P s work-book_part2_ver_4 (1)

6 Hypothesis Testing
-Two Populations

7
Exercise 1:
A random sample of size n = 25 taken from a normal population with σ = 5.2 has a mean
equals 81. A second random sample of size n = 36, taken from a different normal population
with σ = 3.4, has a mean equals 76.
(a) Do the data indicate that the true mean value µ1 and µ2 are different? Carry out a test at
α = 0.01
(b) Find 90% CI on the difference in mean strength

Exercise 2:
Two machines are used for filling plastic bottles with a net volume of 16.0 oz. The fill
volume can be assumed normal with, s1 = 0.02 and s2 = 0.025. A member of the quality
engineering staff suspects that both machines fill to the same mean net volume, whether or
not this volume is 16.0 oz. A random sample of 10 bottles is taken from the output of each
machine with the following results:
(a) Do you think the engineer is correct? Use the p – value approach.
(b) Find a 95% CI on the difference in means.

1

Probability and Statistics Work Book

Exercise 3: (Tutorial 6, No.1)
Two machine are used to fill plastic bottles with dishwashing detergent. The standard
deviations of fill volume are known to be σ1= 0.01 and σ2 = 0.15 fluid ounce for two
machines, respectively. Two random samples of n1 = 12 bottles from machine 1 and n2=10
bottles from machine 2 are selected, and the sample mean fill volumes are x 1 =30.61
x 2 =30.24 fluid ounces. Assume normality.
(i) Test the hypothesis that both machines fill to the same mean volume. Use the P-
value
approach;
(ii) Construct a 90% two-sided CI on the mean difference in fill volume; and
(iii) Construct a 95% two-sided CI on the mean difference in fill volume. Compare and
comment on the width of this interval to the width of the interval in part (ii).

Exercise 4:
To find out whether a new serum will arrest leukemia, 9 mice, all with an advanced stage of
the disease are selected. 5 mice receive the treatment and 4 do not. Survival, in years, from
the time the experiment commenced are as follows:

Treatment 2.1 5.3 1.4 4.6 0.9

No treatment 1.9 0.5 2.8 3.1

At the 0.05 level of significance can the serum be said to be effective? Assume the two
distributions to be of equal variances.

2


A new policy regarding overtime pay was implemented. This policy decreased the pay factor
for overtime work. Neither the staffing pattern nor the work loads changed. To determine if
overtime loads changed under the policy, a random sample of employees was selected. Their
overtime hours for a randomly selected week before and for another randomly selected week
after the policy change were recorded as follows:

Employees: 1 2 3 4 5 6 7 8 9 10 11 12
Before: 5 4 2 8 10 4 9 3 6 0 1 5
After: 3 7 5 3 7 4 4 1 2 3 2 2

Assume that the two population variances are equal and the underlying population is
normally distributed.
(i) Is there any evidence to support the claim that the average number of hours worked
as overtime per week changed after the policy went into effect. Use a P-value
approach in arriving at this conclusion.
(ii) Construct a 95% CI for the difference in mean before and after the policy change.
Interpret this interval.

Exercise 6:
The diameter of steel rods manufactured on two different extrusion machines is being
investigated. Two random samples of sizes n1 = 15 and n2 = 17 are selected, and
respectively. s1 = 0.35 and x2 = drawn s2 = 0.40
x1 = 8.37, Assume that data are 8.68, normal distribution with equal variances.
2 2

(a) Is there evidence to support the claim that the two machines produce rods with different
mean diameters ? Use the p – value approach.
(b) Construct a 95% CI on the difference in mean rod diameter.

3


Exercise 7:
The following data represent the running times of films produced by 2 motion-picture
companies. Test the hypothesis that the average running time of films produced by company
2 exceeds the average running time of films produced by company 1 by 10 minutes against
the one-sided alternative that the difference is less than 10 minutes? Use a = 0.01 and assume
the distributions of times to be approximately normal with unequal variances.

