Numerical Method and Statistic
AQ100-3-3
Chapter 4
TUTORIAL
Probability
1.
Suppose that P(A) = 0.35, P(B) = 0.6 and P(A ∩ B) = 0.27. Find P ( A  B ) .
2.
Suppose that the probability that a construction company will be awarded a certain
contract is 0.25, the probability that it will be awarded second contract is 0.21 and the
probability that it will get both contracts is 0.13. Find the probability that the company
will win at least one of the two contracts.
3.
If P(A|B) =
(a) P(B|A)
2
5
,
1
P(B) = 4 ,
1
P(A) = 3 , find
(b) P(A ∩ B)
4.
A and B are two independent events such that P(A) = 0.2 and P(B) = 0.15.
Evaluate the following probabilities
(a) P(A|B)
(b) P(A ∩ B),
(c) P(A U B)
8.
A ball is drawn at random from a box containing 6 red balls, 4 white balls and 5 blue balls.
Determine the probability that it is (a) red, (b) white, (c) blue, (d) not red, (e) white or
red.
9.
A fair coin is tossed 6 times in succession. What is the probability that at least 1 head
occurs?
10.
In a lot of 1200 of golf balls, 40 have imperfect covers, 32 cannot bounce, and 12 have
both defects. If a ball is selected at random from the lot, what is the probability that the
ball is defective?
11.
The probability that a student passes mathematics is
12.
Two cards are drawn from a well-shuffled ordinary deck of 52 cards. Find the probability
that they are both aces if the first card is (a) replaced, (b) not replaced.
13.
Three balls are drawn successively from the box of Problem 8. Find the probability that
they are drawn in the order red, white and blue if each ball is (a) replaced, (b) not replaced.
2
, and the probability that he passes
3
4
4
English is . If the probability of passing at least one course is , what is the probability
9
5
that he will pass both courses?
1
Numerical Method and Statistic
AQ100-3-3
Chapter 4
15.
One bag contains 4 white balls and 2 black balls; another contains 3 white balls and 5 black
balls. If one ball is drawn from each bag, find the probability that (a) both are white, (b)
both are black, (c) one is white and one is black.
17.
A and B play 12 games of chess of which 6 are won by A, 4 are won by B, and 2 end in a
tie. They agree to play a tournament consisting of 3 games. Find the probability that (a)
A wins all three games, (b) two games end in a tie, (c) A and B win alternatively, (d) B
wins at least one game.
18.
One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls
and 5 black balls. One ball is drawn from the 1st bag and is placed unseen in the second
bag. What is the probability that a ball now drawn from the 2nd bag is black?
A statistics professor classifies his students according to their grade point average (GPA)
and their gender. The accompanying table gives the proportion of students falling into the
various categories.
Gender
Grade Point Average
Under 2.0
2-.0 – 3.0
Over 3.0
Male
0.05
0.25
0.10
Female
0.10
0.30
0.20
What is the probability that when one student is selected at random, the student is a male
given that the GPA of the student selected is over 3.0?
18.
The table below gives the breakdown of students enrolled in the School of Computing
at Staffordshire University.
Software engineering Computer system
Male
15
20
Female
25
30
What is the probability that when one student is selected at random, the student is a female
given that the student is a software engineering major?
19.
Students in a certain town are surveyed to see how many students are unemployed. The
results of the survey are in the table below.
Employed
Unemployed
Total
Males
0.400
0.100
0.50
Females
0.475
0.025
0.50
Total
0.875
0.125
1.00
What is the probability that a randomly selected student is a male given that he is
unemployed?
20.
A manufacturing process requires the use of a robotic welder on each of two assembly
lines. Assembly line A produces 200 units of product per day and the assembly line B
produces 400 units per day. Over a period of time, it has been found that the welder on A
produces 2% defective units, whereas the welder on B produces 5% defective units. If a
unit is selected at random from the total production,
(i)
Represent the above information in a tree diagram.
2
Numerical Method and Statistic
(ii)
(iii)
21.
AQ100-3-3
Chapter 4
What is the probability that the unit has defective welding?
If the unit is found to be defective, what is the probability that the unit came from
assembly line B?
A vendor found out that three companies A, B and C produce machine X. All the companies
equally produce the machines. The percentage of defective machines produced by
company A is 2% and the corresponding percentages for company B and C are 5% and
4.5% respectively.
Represent the above information in a tree diagram.
Hypothesis Testing
1.
A representative of a community group informs the prospective developer of a shopping
center that the average income per household in the area is $25,000. Suppose that for the
type of area involved household income can be assumed to be approximately normally
distributed and that the standard deviation can be accepted as being equal to 2,000, based
on an earlier study. For a random sample of n = 15 household, the mean household income
is found to be x = $24,000.
(a)
Test the null hypothesis that µ = $25,000 by establishing critical limits of the
sample mean in terms of dollars, using the 5 percent level of significance.
(b)
Test the hypothesis by using the standard normal variable z as the test statistic.
2.
The standard deviation of the tube life for a particular brand of ultraviolet tube is known to
be 500 hr, and the operating life of the tubes is normally distributed. The manufacturer
claims that average tube life is at least 9,000 hr. Test this claim at the 5 percent level of
significance by designating it as the null hypothesis and given that for a sample of n = 15
tubes the mean operating life was x = 8,800 hr.
3.
