AHSEC HS 2nd Year Question Paper 2019 Statistics

Total number of pages—8
29T STAT

2019

STATISTICS

Full Marks : 100
Pass Marks : 30

Time : Three hours

The figures in the margin indicate full marks
for the questions.

All Questions are Compulsory.

Total Questions : 25 Nos.

Q. No. 1 carries 1 mark each 1×12 = 12

Q. No. 2 to Q. No. 17 carry 3 marks each 3×16 = 48

Q. No. 18 to Q. No. 25 carry 5 marks each 5×8 = 40

Total = 100

Contd.

1. Answer as directed : 1×12=12
øÚÀ«˙±Ú≈˚±˚˛œ Î◊¬M1 ø˚˛± –
Write down the value of   2x  15 x  3  .
3 2
(a) 1

 3  2x 2  15 x  3  ¬1 ˜±Ú øÚÌ«˚˛ fl¬1±º¬
(b) Verify whether the following function is a probability density function
(p.d.f ) or not. 1
Ó¬˘Ó¬ ø˚˛± Ù¬˘ÚÀȬ± ¸y±øªÓ¬± ‚ÚQ Ù¬˘Ú ˝˚˛ ŒÚ Ú˝˚˛ ¬Û1œé¬± fl¬1±º
f  x   3x 2 , 0  x 1

(c) Establish a relation between the operators  and E. 1
¸—fl¬±1fl¬  ’±1n∏ E 1 ˜±ÊÓ¬ ¤È¬± ¸•§g¬¶ö±¬ÛÚ fl¬1±º
(d) If A and B are two independent events, then P  B A   ? 1
˚ø A ’±1n∏ B ≈Ȭ± ¶§Ó¬La ‚Ȭڱ ˝˚˛, ŒÓ¬ÀôL P  B A   ?
(e) If Var  X   4 , find the value of Var  3X  2  . 1

˚ø Var  X   4 ˝˚˛, ŒÓ¬ÀôL Var  3X  2  1 ˜±Ú øÚÌ«˚˛ fl¬1±º
(f) When a sample is considered large ? 1
¬Œfl¬øÓ¬˚˛± ¤È¬± √õ∂øÓ¬˙«fl¬ ά±„1 ¬ı≈ø˘ ·Ì… fl¬1± ˝˚˛∑
(g) Define Null hypothesis. 1
ø1Mê √õ∂fl¬ä1 ¸—:± ø˚˛±º
(h) If X and Y are two random variables, under what condition

E  XY   E  X  . E  Y  ? 1

˚ø X ’±1n∏ Y ≈Ȭ± ˚±‘ø2Âfl¬ ‰¬˘fl¬ ˝˚˛, øfl¬ ‰¬Ó«¬Ó¬ E  XY   E  X  . E  Y  ∑
(i) Write a point of distinction between sampling error and non-sampling
error. 1
√õ∂øÓ¬‰¬˚˛Ú Sn∏øȬ ’±1n∏ ’√õ∂øÓ¬‰¬˚˛Ú Sn∏øȬ1 ¤È¬± ¬Û±Ô«fl¬… Î◊¬À~‡ fl¬1±º

29T STAT [2]

(j) Define simple random sample. 1
¸1˘ ˚±‘ø2Âfl¬ √õ∂øÓ¬˙«1 ¸—:± ø˘‡±º
(k) Under what condition binomial distribution will be symmetrical ?
1
¬øfl¬ ‰¬Ó«¬Ó¬ ø Z¬Û ¬ıKÈ¬Ú √õ∂øÓ¬¸˜ ˝í¬ı∑
(l) What is meant by a ‘parameter’ ? 1
ë √√õ∂±‰¬˘í ¬ı≈ø˘À˘ øfl¬ ¬ı≈ʱ∑
2. Write down the Newtons’ forward and backward interpolation formulae.
1½+1½ =3
øÚÎ◊¬È¬Ú1 ’·ËªÓ¬œ« ’±1n∏ ¬Û(±»ªÓ«¬œ ’ôLÀ¬ı«˙Ú ¸”S ≈Ȭ± ø˘‡±º

