AHSEC HS 2nd Year Question Paper 2022 Statistics

Total nuntber of pages-1"2

32T STAT

20.22

STATISTICS
Full Marks : L00
Pass Marks : 30

Time : Three hours

The figures in the margin indicate full marks
for the questions.
All Questions are ComPulsory.
Total gu"riions : 25Nos.

Q. No. 7 carries L unrk each "1x12 = 1,2

Q. No, 2 to Q, No. 77 carry 3 tmrks each 3x16 = 48

Q. No. 78 to Q, No. 25 carry 5 nmrks ench 5x8 = 40

Total = ioo

Contd.

1. Answer as directed : 1"x'12=12

Es< fixt:

(n) Write down the value ot N'(ax" *O*".-t).

*(ax" +ar'-'). qF fi'F{ clr
(b) 14/hat clo you mean by arguments ? '

<tft p-+ ffue ft 1otr

(c) Under what condition, Simpson's one-third rule is valid ?
ft 5-$ qtrrrsqs-yqlt$ fu{uq firr sFId sR< {G?
(d) If P (E) = 0, then E is called event. (FilI in the blnnk)
ffi i,1ry = o {r, rscg Es {tr{I ffi csEI E'< r

effirFl,r.wD
@ State the conditir:n under which covariance (X, y) = 0.
-
ft u6 clc"fts covariance (& y) = 0, Umq Gl I

(fl lf X - P(4), tl'ren find the mean of the distribution.
sfr x - p(4) c{, cvrs <&tiFI cxi fi.F{ s{tr

Q) The s.d. of a normal distribution is 20, find the quartile cleviation.
qB efqrlnr <fa-< m++ Rrm zorD-gaf$ &rG fr.f* wtr

(Write True or Fnlx)

sE(r)= o @w cq qauJ futl)
,ln

32r STAT I2l

(n Given the observed value of lZl=Z 15, and table value of
Z =I.96, what is your conclusion in a large sample test ?

frfl qI[E lzl-< fifrfrs {q = 2'15 qFF z-< Elfi-d-.gw ltl 1'961 T{e,
cGn"f {fr$ qtrc 6slTR ffit frl
(il What are the assumptions of 12-test ?
oR-cqra.< "tftsrc \c&{<ac{t ft frt

(k) How many random samples of size n can be drawn from a
population of size N, if sampling is done with replacement ?
N qFI*Ft c{E< "l{l rr eFFFFt ftrl{ qe'$ afrn{ 54- sk{I, {fr
Effit c'j{st F s{t
0 What do you mean by complete enumeration ?
rTnf rlfdl fFre ft Tq.tt

2. Define the oPerator E and show that E = 1+ A. l+2=3

csFrs E-< To frnl oTr ffi1€<l c{ E = 1+ A I

3. State the fundamental theorem of finite differt'nce. Evaluate
A3 (l - x) (1 - sx) (f - or), the interval of difference being
unity' l+2=3
qffis qs<< ffift.+ Eqqtqrd fr.llr{la fi'f'* o<t-
a3 (t - r) (t - sr) (r - ox\, wSEroFFI sF$ I

4. Estimate /(4) from the following data. 3

u+t sql{ "r{l /(4)-< nFl qrsd4 s-{l I

x 1 3
2 5
Y 2 5 7 32

32T STAT t3l Contd.

Or / w4<rl

Using the Lagrange's interpolation formula, find the value of /(5).
sldrq< qq+F {-{ eEfla oR vaq sqr q{{ 'FIl/(5)1El{ fift €rr
,r(1)=4, f (2)=s, f (7)=s, f (8)=4.

6
Find the value of ji *l+ x by Simpsor/s $rd rule. 3
0

,s{-y6rr$ QqF 1-c qrfl{ q'R m fi"fr iqtr
i*.
'Or/qe61

State the Simpson s $th rule and write the conditions of validity of

Simpson's $tt -tu. 1+2=3

&fi-qgql$ e-'Fdq {E6t 86I"r F$ qFF e {get ?<qst cCfiil Ddc{q Eafi
?Ffi I

Define random experiment with suitable examples. 3

qttF{$
"fftsF
qgt S"f{o EqE<"K brycs fr{tt

7. If A and B are two events such ttrat f(a)=0.4, P(AUB) =0.7 and
P(S)= p, for what value of p, A and B are - (i) mutually
(ir) independent ? #;:;
<fra q+ BfiSlt, p(A) = o. 4, P(AU B) = o.7 \fls P(B) = p, p< ftfla
TT;K{6 Aql$ B _ (, 'K*,K qfrgs 1i4 Es6t

