Download math 1111c-probability and statistics

BSc (Hons) Bachelor in Information Systems
Cohort: BIS/11/FT
Examinations for 2012 - 2013 / Semester I
MODULE: PROBABILITY AND STATISTICS
MODULE CODE: MATH 1111C
Duration: 2 Hours 30 Mins
Instructions to Candidates:
1. Answer all questions.
2. Questions may be answered in any order but your answers must show
the question number clearly.
3. Always start a new question on a fresh page.
4. The standard normal and chi square tables are attached.
5. Graph paper will be provided.
6. All questions carry equal marks.
7. Total marks 100.
8. In case of inexact answers, give your answers correct to 3 significant
figures.
This question paper contains 5 questions and 5 pages.
Page 1 of 5
Probability & Statistics (MATH 1111C)
ANSWER ALL QUESTIONS
QUESTION 1: (20 MARKS)
(a) (i) By using a Venn diagram, prove the addition formula of probability for two
events A and B and hence find P(A∩B) given P(AUB) = 0.68, P(A) = 0.6 and
P(B) = 0.2.
(2 + 2 marks)
(ii) What type of events are A and B? Justify your answer.
(1 mark)
(b) There are three sets of traffic lights on Kamal’s journey to work. The
independent probabilities that Kamal has to stop at the first, second and third set
of lights are 0.4, 0.8 and 0.3 respectively. By drawing a probability tree diagram,
(i) Find the probability that Kamal has to stop at each of the first two sets of light
but does not have to stop at the third set.
(2 marks)
(ii) Find the probability that Kamal has to stop at exactly two of the three sets of
lights.
(2 marks)
(iii) Find the probability that Kamal has to stop at the first set of lights, given that
he has to stop at exactly two sets of lights.
(3 marks)
(c) A discrete random variable X has its probability mass function given as
follows:
p(x) =
k ( x + 1 )2, x = 0, 1, 2;
0, otherwise.
(i) Find the value of k.
(3 marks)
(ii) Draw the probability distribution table of X.
(2 marks)
(iii) Find E(X) and Var(X).
(1 + 2 marks)
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Page 2 of 5
Probability & Statistics (MATH 1111C)
QUESTION 2: (20 MARKS)
(a) Major avalanches can be rejected as randomly occurring events. They occur
at a uniform average rate of 8 per year.
(i) Find the probability that more than 3 major avalanches occur in a 3-month
period.
(3 marks)
(ii) Independent of the occurrence of major avalanches, earthquakes occur at an
average rate of 1 per year. Find the probability that exactly 2 major avalanches
and 1 earthquake occur in a 4-month period.
(5 marks)
(b) A continuous random variable X has probability density function given by
f (x) = 1 − ,
0,
0 ≤ x ≤ 2;
otherwise.
(i) Find P ( X > 1.5).
(3 marks)
(ii) Find the mean of X.
(4 marks)
(iii) Find the median of X.
(5 marks)
QUESTION 3: (20 MARKS)
(a) From experience, a cactus grower knows that on average only 40% of cactus
seeds germinate. A cactus seed collector returns from a very dry desert with six
seeds of a previously unknown type of cactus.
(i) Determine the probability that only 1 seed germinates.
(3 marks)
(ii) Determine the probability that at least 2 seeds germinate.
(4 marks)
(iii) Determine the most likely number of germinating seeds.
(4 marks)
(b) The random variable X is normally distributed, with mean and standard
deviation each equal to a. It is given that P(X < 3) = 0.25.
(i) Find the value of a.
(4 marks)
(ii) Hence find P(X < 12).
(5 marks)
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Page 3 of 5
Probability & Statistics (MATH 1111C)
QUESTION 4: (20 MARKS)
(a) When a car is driven under specified conditions of load, tyre pressure and
surrounding temperature, the temperature, y C, generated in the shoulder of the
tyre varies with the speed, x km h-1, according to the linear model y = a + bx,
where a and b are constants. Measurements of y were made at eight different
values of x with the following results:
x
20
30
40
50
60
70
80
90
y
45
52
64
66
91
86
98
104
(Given Σx2 = 28 400, Σy2 =49 278)
(i) Show these data on a scatter diagram.
(2 marks)
(ii) Calculate Σx, Σy and Σxy.
(2 marks)
(iii) Calculate the equation of the regression line of y on x.
(4 marks)
(iv) Estimate the value of y when x = 60.
(1 mark)
(v) Find the product moment correlation coefficient r, between x and y and
comment on its value.
(4 + 1 marks)
(b) The ‘reading age’ of children about to start secondary school is a measure of
how good they are at reading and understanding printed text. A child’s reading
age, measured in years, is denoted by the random variable X, and the mean and
variance of X are denoted by μ and σ2 respectively. The reading ages of a
random sample of 80 children were measured, and the data obtained is
summarized by Σx = 930.4, Σx2 = 11 024.88.
(i) Calculate unbiased estimates of μ and σ2, giving your answers correct to 2
decimal places.
(3 marks)
(ii) Calculate a symmetric 90% confidence interval for μ.
(3 marks)
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Page 4 of 5
Probability & Statistics (MATH 1111C)
QUESTION 5: (20 MARKS)
(a) A random sample of 36 observations is to be taken from a normal distribution
with variance 100. In the past the distribution had a mean of 83.0, but it is
claimed that recently the mean has changed. When the sample is actually taken,
it is found to have a mean of 86.2. Using a 5% level of significance, test the claim
that the mean has changed.
(10 marks)
(b) An ordinary die is tossed 60 times and the following outcomes with their
respective observed frequencies are given below:
Outcome
1
2
3
4
5
6
4
7
16
8
8
17
Observed
frequencies
Test whether the die is fair.
(10 marks)
***END OF QUESTION PAPER***
Page 5 of 5
Probability & Statistics (MATH 1111C)