Time

Company
X1 102 86 98 109 92

X2 81 165 97 134 92 87 114

Exercise 8:
Two companies manufacture a rubber material intended for use in an automotive application.
25 samples of material from each company are tested, and the amount of wear after 1000
cycles are observed. For company 1, the sample mean and standard deviation of wear are
x1 = 20.12mg / 1000cycles and s1 = 1.9mg / 1000cycles
and for company 2, we obtain x2 = 11.64mg / 1000cycles and s2 = 7.9mg / 1000cycles

(a) Do the sample data support the claim that the two companies produce material with
different mean wear? Assume each population is normally distributed but unequal
variances?
(b) Construct a 95% CI for the difference in mean wear of these two companies. Interpret
this interval.


4


Professor A claims that a probability and statistics student can increase his or her score on
tests if the person is provided with a pre-test the week before the exam. To test her theory she
selected 16 probability and statistics students at random and gave these students a pre-test the
week before an exam. She also selected an independent random sample of 12 students who
were given the same exam but did not have access to the pre-test. The first group had a mean
score of 79.4 with standard deviation 8.8. The second group had sample mean score 71.2
with standard deviation 7.9.
(i) Do the data support Professor A claims that the mean score of students who get a
pre-test are different from the mean score of those who do not get a pre test before
an exam. Use the P-value approach and assume that their variances are not equal.
(ii) Construct a 95% CI for the difference in mean score of students who get a pre-test
and those who do not get a pre-test before an exam. Interpret this interval.

Exercise 10:
A vote is to be taken among residents of a town and the surrounding county to determine
whether a proposed chemical plant should be constructed. If 120 of 200 town voters favour
the proposal and 240 of 500 county residents favour it, would you agree that the proportion
of town voters favouring the proposal is higher than the proportion of county voters? Use a =
0.05

5


The rollover rate of sport utility vehicles is a transportation safety issue. Safety advocates
claim that the manufacturer A’s vehicle has a higher rollover rate than that of manufacturer
B. One hundreds crashes for each of this vehicles were examined. The rollover rates were
pA=0.35 and pB=0.25.
(i) By using the P-value approach, does manufacturer A’s vehicle has a higher rollover
rate
than manufacturer B’s?
(ii) Construct a 95% CI on the difference in the two rollover rates of the vehicle.
Interpret
this interval.

Exercise 12:
Professor Rady gave 58 A’s and B’s to a class of 125 students in his section of English 101.
The next term Professor Hady gave 45 A’s and B’s to a class of 115students in his section of
English 101.
(i) By using a 5% significance level, test the claim that Professor Rady gives a higher
percentage of A’s and B’s in English 101 than Professor Hady does. What is
comment?
(ii) Construct a 95% CI on the difference in the percentage of A’s and B’s in English
101 given by this two professors.

6

7

Simple Linear Regression

8
Exercise 1:
The manager of a car plant wishes to investigate how the plant’s electricity usage depends
upon the plant production. The data is given below

Production 4.51 3.58 4.31 5.06 5.64 4.99 5.29 5.83 4.7 5.61 4.9 4.2
(RMmillion)
(x)
Electricity 2.48 2.26 2.47 2.77 2.99 3.05 3.18 3.46 3.03 3.26 2.67 2.53
Usage
(y)

(a) Estimate the linear regression equation Y = β 0 + β1 x
(b) An estimate for the electricity usage when x = 5
(c) Find a 90% Confidence Interval for the electricity usage.

7


Exercise 2:
An experiment was set up to investigate the variation of the specific heat of a certain
chemical with temperature. The data is given below

Temperature oF 50 60 70 80 90 100
(x)

Heat 1.60 1.63 1.67 1.70 1.71 1.71
(y) 1.64 1.65 1.67 1.72 1.72 1.74

(a) Estimate the linear regression equation Y = β 0 + β1 x
(b) Plot the results on a scatter diagram
(c) An estimate for the specific heat when the temperature is 75oF
(d) Find a 95% Confidence Interval for the specific heat.