32T STAT t4I

g. what do you mean by random variable ? write down the difference
between discrete and continuous r.v.
'l+2=3

sqks D-o.<F 1frln fr Wrr Rfu q&-+ DE-s qt+ qRkq stI&T u-drc xtq<
{ef$i Ftttt

Or/qqa1

Define mathematical expectation for discrete r.v. state the addition
1+2=3
theorem of exPectation'
rilFrfrs ?rstFtF rigf fr$lersrFtFr c{l{sq& Stnq oqt

g. Given P(X = *\= x=t,2,....0o' Find a' .f
),
: frfl \flrq e(x = i= j) x=t,2,""o I (vcs a< lla Efis<tl

10. If X and Y are two r.v"s, show that -
v(x+r)= v(x)+v(Y\+2au(x,v)' 3

rfr x qto v1tl lqfr-s DEFF, 6s'cs cq'{e<l c{ -
v (x + v\ = v (x) + v (v) + 2 ou(x, v) I

ll.WritedownthebinomialdistributionwiththePalametersrrandp.Give
t?t,o occurrences of binomialrlistribution' 1+2=3

,? qF$ p $Eqr< fi'"lq <tq6l fr{lfi'E <fi< fA
ffi Emq <tt
Or/w*tt
under what conditions Poisson distribution is the timiting form
of
3
binomial distribution ?
ft u6 ctc,fcs d{E <6{& fr"lr <fi< DiN {qRffcq cqm zl1<l
32T STAT t5 ] CONtd'

12. Find the mean and vadance of standard normal variate. 3

Irds :r{IIlTI sEfGi nr{J qls !T{.r $a Efrs<] t.

Or / w44t

tf X - lV(15, 16), find the probability that X is larger than 1E. 3

<fi x - w(is, ro), X, 1E e<' vts< etr< TglEq fi.f{ rqlt

13. Write down the distinction between parameter and statistic. 3

$Eq qE aG?G< $ws clsl "fieFs filnl

Or / e44t

Explain type-l and type-Il elrols in relating to test of significance. 3

c]efo-q "tfu{ ffir-s deI{ eFm (t) q{ ftq-{ eFFFr (ID str{ <IFfi Gtl
14. Define null hypothesiq alternative hypothesis and level of significance.

Fe ercq, frrq eFEa q<F vlsd qi rr\Eq fiTI r

15. A coin is tossed 1Q000 tirnes and it tirrns up heads 5195 times, discuss if
the coin may be regarded as unbiased. 3
qil {El 10,000 <rr fr(Ft Gtt {i I 5195 <R
{o (elq n's I 1rtrh \m&s {a
s'< 'l{f {FItr{ ?

16. Write down the small sample test statistic for testing the differmce
between two population means. State the assumptions of l-test, 1+2=3

$l qnB< ftt< {d{s, tfr$ oRda, Td qffi {ef{sr tffFr dffit
fr{r r-aAFR qG{F.fi-q{ Bffi{ c<lt

32T STAT t6l

Or/ qlay
Explain briefly the testing significance of single mean for large sample.

cFt fi$< stq,I{"tfu< <tm Tqq aBq.fr q|afN| ,ftst ffi-s <.f{t s-{tt

17. . Write down the causes (sources) of sampling errors.
q&ET4 SF< $F6t (Gqc) C{q BG{ <Ftr

Or/qcfi1
Distinguish between simple random sampling with replacement and
without replacement. 5

ctrEl"F qs qt{rt'F[ ct{t{"| slI&s d&Dr-d< <rd{q EGl"l s-{ I

18. From the following data estimate the no. of students who secured more
than 15 marks. , 5
\5dF{ sqr< el<I 15 {r{*s mfr cqr+ ssr ci?trt Qga6 6f l

Marks 0-10 10-20 20-30 30-40 40-50
({T<)
No. of students 13 25 20 5 2
(Erslx{il)
Or / wQal

Write a short note on General Quadrature formula. Show that
-lr
I u.a* =,.,(sur +SUs -u 1).
o"
2+3=5

flql<q <ft<Rq 1o'6t< e"t<o DX ctm fr"ll r cqls$ m
-
-lr
Io r.ar = ,.,
'o
(su, + 8uo -u_,)

32T STAT 17l Contd.

