Download STAT 319, TERM 042

STAT 319, TERM 072
Home Works for Chapter 1
Problem 1.
Diameters of 30 rivet heads in 1/100 of an inch are given below:
6.72
6.62
6.73
6.66
6.70
6.77
6.75
6.80
6.62
6.72
6.82
6.66
6.72
6.70
6.74
6.70
6.66
6.76
6.72
6.81
6.78
6.64
6.76
6.74
6.74
6.70
6.76
6.68
6.81
6.81
(a)
(b)
(c)
(d)
(e)
(f)
Prepare a Stem-and-leaf plot of the sample, and comment.
What is the mode diameter of the rivets?
Find 93rd percentile and interpret this value.
Do the data satisfy Empirical Rule?
Draw a box plot and comment.
Calculate coefficient of variation and comment on it. Is the variation in the
sample relative to the mean too much? Comment using a scale of [0, n ] .
(g) Calculate coefficient of skewness, and comment on it.
(h) Prepare a relative frequency distribution of the data.
(i) Calculate coefficient of variation and coefficient of skewness from the grouped
data.
(j) Calculate 75th percentile from the grouped data. Find proportion of observations
that are below it, and comment.
(k) What proportion of the observations are below the first quartile for the above
sample?
(l) What proportion of the observations are above the mean?
(m) What proportion of the observations are within 2 standard deviation of the mean?
(n) Calculate z-scores of the smallest and the largest observations in the sample.
What do they add to? Find variance of the z-scores.
Problem 2. The following data are the compressive strengths in pounds per square
inch (psi) of 80 specimens of a new aluminum-lithium alloy undergoing evaluation as
a possible material for aircraft structural elements.
105
245
207
218
160
221 183 186 121 181 180 143 97 154 153 174 120 168 167 141
228 174 199 181 158 176 110 163 131 154 115 160 208 158 133
180 190 193 194 133 156 123 134 178 76 167 184 135 229 146
157 101 171 165 172 158 169 199 151 142 163 145 171 148 158
175 149 87 160 237 150 135 196 201 200 176 150 170 118 149
(a) Construct a frequency distribution and a frequency histogram starting from
70 and the step size 20.
(b) Construct a stem and leaf plot.
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Home Works for Chapter 2
Problem 1. A study was recently done in which 500 people were asked to indicate their
preferences for one of three products. The following table shows the breakdown of the
responses by gender of the respondents.
Gender
Male
Female
Product Preference
B
20
70
A
80
200
C
10
120
Based on these data, what is the probability of selected person is a female and that she prefers
product C?
Problem 2. The managers of a local golf course have recently conducted a study of the
types of golf balls used by golfers based on handicap. A joint frequency table for the 100
golfers covered in the survey is show below:
Handicap
<2
2 < 10
> 10
Strata
5
8
7
Type of Golf Ball
Titleist
Nike
8
3
7
9
8
10
Other
2
10
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a) Based on these data, what is the probability that a player will use a Strata golf ball?
b) Based on these data, if a player has a handicap that is 2 or over, what is the probability
that he or she will use a Nike golf ball?
Problem 3. The Anderson Lumber Company has three sawmills which produce boards of
different lengths. The following table is a joint frequency distribution based on a random
sample of 1,000 boards selected from the lumber inventory.
Sawmill
A
B
C
Board Length
10 ft
100
20
50
8 ft
140
250
160
12 ft
80
100
16
14ft
14
50
20
a) Based on these data, if a board is selected that is 12 feet long, what is the probability that it
was made at sawmill A ?
b) Based on these data, if three boards are selected at random, what the probability that
all three were made at sawmill A ?
Problem 4. Harrison Water Sports has three retail outlets: Seattle, Portland, and Phoenix.
The Seattle store does 50 percent of the total sales in a year, while the Portland store does 35
percent of the total sales. Further analysis indicates that of the sales in Seattle, 20 percent are
in boat accessories. The percentage of boat accessories at the Portland store is 30 and the
percentage is 25 at the Phoenix store. If a sales dollar is recorded as a boat accessory, what is
the probability that the sale was made at the Portland store?
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Home Works for Chapters 3 and 4
Problem 1. A sales rep for a national clothing company makes 4 calls per day. Based on
historical records, the following probability distribution describes the number of successful
calls each day:
Successful Calls
0
1
2
3
4
Probability
0.10
0.30
0.30
0.20
0.10
a) What is the probability that the sales rep will have two successful calls in a two-day
period?
b) Each successful call earns the sales rep $100. Based on the information provided, what is
the expected earnings for a sales rep who makes calls for 10 days?
