Full Text Transcript
Text below was recognized via OCR from scanned pages and may contain minor errors. Mathematical formulas and symbols may be approximate - please refer to the PDF for the exact equations.
Indian Forest Service (Main’ JBNV-B-STSC
Examination, 2024
STATISTICS
Paper - II
Time Allowed : Three Hours Maximum Marks : 200
Question Paper Specific Instructions
Please read each of the following instructions carefully before attempting
questions :
There are EIGHT questions in all, out of which FIVE are to be attempted.
Questions no. 1 and 5 are compulsory. Out of the remaining SIX questions, THREE are
to be attempted selecting at least ONE question from each of the two Sections A and B.
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a
question shall be counted even if attempted partly. Any page or portion of the page left
blank in the Question-cum-Answer Booklet must be clearly struck off.
All questions carry equal marks. The number of marks carried by a question/part is
indicated against it.
Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary and indicate the same clearly.
Answers must be written in ENGLISH only.
JBNV-B-STSC 1
Ql. (a)
(b)
(c)
(d)
JENV-B-STSC
SECTION A
A subgroup of 5 items each are taken from a manufacturing process at a
regular interval. A certain quality characteristic is measured and X and
R values are computed. After 25 subgroups, it is found that = X = 357-50
and ©R = 8-80. If the specification limits are 14-40 + 0-40 and if the
process is in statistical control, what conclusion can you draw about the
ability of the process to produce items within specifications. (Given that,
for subgroup of 5 items, dy = 2-326, co = 0-8407, dg = 0 and dy = 2-11)
Give the relationship between survival function ‘S(t)’, probability density
function ‘f(t)’, and hazard function ‘h(t)’.
Also find : (i) survival function and hazard function for a density
function f(t) = es, t > 0 and (ii) probability density function and hazard
function for the survival function S(t) = exp (— t’).
For a homogeneous Markov chain, define the probability of the first
return to state i in n steps. Compute the same for a Markov chain with
state space {1, 2, 3} and the following transition probability matrix,
when i = 2 andn=3.
Ble wl Ble
So Wl ple
Blo SO ple
In addition, if the initial probability P(X = 1) = > P(X = 2) = . and
P(X = 3) = 7 compute the probability P(X, = 1| Xp # 2).
4
Get an estimate of the integral (a dx through Monte Carlo
‘ x“+e
simulation, using the following random numbers :
0-463, 0-802, 0-607, 0-455, 0-37
0-839, 0-401, 0-029, 0-843, 0-811
(e)
Q2. (a)
(b)
Solve the following linear programming problem graphically.
Maximize 2 = 3x, + 282 + 233
subject to 2x; - 322 6
10x, + 282 + 323 < 40
921 + xg = 10
> 0, 2220 and xg is unrestricted.
Obtain the optimal values 0121 and 25. What is the optimal 2? 8
Suppose that 50 units are placed on life test (without replacement) and
the test is to be truncated after 10 of them have failed. It was further
supposed that the first 10 failure times are 65, 110, 380, 420, 505, 580,
650, 840, 910 and 950 hours. Estimate the mean life of the component
and its failure rate. Also calculate a 90% confidence interval for mean
time 7. (Assume that the exponential model holds) 15
Given that :
2 2
y 10.05 10.95
20 31.41 10-851
21 32-671 11-591
The following table gives the number of missing rivets at aircraft final
inspection :
Aircraft
Number
Number
of missing 3 16 | 14 | 19 11 | 15 11 | 21 | 12 | 23 | 16 0
rivets
Aircraft
Number
Number
of missing | 25 | 15 0 0 14 11 0 10 | 22 | 7 | 28 0
rivets
14 | 15 | 16 | 17 | 18 | 19 20 | 21 | 22 | 23 | 24 | 25
Find ` € , compute trial control charts and plot control chart for . What
value of 0' would you suggest for the subsequent period ? 15
JBNV-B-STSC 3
(c)
Q3. (a)
(b)
(c)
Q4. (a)
JBNV-B-STSC
Define Reliability. Give its basic elements. Since ‘design’ is the main
area of achieving reliability of a product, give the factors that are to be
considered for achieving a reliable design. 10
Solve the following linear programming problem using two-phase
method : 15
Maximize z= 2x, + X9
subject to x, + 2x22
Ky +X_ <4
X1, X_20
Tho cost of a new machine is = 5,000. The running cost of nth year is
given by R, = 500 (n - 1), n = 1, 2 ... . Assuming an interest rate of
5% per year, after how many years will it be economical to replace the
machinery with a new one ? 15
The probability distribution of monthly sales of a certain item is as
follows :
Monthly sales : 0 1 2 3 4 5 6
Probability : 0-04 0-06 0-25 0-35 0-15 0-09 0-06
The cost of carrying inventory is = 20 per unit per month. The current
policy is to maintain a stock of 4 units at the beginning of each month.