For a sample of 50 firms taken from a particular industry the mean number of employees
per firm is 420.4 with a sample standard deviation of 55.7. There are a total of 380 firms
in this industry. Before the data were collected, it was hypothesized that the mean number
of employees per firm in this industry does not exceed 408 employees. Test this hypothesis
at the 5 percent level of significance.
4.
As a commercial buyer for a private supermarket brand, suppose that a random sample of
12 No. 303 cans of string beans at a canning plant. The average weight of the drained beans
in each can is found to be x = 15.97 g, with s = 0.15. The claimed minimum average net
weight of the drained beans per can is 16.0 g. Can this claim be rejected at the 10 percent
level of significance?
5.
An automatic soft ice cream dispenser has been set to dispense 4.00 g per serving. For a
sample of n = 10 servings, the average amount of ice cream is x = 4.05 g with standard
deviation = 0.10 g. The amount being dispensed are assumed to be normally distributed.
3
Numerical Method and Statistic
AQ100-3-3
Chapter 4
Basing the null hypothesis on the assumption that the process is ‘in control’, should the
dispenser be reset as a result of a test at the 10 percent level of significance?
6.
The table below shows the number of employees absent for a single day during a particular
period of time.
Day
Number of absentees
Monday
121
Tuesday
87
Wednesday
87
Thursday
91
Friday
114
Total
500
(a)
(b)
7.
Find the frequencies expected under the hypothesis that the number of absentees is
independent of the day of the week.
Test at the 5% level whether the difference in the observed and expected data are
significant.
300 employees of a company were selected at random and asked whether they were in
favour of a scheme to introduce flexible working hours. The following table shows the
opinion and the departments of the employees.
Department
Production
Sales
Administration
OPINION
In favour
89
53
38
Uncertain
42
36
12
Against
9
11
10
Test whether there is evidence of a significant association between opinion and department
at 5% significance level.
8.
A survey was carried out in a firm of the smoking habits of men and women employees
with the following results:
Men
Women
Smokers
48
27
Non-smokers
58
57
It is required to test whether, at the 5% level of significance, the survey reveals any
difference in the smoking habits of men and women.
9.
A random sample of employees of a large company was selected and the employees were
asked to complete a questionnaire. One question asked whether the employees were in
favour of the introduction of flexible working hours. The following table classifies the
employees by their response and gender i.e. male or female.
RESPONSE
Gender
Male
Female
4
Numerical Method and Statistic
AQ100-3-3
In favour
Not in favour
Chapter 4
57
33
83
27
Test whether there is evidence of a significant association between the response and
gender.
Correlation and Regression
1.
Suppose that a random sample of five families had the following annual income and
savings.
Income (X)
(£’000)
8
11
9
6
5
(a)
(b)
(c)
2.
Savings (Y)
(£’000)
0.6
1.3
1.0
0.7
0.3
Plot this data on an appropriate labelled scatter diagram.
Obtain the least square regression equation of savings (Y) on income (X) and
plot the regression line on a graph.
Estimate the savings if the family income is £ 7000.
As part of an investigation into levels of overtime working, a company decides to tabulate
the number of orders received weekly and compare this with the total weekly overtime
worked to give the following:
Week number
Orders received
Total overtime
1
83
38
2
22
9
3
107
42
4
55
18
5
48
11
6
92
30
7
135
48
8
32
10
9
67
29
10
122
51
Use the method of least squares to obtain a regression line that will predict the level of total
overtime necessary for 100 orders
3.
The following data show weekly prices and also sales of a mail order product over a twomonth period.
Price (£)
8.99
9.50
9.99
10.50
10.99
Sales
496
465
482
459
408
5
Numerical Method and Statistic
11.50
11.99
12.50
12.99
(a)
(b)
(c)
(d)
4.
Chapter 4
382
315
363
309
Plot this data on an appropriate labelled scatter diagram.
Calculate the product moment correlation coefficient.
Obtain the least square regression equation and plot the regression line on your
graph.
Comment on your results.
(a)
Draw a scatter diagram of about 10 points to illustrate the following degree of
linear correlation.
(i)
no correlation
(ii)
weak positive correlation
(iii)
Perfect positive correlation
(iv)
moderately strong negative correlation.
(b)
The data below shows the appraised value and area of home for a sample of
seven homes.
Area (x)
(‘000 square feet )
1.8
1.6
2.5
2.0
1.2
1.5
2.4
(i)
(ii)
5.
AQ100-3-3
value (y)
($’000)
100
96
151
121
83
94
140
Calculate the product moment correlation coefficient between the value
and the area of home.
Calculate the value of coefficient of determination and interpret the value
obtained here.
Two supervisors, Mr A and Mr B are considering the performance of individual employees
according to their opinion of their abilities.
(i)
Find the coefficient of rank correlation for the following Employees Rankings of
10 employees by the two supervisors.
Employees
A
B
C
Ranking by A
2
1
3
Ranking by B
3
2
1
6
Numerical Method and Statistic
D
E
F
G
H
I
J
(ii)
6.
AQ100-3-3
Chapter 4
4
6
5
8
7
10
9
4
6
7
5
9
10
8
Explain what this coefficient show.
A random sample of recent repairs was selected, and the estimated time required for the
repair and the actual time taken were recorded.
Estimated time(mins)
Actual time(mins)
30 90
22 64
15
36
75
61
60
93
40 20
45 33
80
65
45
50
120
90
Calculate
(a)
Spearman’s rank correlation coefficient
(b)
The product moment correlation coefficient
(c)
Explain why the two answers differ.
7