Exercise 3:
An engineer at a semiconductor company wants to model the relationship between the device
HFE (y) and the parameter Emitter - RS ( x1). Data for Emitter - RS was first collected and
a statistical analysis is carried out and the output is displayed in the table given.

Regression Analysis: y = 1075.2 – 63.87x1
Predictor Coef SE Coef T P-value
Constant 1075.2 121.1 8.88 0.000
x1 -63.87 8.002 -7.98 0.000
S = 19.4 R-Sq = 0.78

Analysis of variance
Source DF SS MS F
Regression 1 23965 23965 63.70
Residual 18 6772 376
Total 19 30737

(a) Estimate HFE when the Emitter - RS is 14.5.
(b) Obtain a 95 % confidence interval for the true slope β.
(c) Test for significance of regression for a = 0.05.

8


Exercise 4:
An chemical engineer wants to model the relationship between the purity of oxygen (y)
produced in a chemical distillation process and the percentage of hydrocarbons (x ) that are
present in the main condenser of the distillation unit. A statistical analysis is carried out and
the output is displayed in the table given.

Regression Analysis: y = 74.3 + 14.9x
Predictor Coef SE Coef T P-value
Constant 74.283 1.593 46.62 0.000
x1 14.947 1.317 11.35 0.000
S = 1.087 R-Sq = 87.7%

Analysis of variance
Source DF SS MS F
Regression 1 152.13 152.13 12.86
Residual 18 21.25 1.18
Total 19 173.38

(a) Estimate the purity of oxygen when the percentage of hydrocarbon 1%.
(b) Obtain a 95 % confidence interval for the true slope β.
(c) Test for significance of regression for a = 0.05.

9


Regression methods were used to analyze the data from a study investigating the relationship
between roadway surface temperature (x) and pavement deflection (y). The data follow.

Temperature Deflection Temperature Deflection
x y x y
70.0 0.621 72.7 0.637
77.0 0.657 67.8 0.627
72.1 0.640 76.6 0.652
72.8 0.623 73.4 0.630
78.3 0.661 70.5 0.627
74.5 0.641 72.1 0.631
74.0 0.637 71.2 0.641
72.4 0.630 73.0 0.631
75.2 0.644 72.7 0.634
76.0 0.639 71.4 0.638

(a) Estimate the intercept and slope regression coefficients. Write the estimated
regression line.
(b) Compute SSE and estimate the variance.
(c) Find the standard error of the slope and intercept coefficients.
(d) Show that
(e) Compute the coefficient of determination, R2. Comment on the value.
(f) Use a t-test to test for significance of the intercept and slope coefficients at .
Give the P-values of each and comment on your results.
(g) Construct the ANOVA table and test for significance of regression using the P-value.
Comment on your results and their relationship to your results in part (f).
(h) Construct 95% CIs on the intercept and slope. Comment on the relationship
of these CIs and your findings in parts (f) and (g).

10


The designers of a database information system that allows its users to search backwards for
several days wanted to develop a formula to predict the time it would be take to search.
Actually elapsed time was measured for several different values of days. The measured data
is shown in the following table:

Number of Days 1 2 4 8 16 25
Elapsed Time 0.6 0.79 1.36 2.26 3.59 5.39
5

(i) Estimate the intercept and slope regression coefficients. Write the
estimated regression line.
(ii) Compute SSE and estimate the variance.
(iii) Find the standard error of the slope and intercept coefficients.
(iv) Show that
(v) Compute the coefficient of determination, R2. Comment on the value.
(vi) Use a t-test to test for significance of the intercept and slope coefficients at
. Give the P-values of each and comment on your results.
(vii) Construct the ANOVA table and test for significance of regression using the
P-value. Comment on your results and their relationship to your results in
part (vi).
(viii)Construct 95% CIs on the intercept and slope. Comment on the relationship
of these CIs and your findings in parts (vi) and (vii).

