exceed 196 km?
c. If 2 cars are sampled, find the distribution of the average speed. Also find the 95 th
percentile of the distribution of the average speed.
d. If 30 cars are sampled, find the distribution of the average speed. Also find the 95 th
percentile of the distribution of the average speed.
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STAT 319, TERM 091, HW 091
Chapter 9
1. Police records that the speed of cars on the Kwinana Freeway is normally distributed. A
sample of 2 cars provide a mean speed of x  100 km/hr with a standard deviation of 10 km/hr.
Construct a 95% confidence interval for the average speed.
2. Police records that the speed of cars on the Kwinana Freeway has a standard deviation of 14
km/hour. A sample of 49 cars provides a mean speed of x  118 km/hr (with a standard
deviation of 16 km/hr). Construct a 95% confidence interval for the average speed.
3. Police records that the speed of cars on the Kwinana Freeway has a normal distribution with a
standard deviation of 14 km/hour. A sample of 2 cars provides a mean speed of x  118 km/hr
(with a standard deviation of 16 km/hr). Construct a 95% confidence interval for the average
speed.
4. Police records that the speed of cars on the Kwinana Freeway is exponentially decreasing. A
sample of 36 cars provide a mean speed of x  100 km/hr with a standard deviation of 12
km/hr. Construct a 95% confidence interval for the average speed.
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STAT 319, TERM 091, HW 091
Chapter 10
1. Police records that the speed of cars on the Kwinana Freeway is normally distributed. A
sample of 2 cars provide a mean speed of x  122 km/hr with a standard deviation of 10 km/hr.
Test at 5% level of significance if the average speed exceeds 120 km/hr.
2. Police records that the speed of cars on the Kwinana Freeway has a standard deviation of 14
km/hour. A sample of 49 cars provides a mean speed of x  122 km/hr (with a standard
deviation of 16 km/hr).
a. Test at 5% level of significance if the average speed exceeds 120 km/hr.
b. Calculate p-vale of the test and explain it.
3. Police records that the speed of cars on the Kwinana Freeway has a normal distribution with a
standard deviation of 14 km/hour. A sample of 2 cars provides a mean speed of x  122 km/hr
(with a standard deviation of 16 km/hr). Test at 5% level of significance if the average speed
exceeds 120 km/hr.
4. Police records that the speed of cars on the Kwinana Freeway is exponentially decreasing. A
sample of 36 cars provide a mean speed of x  122 km/hr with a standard deviation of 12
km/hr. Test at 5% level of significance if the average speed exceeds 120 km/hr.
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STAT 319, TERM 091, HW 091
Chapter 11
The rainfall ( x ) in inches affect the yields ( y ) of wheat (bushels per acre). The following data
has been collected:
x
y
12.9
7.2
11.3
18.6
8.8
10.3
15.9
13.1
62.5
28.7
52.2
80.6
41.6
44.5
71.3
54.4
a. Estimate the coefficient of correlation by preparing two columns of z (x ) and z ( y ) and then
using the formula
r
1
 z (x )z ( y ) .
n 1
b. Estimate the slope by the following formula
b r
sy
sx
.
c. Estimate the points ( x , y ) and (x  r , y  br ) , and join them to estimate the line of best fit.
Answer: The line ˆ ( x )  0.23  4.42x .
d. Estimate the errors e  y  ˆ ( x ) , square them, and hence derive Sum of Squares due to
Errors.
Alternative formulas should also be credited.
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STAT 319, TERM 091, HW 091