Problem 2. The following probability distribution has been assessed for the number of
accidents that occur in a mid-western city each day:
Accidents
0
1
2
3
4
Probability
0.25
0.20
0.30
0.15
0.10
a) Based on this distribution, what the expected number of accidents in a given day?
b) Based on this probability distribution, what the standard deviation in the number of
accidents per day?
Problem 3. A random variable Y, which represent the weight (in ounces) of an
article, has probability density function given by
 y 8

f ( y )  10  y
 0

for 8  y  9
for 9  y  10
otherwise
a) Calculate the mean of the random variable X.
b) The manufacturer sells the article for a fixed price of $2.00. He guarantee to refund
the purchase money to any customer who finds the weight of his article to be less than
8.25 ounces. His cost of production is related to the weight of the article by the
relation is x/15 + 0.35. Find the expected profit per article.
Problem 4. A college professor always finishes his lectures within 2 minutes after the
bell rings to end the period and the end of the lecture. Let X = the time that
elapses between the bell and the end of the lecture and suppose the pdf of X is
kx 2

f ( x)  

0
0 x2
otherwise
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a. Find the value of k.
b. What is the probability that the lecture ends within 1minutes of the bell
ringing?
c. What is the probability that the lecture continues beyond the bell for
between 60 and 90 seconds?
d. What is the probability that the lecture continues for at least 90 seconds
beyond the bell?
Home Works for Chapter 5
Problem 1. If the probability that a fluorescent light has a useful life of at least 500
hours is 0.85, find the probabilities that among 20 such lights
(a) 18 will have a useful life of at least 500 hours.
(b) at least 15 will have a useful life of at least 500 hours.
(c) at most 10 will not have a useful life of at least 500 hours.
Problem 2. Suppose that 20% of all copies of a particular textbook fail a certain
binding strength test. Let X denote the number among 15 randomly selected copies
that fail the test.
(a1) What is the distribution of X ?
(a) Complete the probability and cumulative probability distribution for the
number of failures.
(b) Draw the probability and cumulative probability histograms.
(c) Find the probability that at most 8 fail the test.
(d) Find the probability that exactly 8 fail the test.
(e) Find the probability that at least 8 fail.
(f) Find the probability that between 4 and 7, inclusive, fail the test.
Problem 3. In the inspection of tinplate produced by a continuous electrolytic
process, 0.2 imperfections are spotted on the average per minute. Find the
probabilities of spotting
(a) one imperfection in 3 minutes.
(b) at least 2 imperfections in 5 minutes.
(c) at most one imperfection in 15 minutes.
Problem 4. A foundry ships engine blocks in lots of 20. Three items are selected and
tested. If the lot actually contains five defective items, find the probability that
there will be at least 2 defective blocks in the sample?
Problem 5. Each of 12 refrigerators of a certain type has been returned to a distributor
because of the presence of a high-pitched oscillating noise when the
refrigerator is running. Suppose that 5 of these 12 have defective compressors
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and the other 7 have less serious problems. If they are examined in random
order, let X = the number among the first 6 examined that have a defective
compressor. Compute the following:
a. P(X = 1)
b. P( X  4)
c. P(1  X  3)
Home Works for Chapter 6
1. A transistor has an exponential time to failure distribution with mean time to
failure of β = 20,000 hours.
(a) What is the probability that the transistor fails by 30,000 hours?
(b) The transistor has already lasted 20, 000 hours in a particular application.
What is the probability that it fails by 30, 000 hours?
(c) Comment on the probability in (a)
if Beta = 10000;20000;30000;40000;50000;60000
2. The lifetime X (in hours) of the central processing unit of a certain type of
microcomputer is an exponential random variable with parameter 0.001. What is
the probability that the unit will work at least 1,500 hours?
3. The lifetime (in hours) of the central processing unit of a certain type of
microcomputer is an exponential random variable with mean β = 1000.
(a) What is the probability that a central processing unit will have a lifetime of at
least 2000 hours?
(b) What is the probability that a central processing unit will have a lifetime of at
most 2000 hours
4. Let X denote the number of flaws along a 100-m reel of magnetic tape. Suppose X
has approximately a normal distribution with 5 = µ and 5 = σ . Calculate the
probability that the number of flaws is
(a) between 20 and 30.
(b) at most 30.
(c) less than 30.
(d) not more than 25.