11


8 Multiple Linear Regressions

9
Exercise 1:
Given the data:

Test Number y x1 x2
1 1.6 1 1
2 2.1 1 2
3 2.4 2 1
4 2.8 2 2
5 3.6 2 3
6 3.8 3 2
7 4.3 2 4
8 4.9 4 2
9 5.7 4 3
10 5 3 4

(a) Fit a multiple linear regression model to these data.

12


Exercise 2:
Given the data:

Observation Number Pull Strength y Wire Length x1 Die Height x2
1 9.95 2 50
2 24.45 8 110
3 31.75 11 120
4 35.00 10 550
5 25.02 8 295
6 16.86 4 200
7 14.38 2 375
8 9.60 2 52
9 24.35 9 100
10 27.50 8 300
11 17.08 4 412
12 37.00 11 400
13 41.95 12 500
14 11.66 2 360
15 21.65 4 205
16 17.89 4 400
17 69.00 20 600
18 10.30 1 585
19 34.93 10 540
20 46.59 15 250
21 44.88 15 290
22 54.12 16 510
23 56.63 17 590
24 22.13 6 100
25 21.15 5 400

(b) Fit a multiple linear regression model to these data.

13


Exercise 3:
A study was performed to investigate the shear strength of soil (y) as it related to depth in
meter (x1) and percentage moisture content (x2). Ten observations were collected and the
following summary quantities obtained:

n = 10, ∑x i1 = 223, ∑x i2 = 553,∑y i = 1,916,

∑x 2
i1 = 5,200.9, ∑ x = 31,729,
2
i2 ∑x x
= 12,352,i1 i 2

∑x i1 i y = 43,550.8, ∑ x y = 104,736.8, ∑ y = 371,595.6
i2 i
2
i

(a) Estimate the parameters to fit the multiple regression models for these data.
(b) What is the predicted strength when x1=18meter and x2= 43%.

14


Exercise 4:
A set of experimental runs were made to determine a way of predicting cooking time y at
various levels of oven width x1, and temperature x2. The data were recorded as follows:

y x1 x2
6.4 1.32 1.15
15.05 2.69 3.4
18.75 3.56 4.1
30.25 4.41 8.75
44.86 5.35 14.82
48.94 6.3 15.15
51.55 7.12 15.32
61.5 8.87 18.18
100.44 9.8 35.19
111.42 10.65 40.4

(b) Estimate and the standard errors of the regression coefficients.
(c) Test for significance of and .
(d) Predict the useful range when brightness = 80 and contrast = 75. Construct a 95% PI.
(e) Compute the mean response of the useful range when brightness = 80 and contrast = 75.
Compute a 95% CI.
(f) Interpret parts (d) and (e) and comment on the comparison between the 95% PI and 95%
CI.

15


An article in Optical Engineering (“Operating Curve Extraction of a Correlator's Filter,” Vol.
43, 2004, pp. 2775–2779) reported the use of an optical correlator to perform an experiment
by varying brightness and contrast. The resulting modulation is characterized by the useful
range of gray levels. The data are shown

Brightness (%): 5 6 6 10 10 10 50 57 54
4 1 5 0 0 0
Contrast (%): 5 8 7 50 65 80 25 35 26
6 0 0
Useful range (ng): 9 5 5 11 96 80 15 14 25
6 0 0 2 5 4 5

(b) Estimate and the standard errors of the regression coefficients.
(c) Test for significance of and .
(d) Predict the useful range when brightness = 80 and contrast = 75. Construct a 95% PI.
(e) Compute the mean response of the useful range when brightness = 80 and contrast = 75.
Compute a 95% CI.
(f) Interpret parts (d) and (e) and comment on the comparison between the 95% PI and 95%
CI.

