Download MTE-11

MTE-11
Assignment Booklet
Bachelor’s Degree Programme (B. Sc.)
Probability and Statistics
(01 July, 2008 –30 June, 2009)
School of Sciences
Indira Gandhi National Open University
New Delhi
Dear Student,
Please read the section on assignment in the Programme Guide for elective Courses that we sent you after
your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous
evaluation, which would consist of one tutor-marked assignment for this course. This assignment is in
this booklet.
Instructions for Formating Your Assignments
Before attempting the assignment please read the following instructions carefully.
1) On top of the first page of your answer sheet, please write the details exactly in the following format:
ROLL NO.:……………………………………………
NAME :……………………………………………
ADDRESS :……………………………………………
……………………………………………
……………………………………………
COURSE CODE:
…………………………….
COURSE TITLE :
…………………………….
ASSIGNMENT NO. ………………………….…
STUDY CENTRE:
………………………..…..
DATE.………………………….……….
PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND
TO AVOID DELAY.
2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.
3) Leave 4 cm margin on the left, top and bottom of your answer sheet.
4) Your answers should be precise.
5) While solving problems, clearly indicate which part of which question is being solved.
6) This assignment is valid only upto 30 June, 2009. If you have failed in this assignment or fail to
submit it by 30 June, 2009, then you need to get the assignment for the year 2009 – 2010 and submit
it as per the instructions given in the programme guide.
We strongly suggest that you retain a copy of your answer sheets.
Wish you good luck.
2
TUTOR MARKED ASSIGNMENT
(To be done after studying all the four Blocks)
Course Code: MTE-11
Assignment Code: MTE-11/TMA/2008-09
Maximum Marks: 100
Q.1 (a)
The following table shows the number of children, out of a sample of 500, with the heights in
the given range:
Height in cm.
105-108
108-111
111-114
114-117
117-120
120-123
123-126
126-129
129-132
132-135
Number of
Children
50
35
82
105
66
48
24
16
48
26
Draw a cumulative frequency ogive curve to represent the given data. From your
curve estimate
(i) the median height,
(ii) the upper and lower quartile heights,
(iii) the number of children in this sample with height 120 cm or more.
(b)
The weight in grams of 50 apples picked out at random from a consignment is as
follows:
106, 107, 76, 82, 109, 107, 115, 93, 187, 95,
123, 125, 111, 92, 86, 70, 126, 68, 130, 129,
139, 119, 115, 128, 100, 186, 84, 99, 113, 204,
111, 141, 136, 123, 90, 115, 98, 110, 78, 90,
107, 81, 131, 75, 84, 104, 110, 80, 118, 82,
(i)
Form the grouped frequency table by dividing the variate range into intervals of equal
width, each corresponding to 20 grams in such a way that the mid-value of first class
corresponds to 70 grams.
(ii)
Find the mean, median, standard deviation and mean deviation of the weights of
the apples.
(iii) Also, find the proportion of the apples with weight more than 150 grams.
Q.2 (a)
(5)
(10)
Calculate the coefficient of correlation from the following data:
X: 1
Y: 9
2
8
3
10
4
12
5
11
6
13
7
14
8
16
9
15
Also write down the equations of the lines of regression and obtain an estimate of Y which
corresponds the X = 6.2.
(7)
3
(b) The face cards are removed from a full pack of cards. Out of the remaining cards, 4 cards are
drawn at random. What is the probability?
(i) that they belong to different suits? and
(ii) that the 4 cards drawn belong to different denominations?
(c)
(d)
Q.3 (a)
(b)
(3)
A Company has two plants to manufacture an item. Plant I manufactures 70% of the items
and Plant II manufactures 30%. At Plant I, 80% of the items are rated standard quality and at
Plant II, 90% of the items are rated standard quality. An item is picked up at random and is
found to be of standard quality. What is the probability that it has come from Plant I?
(3)
A sample space has five outcomes S   e1 , e 2 , e 3 , e 4 , e 5  with the probabilities of
elementary events
2
1
1
1
1
P(e1 )  , P(e 2 )  , P(e 3 )  , P(e 4 )  , P(e 5 )  .
6
12
6
3
12
Let A = {e3, e4} and B= {e1, e4}, then show that A and B are statistically independent.
