UNIVERSITY OF TECHNOLOGY, JAMAICA
FACULTY OF SCIENCE AND SPORT
SCHOOL OF MATHEMATICS AND STATISTICS
FINAL EXAMINATION, SEMESTER 2
________________________________________________________________________
Module Name : Probability and Statistics
Module Code : STA2002
Date : March 2010
Theory/ Practical : Theory
Group : BENG 2 (Chemical)
Duration : 2 Hours
________________________________________________________________________
INSTRUCTIONS
1. Answer ANY FOUR(4) questions.
2. This examination paper consists of SIX (6) questions and three (3)
printed pages.
3. Please begin the answer for each question on a new page.
3. Mathematical instruments and silent electronic calculators may be used
for this paper.
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
1
QUESTION NO.1
(a) A large group of people is to be tested for two compounds in their urine. It is
thought that 20% of the people possess compound A alone, 30% possess
compound B alone, 10% possess both compounds, and the remanding have
neither compound. For one person chosen at random from this group, find these
probabilities:
(i) that the person has at least one compound.
(ii) that the person has neither compound.
(iii) that the person has both compounds, given that he has compound B.
[2+1+3marks]
(b) If A and B are independent events, show that A and B are also independent.
[3marks]
(c) At a machine center there are three automatic screw machines. Machine A, B, C
manufacture 35%, 40%, and 25% of the total output respectively. Of their outputs
4%, 2%, and 5% respectively, are defective screws.
(i) If a screw is randomly picked from inventory, what is the probability that
it will be defective?
(ii) If a screw is picked and found to be defective, what is the probability that
it was produced by machine C?
[4+2 marks]
QUESTION NO.2
(a) The continuous random variable X, has probability density function
kxe3 x 0 x
f ( x)
0 elsewhere
(i) Find the value of k.
(ii) Find E(X). [3+4 marks]
(b) The discrete random variable X is binomially distributed with p.d.f given by
n
P( X x) p x (1 p) n x for x = 0, 1, 2, … , n.
x
Show that
(i) E(X) = np,
(ii) V(X) = np( 1 – p ) [3+ 5 marks]
GO ON TO NEXT PAGE
2
QUESTION NO.3
(a) A study investigating the marks (x) obtained in a Semester 1 Chemistry
examination and the marks (y) obtained in the following Semester 2 examination
by a group of nine students was carried out by a university. It is given that
x 567, y 552, xy 36261, x 2
37777, y 2 36112.
(i) Find, by calculation, the equation of the regression line of Y on X.
(ii) Tenth student obtained a mark of 60 in the Semester1 examination but was
absent from the Semester 2 examination. Estimate the mark that this
student would have obtained in the Semester 2 examination.
(iii) Find, the coefficient of determination, r2, and interpret your answer.
[3+1+3marks]
(b) Calculate the 95% confidence interval of the mean for nitrate ion concentration
given that the sample mean is 0.60 gml-1 , the standard deviation 0.0175 gml-1 and
the sample size is 50. [3 marks]
(c) An advertising consulting states that 25% of the compound produced by a
machine are impure. In order to test this assertion, a random sample of 150
compounds were taken and 34 were found to be impure. Test whether these
results provide significant evidence, at the 5% level, that the consultant was
incorrect. State clearly your null and alternative hypotheses.
[5marks]
QUESTION NO.4
(a) Find the cumulative distribution function for
x x2
2
e 2t , t 0, x 0
f ( x) t 2 .
0,
otherwise
[5 marks]
(b) A continuous random variable X having probability density function
( x )2
1
f ( x) e 2 for x
2
2
is normally distributed with mean and variance 2 .
Show that
(i)
f ( x) dx 1
’
(ii ) E ( X ) .
[4+6 marks]
GO ON TO NEXT PAGE
3
QUESTION NO.5
(a) In a comparison of two methods for the determination of potassium in rye gas,
the following results (mgkg-1 K) were obtained:
Method 1: mean = 1.48; standard deviation 0.28
Method 2: mean = 2.33; standard deviation 0.31
For each method five determinations were made.
Do these methods give results having means which differ significantly at the 0 05
level of of significance?
[6 marks]
(b) Write down a multiple regression equation with two independent variables and
explain the meaning of all the terms in the equation.
[5 marks]
(c) The quality control engineer at Bethel Steel is interested in estimating the tensile
strength of steel wire based on its outside diameter and the amount of iron in
the steel . As an experiment , she selected thirty-five pieces of wire , measured the
outside diameters ,and determined the iron content . Then she measured the
tensile strength of each piece . The results of the first four were :
Pieces Tensile Outside Amount
Strength(psi) Diameter(cm), Iron(units)
Y X1 X2
A 11 .03 6
B 9 .02 5
C 16 .04 8
D 12 .03 7
Suppose the multiple regression equation is Y1 = -0.5 + 20X1 + 1X2 .
(i) Based on the equation , what is the predicted tensile strength of a steel wire
having an outside diameter of 0.35 cm and 6.4 units of iron ?
(ii) Interpret the value of b1 in the equation . [3+1marks]
QUESTION NO.6
(a) The random variable X is such that X ~ Po( ). Show that the moment generating
function of X is e ( e 1) . Hence, obtain the mean and variance of X.
t
[8 marks]
(b) The random variable Y is independent of X, and Y~ Po( ) . Given that T = X+Y,
Find the moment generating function of T. Hence identify the distribution of T.
[4 marks]
(c) A chemist dissolves sodium ions and nitrate ions in a volume of water. The
average concentration of the sodium ions is 0.20 per milliliter(ml) and that of the
nitrate ions is 0.05 per milliliter(ml). The random variables X and Y denote the
concentration of sodium ions and nitrate ions, respectively, contained in a
randomly selected 1 millimeter (1ml) volume of water . Assume that X and Y
follow the Poisson distribution. Determine, correct to three decimal places,
P(X +Y = 0).
[3 marks]
END OF PAPER
4
USEFUL FORMULAE
The product moment coefficient of correlation
n XY X Y
r
n X 2 X 2 n Y 2 Y 2
Least squares regression line, Y a bX ,
n XY X Y Y b X
b a n
n X 2 X
2
n
Confidence interval
s
xz
n
Test statistic
p p
Z cal
p(1 p)
n
(n1 1)( s1 ) (n2 1)( s 2 )
2 2
sp
2
Pooled variance
n1 n2 2
X1 X 2
Two- Sample Test of Means t
2 1 1
s p
n1 n2
13. PROBABILITY
(a) P( AUB) P( A) P( B) P( A B) (Non - mutually exclusive events)
(b) P( AUB) P( A) P( B) (Mutually exclusive events)
(c ) P( AUBUC) P( A) P( B) P(C ) P( A B) P( B C ) P( A C ) P( A B C )
P( A B)
(d) P( A / B) ( Conditional events)
P( B)
P( A B)
(e) P( B / A)
P( A)
(f) P( A and B) P( A) P( B) (Independent events)
(g) P( A) 1 P( A)
Bayes’ Theorem
P( A / Bi ) p( Bi )
(h) P( Bi / A)
P( A / B1 ) P( B1 ) P( A / B2 ) P( B2 ) P( A / Bk ) P( Bk )