16


A study was performed on wear of a bearing y and its relationship to x1 = oil viscosity and
x2 = load. The following data were obtained:

x 1.6 15.5 22.0 43.0 33.0 40.0
1
x 85 816 1058 120 135 111
2 1 1 7 5
y 29 230 172 91 113 125
3

(a) Fir a multiple regression model to these data.
(b) Estimate σ2 and the standard errors of the regression coefficients.
(c) Use the model to predict wear when x1 = 25 and x2 = 1000.
(d) Fit a multiple regression model with an interaction term to these data.
(e) Estimate σ2 and se(βj) for this new model. How did these quantities change? Does
this tell you anything about the value of adding the interaction term to the model?
(f) Use the model in (d), to predict when x1=25 and x2=1000. Compare this prediction
with the predicted value from part (c) above.

17


9 Factorial Experiments
– 22 Factorial design

Exercise 1:
An engineer is investigating the thickness of epitaxial layer which will be subject to two
variations in A, deposition time (+ for short time, and – for long time) and two levels of B,
arsenic flow rate (- for 55% and + for 59%). The engineer conduct 22 factorial design with n
= 4 replicates. The data are as follow:

18


Arsenic Level
B– B+
(Low - 55%) (High – 59%)

Deposition Time
14.037 13.880
14.165 13.860
A - (Long) 13.972 14.032
13.907 13.914

14.821 14.888
14.757 14.921
A + (Short) 14.843 14.415
14.878 14.932

a) Construct the 2 X 2 factorial design table.
b) Find the estimate of all effects and interaction.
c) Construct the ANOVA table for each effect, test the null hypothesis that the effect is
equal to 0.

A two factor experimental design was conducted to investigate the lifetime of a component
being manufactured. The two factors are A (design) and B (cost of material). Two levels ((+)
and (-)) of each factor are considered. Three components are manufactured with each
combination of design and material, and the total lifetime measured (in hours) is as shown in
table below

19


Total lifetime of 3
Design Material
AB components
Treatment A B
(in hours)
Combination
(1) - - + 122
a
+ - - 60
b - + - 120
ab + + + 118

(a) Perform a two way analysis of variance to estimate the effects of design and material
expense on the component life time if the sum squares of total are 1050.
(b) Based on your results in part (a), what conclusions can you draw from the factorial
experiment?
(c) Indicate which effects are significant to the lifetime of a component.
(d) Write the least square fitted model using only the significant sources.

Exercise 3:
An engineer suspects that the surface finish of metal parts is influenced by the type of paint
used and the drying time. He selected three drying times – 20, 25, and 30 minutes and used
two types of paint. Three parts are tested with each combination of paint typoe and drying
time. The data are as follow:

Drying Time (min)
Paint 20min 25min 30min
ICI 74 73 78
64 61 85
50 44 92
NIPPON 92 98 66
20
86 73 45
68 88 85


(a) Compute the estimates of the effects and their standard errors for this design.
(b) Construct two-factor interaction plots and comment on the interaction of the factors.
(c) Use the t ratio to determine the significance of each effect with .Comment on
your findings.
(d) Compute an approximate 95% CI for each effect. Compare your results with those in
part (c) and comment.
(e) Perform an analysis of variance of the appropriate regression model for this design.
Include in your analysis hypothesis tests for each coefficient, as well as residual

An experiment involves a storage battery used in the launching mechanism of a shoulder-
fired ground-to-air missile. Two material types can be used to make the battery plates. The
objective is to design a battery that is relatively unaffected by the ambient temperature. The
output response from the battery is effective life in hours. Two temperature levels are
selected, and a factorial experiment with four replicates is run. The data are as follows:

21


Temperature (°F)
Material Low High
1 13 15 2 70
0 5 0
74 18 8 58
0 2
2 13 11 9 10
8 0 6 4

(a) Compute the estimates 16 16 8 60
of the effects and their 8 0 2
standard errors for this
design.
your findings.
analysis. State your final conclusions about the adequacy of the model. Compare your
results to part (c) and comment.