(2)
Using the distribution found in Q.1 (b), compute
(i) moment coefficient of skewness.
(ii) moment coefficient of kurtosis.
A discrete random variable has the probability distribution
X
:
0
1
2
3
4
P (X = x) :
a
3a
5a
7a
9a
(6)
5
11a
6
13a
7
16a
Determine
(i) a
(ii) P (X  5 )
(iii) The value of X such that P (X  x )  0.7 .
(c)
Q.4 (a)
(6)
Let X 1 and X 2 are independent Poisson variates with means 1 and 2 respectively, find
P (X1  X 2  4).
(3)
The marks of a class are normally distributed with means = 200 and s. d. =10.
(i) Find the number of students in a class of 100 getting marks between 200 and 220,
between 190 and 215, above 230, below 180.
(ii) Assuming that there are only 84 seats in the next class, calculate the minimum number of
marks a candidate should secure in order to be eligible for promotion.
(6)
(b)
The joint probability density function of the two-dimensional random variable (X, Y)
is given by :
8
 x y , 1 x  y  2
f ( x , y)   9

, elsewhere
0
(i) Find the marginal density functions of X and Y,
(ii) Find the conditional density functions of Y given X = x, and conditional density function
of X given Y = y.
(4)
4
(c)
If X is the number scored in a throw of an unbiased die, show that the chebychev’s inequality
gives P [| X   | > 2.5] < 0.47, where  is the mean of X, while the actual probability is
zero.
(5)
Q.5 (a)
A six faced die is thrown and the expectation that in 10 throws it will give five
even numbers is twice the expectation that it will give four even numbers. How many times
in 10, 000 sets of 10 throw each, would you expect it to give no even number?
(5)
(b)
A genetical studies says that children having one parent of blood group M and the other
parent of blood group N will always be one of the three blood groups M, MN, N, ; and that
the average number of children in these groups will be in the ratio 1 : 2 : 1. The reports on an
experiment states as follows : “Of 162 children having one M parent, and one N parent, 28 %
were found to be of group M, 42% of MN and the rest of the group N”. Test the hypothesis
that the data in the report confirms to the expected genetic ratio 1: 2: 1 at the 5% level of
significance?
(5)
(c)
The heights of 10 males of a given locality are found to be 70, 67, 62, 68, 61, 68, 70, 64, 64,
66 inches. Is it reasonable to believe that the average height is greater than 64 inches? Test at
5% significance level, assuming that for 9 degree of freedom P (t >1.83) = 0.05.
(5)
Q.6 (a)
X 1 , X 2 , and X 3 is a random sample of size 3 from a population with mean value  and
variance  2 . T1 , T2 , T3 are the estimator’s used to estimate mean value  , where
T1  X1  X 2  X 3 , T2  2X1  3X 3  4X 2 , and T3  (X1  X 2  X 3 ) / 3
(i)
(ii)
(iii)
(iv)
(b)
Are T1 and T2 unbiased estimators?
Find the value of  such that T3 is unbiased estimator for  .
With this value of  is T3 a consistent estimator?
Which is the best estimator?
Given the probability density function
f(x : )  [{1  (x  ) 2 }] 1;    x  ,      
show that the Cramer-Rao lower bound of variance of an unbiased estimator of  is
where n is the size of the random sample from this distribution.
(c)
(5)
2
,
n
(5)
Prove that the maximum likelihood estimate of the parameter  of a population having
density function:
2
(  x ), 0  x  
2
for a sample of unit size is 2x, where x is the sample value. Also, show that the estimate is
biased.
(5)
Q.7 State whether the following statements are true or false? Give reasons for your answers.
(i)
The regression coefficient of X on Y is 3.2 and that of Y on X is 0.8.
(ii)
A certain dice was thrown 600 times and a 3 or 4 was obtained 205 times. On the assumption
of random throwing, this data indicate an unbiased die.
(iii) For a Poisson distribution with parameter , 1 is consistent estimator of 1  , where X is
X
the mean of a random sample for the given population.
5
(iv)
From the population consisting of 5 items, the total number of possible samples of size 2
when the sampling is without replacement, when ordering is important and when it is ignored
are 60 and 10, respectively.
(v)
If the coefficients of variation of two series are 75% and 90%, with S. D. 15 and 18
respectively, then both the series have same mean.
(10)
6