Assuming that the cost of shortage is proportional to both time and
quantity short, obtain the attributed range of cost of a shortage of one
item for one unit of time. 10
The following data is on defective electronic machine parts from a
standard brand of manufacturer. Sample sizes and number of defectives
in each sample are given below. Using suitable control limits, test
whether the process is statistically controlled or not. 15
Sample Size 115 | 220 | 210 | 220 | 220 | 255 | 440
Number of defectives 15 18 23 22 18 15 44
Sample Size 365 255 300 280 330
Number of defectives 47 13 33 42 46
(b)
Two companies A and B manufacture similar products and they are
competing for an increased market share. The payoff matrix shown in
the following table, describes the increase in market share of company A
and decrease in market share of company B.
(c)
JBNV-B-STSC
Company B > | Give Decrease | Maintain | Increase
Coupons | Price Present | Advertising
Company A Strategy
Give 2 _9 4 1
Coupons
Decrease Price 6 1 12 3
Maintain Present
Strategy se - id f
Increase Advertising 2 -3 7 Z
Determine the optimal strategies for both the manufacturers and the
value of the game. 15
Mention the names of different sets of tables under Dodge-Romig
system. Explain the CSP-1 as developed by Dodge. 10
SECTION B
Q5. (a) Define Crude Death Rate. Write merits and demerits of Crude Death
Rate. Is Crude Death Rate an accurate measure of the mortality of
population of a country ? If yes, explain why. If not, how will you modify
it to give reliable results ? 8
(b) | What do you mean by stable and stationary population ? Derive the age
distribution of a life table for stationary population. 8
(c) | Explain the system of collection and publication of Industrial Statistics
in India. 8
(d) Explain ratio to trend method for measurement of seasonal variation in
a time series data. Discuss its merits and demerits. 8
(e) If L(p) and P(q) represent Laspeyres’ Index Number for prices and
Paasche’s Index Number for quantities respectively, show that
where Vo; is the value index number.
Lip) _ Pe)
Hence or otherwise show that =
L(q) P(q)
Also, show that if A(p) = 5 Lp) + P(q)], then A(p) is greater than the
Fisher’s Ideal Index Number. 8
Q6. (a) For an autoregressive series of order two, namely
Yix2 + aYtit+ byt= eto, |b] <1
show that the correlogram Py of order k is given by
k sin(k6+y)
i = a) pea
sin y
1+p? a
where tan y = 5 tan 0, cos 0 =— — 15
1-p 2p
(b) Explain an ARMA (p, q) process and ARIMA (p, d, q) process. How does
Box-Jenkins (BJ) method facilitate the identification of ARMA or
ARIMA model ? 10
JBNV-B-STSC 6
Qi.
(c)
(a)
(b)
(c)
Consider the general linear model given by
Y=XBP+U
where Y is an (n x 1) vector observation on the explained variable
X is an (n x k) matrix of observations on the explanatory variable
U is an (n x 1) vector of error
B is (k x 1) vector of parameters to be estimated
Assume that U ~ N(0, o I) is independently and identically distributed
and rank (X) =k <n.
A
Show that B is the least square estimator of B and is Best Linear
Unbiased Estimator (BLUE) of B. Test the significance of the Null
Hypothesis Hg : B = Bo. 15
Define the force of mortality and show that its approximate value is
d, ,+d,
= > 2 1.
Hx 21, x
If U(x) = 100V100—x, find (84) exactly and by using approximate
method. 15
State the scaling procedures developed for psychological research.
Explain z-score and linear deviate score in brief.
The following table gives the mean and standard deviations of scores
and the raw scores of two students in a number of examinations.
Compare the overall performance of the two students. Give your
conclusions. 15
Raw Score of
Examinations Mean Standard Students
Deviation I Tr
A 155-7 26-4 195 162
B 33-7 8-2 20 54
C 54:5 9-3 39 72
D 87-1 25:8 139 84
E 24-8 6-8 41 25
What do you mean by projection of population ? Explain how you would
do adjustment of migratory disturbances while projecting population. 10
JBNV-B-STSC 7
Q8. (a)
(b)
(c)
JBNV-B-STSC
Consider the following extended Keynesian model of income determination :
Consumption function: C;,= 1+ Bo ¥,—- 83 Ty + Ure
Investment function : Ip=O)+ 0, Yp_1+ Ust
Taxation function 2) pay atm, Y; + Use
Income identity >: Yp=C,+1,+G,
where C = Consumption expenditure
Y = Income
I = Investment
T = Taxes
G = Government expenditure
U's = the disturbance terms
In the model, the endogeneous variables are C, I, T and Y and the
predetermined variables are G and Y; _ }.
By applying the order condition, check the identifiability of each of the
equations in the system and of the system as a whole ? Write your
conclusion.
Define Gross reproduction rate and Net reproduction rate. Distinguish
between these two reproduction rates. Which one attains greater value ?
Why ? Which one can be used for forecasting future population changes ?
Why ?
Explain quasi-stable population. Stating the underlying assumptions,
derive the age distribution function for quasi-stable population with
changing mortality.
15
15
10