Exercise 5:

22


An article in the IEEE Transactions on Semiconductor Manufacturing (Vol. 5, 1992, pp.
214-222) describes an experiment to investigate the surface charge on a silicon wafer. The
factors thought to influence induced surface charge are cleaning method (spin rinse dry or
SRD and spin dry or SD and the position on the wafer where the charge was measured. The
surface charge ( X1011 q/cm3) response data are shown.

Test Position
L R
1.66 1.84
Cleaning SD 1.90 1.84
Method 1.92 1.62
-4.21 -7.58
SRD -1.35 -2.20
-2.08 -5.36

(a) Compute the estimates of the effects and their standard errors for this design.
your findings.
analysis. State your final conclusions about the adequacy of the model. Compare your
results to part (c) and comment.

23

10

Concept of Probability

00
Learning Outcome:

The students should be able to understand the basic concept of probability, sample space,
probability of events, counting rule; conditional probability; multiplication rule and Bayes
theorem

Exercise 1:
Each message in a digital communication system is classified as to whether it is received
within the time specified by the system design. If 3 messages are classified, what is an
appropriate sample space for this experiment?

Exercise 2:
A digital scale is used that provide weights to the nearest gram. Let event A: a weight
exceeds 11 grams, B: a weight is less than or equal to 15 grams, C: a weight is greater than or
equal to 8 grams and less than 12 grams.
What is the sample space for this experiment? and find

(a) A U B (b) A’ (c) A ∩ B

(d) (A U C)’ (e) A ∩ B ∩ C (f) B’ ∩ C

24


Exercise 3:
Samples of building materials from three suppliers are classified for conformance to air-
quality specifications. The results from 100 samples are summarized as follows:

Conforms
Yes No
R 30 10
Supplier S 22 8
T 25 5

Let A denote the event that a sample is from supplier R, and B denote the event that a sample
conforms to the specifications. If sample is selected at random, determine the following
probabilities:

(a) P(A) (b) P(B) (c) P(B’)
(d) P(AUB) (e) P(A ∩ B) (f) P(AUB’)
(g) P( A B) (h) P( B A)

25



The compact discs from a certain supplier are analyzed for scratch and shock resistance. The
results from 100 discs tested are summarized as follows:

Scratch
Resistance
High Low
High 30 10
Shock Medium 22 8
Resistance Low 25 5

Let A denote the event that a disc has high shock resistance, and B denote the event that a
disc has high scratch resistance. If sample is selected at random, determine the following
probabilities:

(a) P(A) (b) P(B) (c) P(B’)
(d) P(AUB) (e) P(A ∩ B) (f) P(AUB’)
(g) P( A B) (h) P( B A)

Exercise 5:
The reaction times ( in minutes) of a reactor for two batches are measured in an experiment.
a) Define the sample space of the experiment.
b) Define event A where the reaction time of the first batch is less than 45 minutes and event
B is the reaction time of the second batch is greater than 75 minutes.
c) Find A U B, A ∩ B and A’
d) Verify whether events A and B are mutually exclusive.

26


When a die is rolled and a coin is tossed, use a tree diagram to describe the set of possible
outcomes and find the probability that the die shows an odd number and the coin shows a
head.


A bag contains 3 black and 4 while balls. Two balls are drawn at random one at a time
without replacement.

(i) What is the probability that a second ball drawn is black?
(ii) What is the conditional probability that first ball drawn is black if the second ball is
known to be black?

Exercise 8:
An oil-prospecting firm plans to drill two exploratory wells. Past evidence is used to assess
the possible outcomes listed in the following table:

Event Description Probability

A Neither well produces oil or gas 0.80
B Exactly one well produces oil or gas 0.18
C Both wells produce oil or gas 0.02

Find and give description for

27


P ( A ∪ B ), P ( B ∪ C ) and P ( B' )

Exercise 9:
In a residential suburb, 60% of all households subscribe to the metro newspaper published in
a nearby city, 80% subscribe to the local paper, and 50% of all households subscribe to both
papers. Draw a Venn diagram for this problem.
If a household is selected at random, what is the probability that it subscribes to
a) at least one of the two newspapers
b) exactly one of the two newspapers

Exercise 10:
In a student organization election, we want to elect one president from five candidates, one
vice president from six candidates, and one secretary from three candidates. How many
possible outcomes?