19. A bag contains 5 white and 2 black balls. Another bag contains 4 white
and 4 black balls. One ball is transferred at random from the first bag to
the second bag and then a ball is drawn randomly from the second bag'
What is the probability that the ball is white ? 5

q$ 4Bl T'dt <q
.sbt c{FlE 5A <tfi qrd 2fi :p'qt <q \Ttcql q{&ls 4Bt <'tl
qtcsr aq{&R ,m fifrsdtfr {tEF{sst6{ aFr <-q q-{F iqt <? qs '44 ({
fi-ffiR "r< o5&-+ura t{ cqr redt <ttl <E R!-{Fl q-elRq firyFt
Or/qt1a1

-
For any two events A and B, prove that

P(AnB)< P(A)< P(AuB)< r(a)+ r(r)
ffi fi 6at a q+ B< <fr< E{l6l <x{l R -
P(AnB)< P(A) < P(AUB)< r(a)+ r(a)

The r.v. X has the P.m.f. -

tqk{Eq'$ x-<TutFqEIFFI-

X--x 0 1 2 3
p(x\ I a l 24

Find the expected value of v --(X -t)2.

v =(x -t)2 < fllEfu avtP'fi fi"f{ ffit

27. Show that mean and variance of Poisson distribution are equal'
cc{srl (3 dTS <til-{ {t{l q+ efrFH ir5lFl I

32T STAT t8l

Orl wtal

Derive the p.d.f. of standard normal variate 5

{Frs erfiffii EErfi q;N +-+< fi.f* cIr

22. A sample of 400 observations has a mean 95. Could it be a random sample
from a population with mean 98 and s.d. 12 ? 5

4008frfrsfirt{r 95rltql g8q-$qEafiE-ffi u erslrnE<fi-{trss-{ elltt
<lI&+ uE+ Errar

23. Ten individuals are chosen from a population and their heights in inches
are found to be 63, 65, 66, 67, 68, 69,70,71,,70,77.

(Assume that the population distribution of height is normal)

O Estimate SE of sanlple mean.

dil Does the observed mean differ significantly from 70 inches ?
[Given t6.s5 Q)=22621 2+3=5

qB q{ qq&< ,m 10 wr rflffi ffiA s-{ {q I fr{p< 66-e-t 63,6s,66,62,68,

69, 70, 71,, 70, 71, t

(fu< qnE <&{ltl afiFi fi {R 6dl<l s'6')
(n gGr.f lt$< Tl{s sF qRidFl <5:rtl
(i) fiAfrE {l{ittsl 70 ElcB< "l<l &E ot
(ftfl \$(q ,o.os (9) =2262)

32T STAT le1 Contd.

Or/qcl41

From the following table compute the test statistic for testing the
independence of two attributes A and B. 5

Effi
s-ffi qft-sF,Frt A qFF B ltI e.t<r Elstr{ {ft+t< <lrs qfter fift <r r

A1 A2 Total ({b)
B

B1 10 30 40
B2 18 52 70

Total ({5) 28 82 110

24. Explain the principal steps in sample survey. 5

afra"f {frs'R e$q 4{n q{s <d{t cl I

Or/qct41

What are the limitations of simple random sampling ? Show that in simple
random sampling with replacement the sample mean f is an unbiased
estimate of population mean f, . 2+3=5

:r+d {I&-s aGETd< ff{r<hst q{qft ftr cq'.I€<l 6{ c13sqtqn< irqd {rt&s
aGEsdE s&c..rflv t r{& dv X < q-dfus qFFqsl
25. What do you mean by stratified random sampling ? From the following
data on stratified random sampling estimate the population -"-.r*O=U

Size of stratum Size of samples Sample means
MO 10 96.8

400 10 862
110 05 22L.0

32T STAT t 10l

sATs ttTe,6 dfuR{ $lra fr Wr? sEE sfrTs nqes oGEtr;K sal< erct
qqB rt$ q-E-ap +<tt

gqq qr{,r< ffiF{< wsFr q&F{ntq]
440 10 96.8
400 10 862
110 05 221.0

32T STAT [11]

[12] 2.5+
32T STAT