(e) at most 10
5. A machining operation produces steel shafts having diameters that are normally
distributed with a mean of 1.005 inches and a standard deviation of 0.01 inch.
Specifications call for diameters to fall within the interval 1.00 ± 0.02 inches.
(a) What percentage of the output of this operation will fail to meet
specifications?
(b) Comment on the percentage in (a) if σ increases.
6. The weekly amount spent for maintenance and repairs in a certain company has
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approximately a normal distribution with a mean of $400 and a standard deviation
of $20.
(a) If $450 is budgeted to cover repairs for next week, what is the probability that
the actual costs will exceed the budgeted amount?
(b) Comment on the probability in part (a) if µchanges, keeping σfixed.
(c) Comment on the probability in part (a) if σchanges, keeping µfixed.
7. Consider a binomial random variable with 20 trials and success probability 0.45.
Using the normal approximation
(a) Compute the probability that it is at least equal to 3.
(b) Compute the probability that it is at most equal to 16.
(c) Compute the probability that it is equal to 3.
(d) Compute the probability that it is equal to 12.
Home Works for Chapter 8
1. A random sample of size 100 is taken from an infinite population having a mean,
76 and a variance, 256. What is the probability that the sample mean will be between
75 and 78?
2. A wire- bonding process is said to be in control if the mean pull-strength is 10
pounds. It is known that the pull-strength measurements are normally distributed with
a standard deviation of 1.5 pounds. Periodic random samples of size 4 are taken from
this process and the process is said to be “out of control” if a sample mean is less than
7.75 pounds. Comment.
3. The weights of ball bearings have a distribution with a mean of 22.40 ounces and
a standard deviation of 0.048 ounces. If a random sample of size 49 is drawn from
this population, find the probability that the
(a) sample mean lies between 22.36 and 22.41,
(b) sample mean is more than 22.38,
(c) sample mean is no more than 22.43,
(d) sample mean is greater than or equal to 22.41.
4. A random sample of size 100 is taken from an infinite population having a mean,
76 and a variance, 256. What is the probability that the sample mean will be between
75 and 78?
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Chapter 9: Interval Estimation
Problems on True Mean
1. The article "Ozone for removal of acute toxicity from logyard run-off" (M. Zenaitis
and S Duff, 2202, Ozone Science and Engineering, 83-90) presents chemical analyses
of runoff water from sawmills in British Columbia. Included were measurements of
pH for six water specimens: 5.9, 5.0, 6.5, 5.6, 5.9, 6.5. Experience says that these type
of pH contents follow normal distribution. Derive a 95% confidence interval for the
true mean pH. Would you modify the method of estimation if sample size were large?
(T)
2. The pH content of runoff water from sawmills in British Columbia is considered.
Six water specimens produce a mean of 5.5. Experience says that these type of pH
contents follow normal distribution with standard deviation 0.81. Derive a 95%
confidence interval for the true mean pH. Would you modify the method of estimation
if sample size were large? (RT)
3. The pH content of runoff water from sawmills in British Columbia is considered.
Thirty water specimens produce a mean of 5.5. Experience says that these type of pH
contents follow normal distribution with standard deviation 0.81. Derive a 95%
confidence interval for the true mean pH. Would you modify the method of estimation
if sample size were large? (CLT)
4. The pH content of runoff water from sawmills in British Columbia is considered.
Thirty water specimens produce a mean of 5.5 and standard deviation 0.81. Derive a
95% confidence interval for the true mean pH. (ST)
Problem on True Proportion
5. Concentrations of atmospheric pollutants such as carbon monoxide (CO) can be
measured with a spectrophotometer. In a calibration test, 40 measuremenst were taken
of a laboratory gas sample that is known to have a CO concentration of 80 parts per
million (ppm). A measurement is considered to be satisfactory if it is within 5 ppm of
the true concentration. Of the 40 measurements, 36 were satisfactory.
a. What proportion of the sample measurements were satisfactory?
b. Derive a 95% confidence interval of the proportion of measurements made by
this instrument that will be satisfactory.
c. How many measurements must be taken to specify the proportions of
satisfactory measurements to within 0.10 with 95% confidence?
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Problem on Difference in Two Proportions
6. In a test of the effect of dampness on electric connection, 100 electric connections
were tested under
damp conditions and 150 were tested under dry conditions. Twenty of the damp
connections failed and only 10 of the dry ones failed. Derive a 95% confidence
interval for the difference between the proportions of connections that fail when damp
as opposed to dry.