Exercise 11:
Suppose each student is assigned a 5 digit number. How many different numbers can be
created?

Exercise 12:
A chemical engineer wishes to conduct an experiment to determine how these four factors
affect the quality of the coating. She is interested in comparing three charge levels, five

28


density levels, four temperature levels, and three speed levels. How many experimental
conditions are possible?

A menu has five appetizers, three soup, seven main course, six salad dressings and eight
desserts. In how many ways can
a) A full meal be chosen?
b) A meal be chosen if either and appetizer or a soup is ordered, but not both?

Exercise 14:
Ten teaching assistants are available to grade a test of four questions. Wish to select a
different assistant to grade each question (only one assistant per question). How many
possible ways can the assistant be chosen for grading?

Exercise 15:
Participant samples 8 products and is asked to pick the best, the second best, and the third
best. How many possible ways?

Exercise 16:
Suppose that in the taste test, each participant samples eight products and is asked to select
the three best products. What is the number of possible outcomes?

29


Exercise 17:
A contractor has 8 suppliers from which to purchase electrical supplies. He will select 3 of
these at random and ask each supplier to submit a project bid. In how many ways can the
selection of bidders be made?

Exercise 18:
Twenty players compete in a tournament. In how may ways can
a) rankings be assigned to the top five competitors?
b) the best five competitors be randomly chosen?

Exercise 19:
Three balls are selected at random without replacement from the jar below. Find the
probability that one ball is red and two are black.

Exercise 20:

30


A university warehouse has received shipment of 25 printers, of which 10 are laser printers
and 15 are inkjet models. If 6 of these 25 are selected at random by a technician, what is the
probability that exactly 3 of those selected are laser printers?

Exercise 21:
There are 17 broken light bulbs in a box of 100 light bulbs. A random sample of 3 light bulbs
is chosen without replacement.
a) How many ways are there to choose the sample?
b) How many samples contain no broken light bulbs?
c) What is the probability that the sample contains no broken light bulbs?
d) How many ways to choose a sample that contains exactly 1 broken light bulb?
e) What is the probability that the sample contains no more than 1 broken light bulb?

An agricultural research establishment grows vegetables and grades each one as either good
or bad for taste, good or bad for its size, and good or bad for its appearance. Overall, 78% of
the vegetables have a good taste. However, only 69% of the vegetables have both a good
taste and a good size. Also, 5% of the vegetables have a good taste and a good appearance,
but a bad size. Finally, 84% of the vegetables have either a good size or a good appearance.
a) if a vegetable has a good taste, what is the probability that it also has a good size?
b) if a vegetable has a bad size and a bad appearance, what is the probability that it has a
good taste?

31


Exercise 23:
A local library displays three types of books entitled “Science” (S),
“Arts” (A), and “Novels” (N). Reading habits of randomly selected
reader with respect to these types of books are

Read regularly S A N S∩A S∩N A∩N S∩A∩N
Probability 0.14 0.23 0.37 0.08 0.09 0.13 0.05

Find the following probabilities and interpret
a) P( S | A )
b) P( S | A U N )
c) P( S | reads at least one )
d) P( S U A | N)

A batch of 500 containers for frozen orange juice contains 5 that are defective. Two are
selected at random, without replacement, from the batch. Let A and B denote that the first
and second selected is defective respective
a) Are A and B independent events?
b) If the sampling were done with replacement, would A and B be independent?

