CATHOLIC UNIVERSITY OF RWANDA
FACULTY OF EDUCATION
Options: EMC-EMB-MCS
EXERCISES:PROBABILITY AND STATISTICS II / INFERENTIAL STATISTICS
April 12, 2025
Question 1
In the following multiple choice questions, circle the correct answer:
A. A numerical value used as a summary measure for a sample, such as sample mean, is known as a
a. population parameter
b. sample parameter
c. sample statistic
d. population mean
e. None of the above answers is correct.
B. Since the population size is always larger than the sample size, then the sample statistic
a. can never be larger than the population parameter
b. can never be equal to the population parameter
c. can never be zero
d. can never be smaller than the population parameter
e. None of the above answers is correct.
C. µ is an example of a
a. population parameter
b. sample statistic
c. population variance
d. mode
e. None of the above answers is correct.
The following data show the number of hours worked by 200 statistics students.
Data set A
Number of Hours
0-9
Frequency
40
10 - 19
50
20 - 29
30 - 39
70
40
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D. Refer to Data set A the class width for this distribution
a. is 9
b. is 10
c. is 11
d. varies from class to class
e. None of the above answers is correct.
E. Refer to Data set A, the number of students working 19 hours or less
a. is 40
b. is 50
c. is 90
d. can not be determined without the original data
e. None of the above answers is correct.
F. Refer to Data set A,the relative frequency of students working 9 hours or less
a. is 0.2
b. is 0.45
c. is 40
d. can not be determined from the information given
e. None of the above answers is correct.
G. Refer to Data set A The cumulative relative frequency for the class of 10 - 19
a. is 90
b. is 0.25
c. is 0.45
d. can not be determined from the information given
e. None of the above answers is correct.
Question 2(⋆)
A lecturer gives a science test to two classes and calculates the results as follows:
Class A - average mark 36%
Class B - average mark 40%
The lecturer reports to her Head of Department that the average mark over the two classes must be 38%.
The Head of Department disagrees, who is right? Justify your answer
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Question 3
Circle the correct answer:
(a) Is the variance measured in the same units as the mean?
(b) Is the mean measured in the same units as the median?
(i)YES
(i)YES
(c) Is the standard deviation measured in the same units as the mode?
(ii)NO
(ii)NO
(i)YES
(ii)NO
Question 4(⋆)
A medical survey was conducted in order to establish the proportion of the population which was infected
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with cancer. The results indicated that 40% of the population was suffering from the disease. A sample of
6 people was later taken and examined for the disease. Find the probability that the following outcomes
were observed
a) At most two people had the disease
b) At least two people had the disease
c) Three or four people had the disease
Question 5
Explain two types of errors that can be made when you make your decision about whether or not to
reject a null hypothesis
Question 6(⋆)
What is meant by the “Power of a test”?
Question 7
A sample of size 16 is taken from the distribution of X ∼ N(µ, 32) and a hypothesis test is carried out
at 1% level of significance. On the basis of the value of the sample mean x the null hypothesis µ = 100
is rejected in favor of the alternative hypothesis µ > 100 . What can be said about x ?
Question 8(⋆)
One measure of personal fitness is the time taken for an individual’s pulse rate to return to normal after
strenuous exercise; the greater the fitness, the shorter the time. Following a short program of strenuous
exercise recorded his pulse rates P at time t minutes after he had stopped exercising. Norman’s results
are given in the table below:
t
0.5
1
1.5
2
3
4
5
p 125 113 102 94 81 83 71
i. Draw a scatter diagram to present this information
ii. Find the equation of regression line of P on t for Norman’s data
iii. Use the above information to estimate Norman’s rate 2.5 minutes after stopping the exercises program.
Question 9
Each year trainees throughout the country sit a test. Over a period of time, it has been established that
the marks can be modeled by a normal distribution with the mean 70 and standard deviation 6. This
year, it was thought that trainees from a particular country did not perform as well as expected. The
marks of a random sample of 25 trainees from the country were scrutinized and it was found that their
mean mark was 67.3. Does this provide evidence, at the 5% significance level, that trainees from this
country did not perform well as expected?
Question 10
The random variable X follows a normal distribution with mean 1000 and variance 100. When X takes
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values between 1005 and 1010, between which values does the standardised normal variable Z lie?
Question 11
Given that the variable X follows a the normal distributionX ∼ N(151, 152), calculate:
(a) P(120 ≤ x ≤ 155)
(b) P(x ≥ 185)
Question 12(⋆)
An electrical firm manufactures bulbs that have lifetimes with mean µ = 800, σ = 40 hours. Find the
probability that a random sample of size 41 will have an average life of less than 795 hours.
Question 13
Consider an experiment of tossing a coin 5 times and on each toss observing whether the coin lands with
a head or a tail on its upward face. Suppose we are interested in the counting of the number of heads
appearing during the 5 tosses. Does this experiment have the properties of the binomial distribution?
Question 14
Dishwasher powder is poured into the cartons in which it is sold by an automatic dispensing machine
which is set to dispense 3 kg of powder into each carton. In order to check that the dispensing machine is
working to an acceptable standard (i.e. does not need adjustment), a production engineer takes a random
samples of 40 cartons and weighs them. It is found that the mean weight of the sample is 3.005 kg. It
is known that the dispensing machine operates with a variance of 0.0152kg2 and that the manufacturer
of the powder is willing to rely on a 5% level of significance. Does the sample provide the engineer with
sufficient evidence that the machine is operating acceptably and so does not need adjustment?
(Answer: We must reject the null hypothesis and conclude that the machine is not operating acceptably
and needs adjustment.)
Question 15
A manufacturer of electronic equipment has developed a circuit to feed current to a particular component
in a computer display screen. While the new design is cheaper to manufacture, it can only be adopted
for mass production if it passes the same average current to the component. In tests involving the two
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circuits, the following results are obtained.
T estNbr
1
Circuit1 − Current(mA)
80.1
Circuit2 − Current(mA)
80.7
2
82.3
81.3
3
84.1
84.6
4
82.6
81.7
5
85.3
86.3
6
81.3
84.3
7
83.2
83.7
8
81.7
84.7
9
82.2
82.8
10
81.4
84.4
11
12
85.2
84.9
On the assumption that the populations from which the samples are drawn have equal variances, should
the manufacturer replace the old circuit design by the new one? Use the 5% level of significance.
(Answer: We conclude that we cannot reject the null hypothesis in favour of the alternative.)
Question 16(⋆)
In an experiment to determine the most advantageous position in a machine to mount an electronic
component which may be prone to failure due to excessive heat build-up, 300 machines are tested with
100 randomly chosen examples of the component in each of 3 positions. The results obtained were as
follows.
Position
1
2
3
Failure
40
30
50
120
Non − failure
60
70
50
180
100
100
100
300
Column
T otals
Row
T otals
Use a χ2-test at the 5% level of significance to determine whether component failure is related to mounting position.
Question 17
A factory runs three machines producing compression joints over a period of three daily shifts. In order
to check whether there is any variability in the joints produced a random sample of joints produced is
checked and the failures recorded. The results are given in the table below.
Machines
Shift
1
2
3
1
40
28
34
102
2
27
39
32
98
3
45
26
29
100
112
93
95
300
Column
T otals
5
Row
T otals
Use a χ2-test at the 5% level of significance to determine whether joint failure is related to the machine
and the shift during which they were produced.
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