Problem on Difference in Two Means
7. A machine is used to fill plastic bottles with bleach. A sample of 18 bottles had a
mean fill volume 0f 2.006L and a standard deviation of 0.011L. The machine was
then moved to another location. A sample of 10 bottles filled at the new location had a
mean fill volume of 2.001L and a standard deviation of 0.012L. It is believed that
moving the machine may have changed the (true) mean fill volume, but it is unlikely
to have changed the (true) standard deviation. Assume that both samples come from
approximately normal populations. Derive a 95% confidence interval for the
difference between the (true) mean fill volumes at the two locations.
8. A sample of 120 pieces of yarn had mean breaking strength 6.2N and standard
deviation 0.6N. A new batch of yarn was made, using new raw materials from a
different vendor. In a sample of 70 pieces of yarn from the new batch, the mean
breaking strength 5.7 N and standard deviation was 1.2N. Find a 95% confidence
interval in mean breaking strength between the two types of yarn.
Chapter 10: Tests of Hypotheses
Problems on True Mean
1. The article "Ozone for removal of acute toxicity from logyard run-off" (M. Zenaitis
and S Duff, 2202, Ozone Science and Engineering, 83-90) presents chemical analyses
of runoff water from sawmills in British Columbia. Included were measurements of
pH for six water specimens: 5.9, 5.0, 6.5, 5.6, 5.9, 6.5. Experience says that these type
of pH contents follow normal distribution.
1.
2. a. Test the hypothesis that true pH content is more than 5.5.
3. b. Test the hypothesis that true pH content is less than 5.5.
4. c. Test the hypothesis that true pH content is 5.5.
Would you modify the method of testing if sample size were large? (T)
2. The pH content of runoff water from sawmills in British Columbia is considered.
Six water specimens produce a mean of 5.5. Experience says that these type of pH
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contents follow normal distribution with standard deviation 0.81. Test the claim that
true pH content is more than 5.5. Would you modify the method of estimation if
sample size were large? RT)
3. The pH content of runoff water from sawmills in British Columbia is considered.
Thirty water specimens produce a mean of 5.5. Experience says that these type of pH
contents follow normal distribution with standard deviation 0.81. Test the claim that
true pH content is more than 5.5. Would you modify the method of estimation if
sample size were large? (CLT)
4. The pH content of runoff water from sawmills in British Columbia is considered.
Thirty water specimens produce a mean of 5.5 and standard deviation 0.81. Test the
claim that true pH content is more than 5.5. (ST)
Problem on True Proportion
5. Concentrations of atmospheric pollutants such as carbon monoxide (CO) can be
measured with a spectrophotometer. In a calibration test, 40 measurements were taken
of a laboratory gas sample that is known to have a CO concentration of 80 parts per
million (ppm). A measurement is considered to be satisfactory if it is within 5 ppm of
the true concentration. Of the 40 measurements, 36 were satisfactory. Test the claim
that the proportion of satisfactory measurements is more than 0.80.
Chapter 11: Simple Linear Regression
Inertial weight (tons) and fuel economy (in mi/gal) were measured for a sample of 7
diesel trucks. Obviously mileage is the main variable of interest. The results are
presented in the following table.
weight
Mileage
8.25
7.55
24.25
4.95
25.50
5.05
12.50
6.25
28.75
4.45
21.50
4.55
14.75
6.55
a. Draw a scatter diagram for the sample. Comment on the degree of linearity
among truck mileage and weight. Do you recommend fitting a line?
b. Estimate regression coefficients and explain them.
c. Find standard error of estimate of the slope.
d. Estimate the least squares line that governs truck mileage and truck weight.
e. Plot the line of best fit.
f. Predict the truck mileage for trucks with a weight of 15 tons.
g. Estimate 7 errors and hence or otherwise calculate SSE.
h. Decompose MCSS (Mean Corrected Sum of Squares of the mileages) into two
natural parts, and explain each part.
i. Quantify the linear association between truck mileage and truck weight.
j. Find the total variation in truck mileage that can be attributed linearly to truck
weight.
k. What proportion of variability in truck mileage is explained by truck weight?
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l.
m.
n.
o.
p.
Explain the correlation coefficient.
Estimate the average truck mileage for trucks weighing 15 tons.
Estimate the average mileage for trucks weighing about the sample average.
Test if a linear relationship exists between the truck mileage and truck weight?
Find a 95% confidence interval for the expected truck mileage for trucks
weighing about 15 tons.
q. Find the coefficient of determination and interpret them.
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