32


Exercise 25:
Everyday (Mon to Fri) a batch of components sent by a first supplier arrives at certain
inspection facility. Two days a week, a batch also arrives from a second supplier. Eighty
percent of all batches from supplier 1 pass inspection, and 90% batches of supplier 2 pass
inspection. On a randomly selected day, what is the probability that two batches pass
inspection?

Exercise 26:
The probability is 1% that an electrical connector that is kept dry fails during the warranty
period of a portable computer. If the connector is ever wet, the probability of a failure during
the warranty period is 5%. If 90% of the connectors are kept dry and 10% are wet, what
proportion of connectors fail during the warranty period?

33


Exercise 27:
Computer keyboard failures are due to faulty electrical connects (12%) or mechanical defects
(88%). Mechanical defects are related to loose keys (27%) or improper assembly (73%).
Electrical connect defects are caused by defective wires (35%), improper connections (13%)
or poorly welded wires (52%). Find the probability that a failure is due to

i. loose keys
ii. improperly connected or poorly welded wires.

Exercise 28:
During a space shot, the primary computer system is backed up by two secondary systems.
They operate independently of one another, and each is 90% reliable. What is the probability
that all three systems will be operable at the time of the launch?

34


Exercise 29:
A store stocks light bulbs from three suppliers. Suppliers A, B, and C supply 10%, 20%, and
70% of the bulbs respectively. It has been determined that company A’s bulbs are 1%
defective while company B’s are 3% defective and company C’s are 4% defective. If a bulb
is selected at random and found to be defective, what is the probability that it came from
supplier B?

Exercise 30:
A particular city has three airports. Airport A handles 50% of all airline traffic, while airports
B and C handle 30% and 20%, respectively. The rates of losing a baggage in airport A, B and
C are 0.3, 0.15 and 0.14 respectively. If a passenger arrives in the city and losses a baggage,
what is the probability that the passenger arrives at airport A?

35


Exercise 31:
A company rated 75% of its employees as satisfactory and 25% unsatisfactory. Of the
satisfactory ones 80% had experience, of the unsatisfactory only 40%. If a person with
experience is hired, what is the probability that (s)he will be satisfactory?

Exercise 32:
In a certain assembly plant, three machines, B1, B2, B3, make 30%, 45% and 25%,
respectively, of the products. It is known from past experience that 2%,3% and 2% of the
products made by each machine, respectively, are defective. Now, suppose that a finished
product is randomly selected.

i. What is the probability that it is defective?

36


ii. If a product was chosen randomly and found to be defective, what is the probability
that it was produced by machine B3?

Three machines A, B and C produce identical items of their respective output 5%, 4% and
3% of the items are faulty. On a certain day A has produced 25%, B has produced 30% and
C has produced 45% of the total output. An item selected at random is found to be faulty.
What are the chances that it was produced by C?

Suppose that a test for Influenza A, H1N1 disease has a very high success rate: if a tested
patient has the disease, the test accurately reports this, a ’positive’, 99% of the time, and if a
tested patient does not have the disease, the test accurately reports that, a ’negative’, 95% of
the time. Suppose also, however, that only 0.1% of the population have that disease.

37


(i) What is the probability that the test returns a positive result?
(ii) If the patient has a positive, what is the probability that he has the disease?
(iii) What is the probability of a false positive?

Exercise 35:
An insurance company charges younger drivers a higher premium than it does older drivers
because younger drivers as a group tend to have more accidents. The company has 3 age
groups: Group A includes those less than 25 years old, have a 22% of all its policyholders.
Group B includes those 25-39 years old, have a 43% of all its policyholders, Group C
includes those 40 years old and older, have 35% of all its policyholders. Company records
show that in any given one-year period, 11% of its Group A policyholders have an accident.
The percentages for groups B and C are 3% and 2%, respectively.
(a) What is the probability that the company’s policyholders are expected to have an
accident during the next 12 months?
(b) Suppose Mr. Chong has just had a car accident. If he is one of the company’s
policyholders, what is the probability that he is under 25?

38

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