3. Show that 3
Œ‡≈›ª± Œ˚
⎧  f x  ⎫
 log f x   log ⎨ 1  ⎬
⎩ f x  ⎭

4. What is interpolation ? Write down the general quadrature formula and
identify different terms in it. 1+2=3

’ôLÀ¬ı«˙Ú ¬ı≈ø˘À˘ øfl¬ ¬ı≈ʱ∑ ¸±Ò±1Ì ¬ı·«œfl¬1Ì ¸”SÀȬ± ø˘‡± ’±1n∏ ˝◊˚˛±Ó¬ ¬ı…ª˝+Ó¬ ¬Û¸˜”˝1
¬Ûø1‰¬˚˛ ø˚˛±º

5. From the table given below, estimate f 2 . 3

Ó¬˘Ó¬ ø˚˛± Ó¬±ø˘fl¬±1 ¬Û1± f 2 1 ˜±Ú øÚÌ«˚˛ fl¬1±º

x : 1 2 3 4 5
f  x  : 7  13 21 37

29T STAT [3] Contd.

6. State the addition rule of probability for any two events. What form does
this result take when the events are (i) independent and (ii) mutually
exclusive ? 3
ø˚Àfl¬±ÀÚ± ≈Ȭ± ‚Ȭڱ1 ¬ı±À¬ı ¸y±øªÓ¬±1 Œ˚±· ¸”SÀȬ± ø˘‡±º ˚ø ‚Ȭڱ ≈Ȭ± (i) ¶§Ó¬La ’±1n∏
(ii) ¬¬Û1•Û1±ôL1 ˝˚˛, ŒÓ¬øÓ¬˚˛± ¸”SÀȬ± Œfl¬ÀÚ ˝í¬ı∑

kx
7. If P  x   ; x  1, 2, 3 is the p.m.f of X , find k and E  X  . 3
3

˚ø X ‰¬˘fl¬1 ¸y±øªÓ¬± ˆ¬1 Ù¬˘Ú P  x   kx ; x  1, 2, 3 ˝˚˛, ŒÓ¬ÀôL k ’±1n∏ E  X  1
3
˜±Ú Î◊¬ø˘›ª±º

1 1 1
8. Given P  A   , P B   and P  A  B   . 3
2 4 5

ø˚˛± ’±À P  A   1 , P B  
1
’±1n∏ P  A  B   1 º
2 4 5
Find [Î◊¬ø˘›ª±]
(i) P  A B 
(ii) P  A B 

1
9. If X has Poisson distribution and P  X  0   , what is E  X  ?
2
(Given log e 2  0  693 ) 3

˚ø√ ˚±‘ø2Âfl¬ ‰¬˘fl¬ X Œ˚˛ ¬Û˚˛‰“¬ ¸y±øªÓ¬± ¬ıKÈ¬Ú ˜±ÀÚ ’±1n∏ P  X  0   1 , E  X  1 ˜±Ú
2
øfl¬ ˝í¬ı∑ [ø˚˛± ’±À log e 2  0  693 ]

10. If X ~ B  n , p  , then find Var  X  . 3

˚ø X ~ B  n , p  ˝˚˛, Var  X  1 ˜±Ú Î◊¬ø˘›ª±º

11. Define random experiment with two examples. 3
¬≈Ȭ± Î◊¬±˝1̸˝ ˚±‘ø2Âfl¬ ¬Û1œé¬±1 ¸—:± ø˘‡±º

29T STAT [4]

12. If A, B and C are three mutually exclusive and exhaustive events and
3 P C   2 P  A   P B  , then find the value of P C  . 3
˚ø A, B ’±1n∏ C øÓ¬øÚȬ± ¬Û1¶Û1 ¬ıø˝ˆ«”¬Ó¬ ’±1n∏ ¸•Û”Ì« ‚Ȭڱ ˝˚˛ ’±1n∏
3 P C   2 P  A   P B  , ŒÓ¬ÀôL P C  1 ˜±Ú øÚÌ«˚˛ fl¬1±º

13. What do you mean by stratified random sampling ? For the following data
on stratified random sampling, estimate the population mean. 2+1=3
ô¶1œfl‘¬Ó¬ ˚±‘ø2Âfl¬ √õ∂øÓ¬‰¬˚˛Ú ¬ı≈ø˘À˘ øfl¬ ¬ı≈ʱ∑ Ó¬˘Ó¬ ø˚˛± ô¶1œfl‘¬Ó¬ ˚±‘ø2Âfl¬ √õ∂øÓ¬‰¬˚˛Ú1 Ó¬Ô…¸˜”˝1
¬Û1± ¸˜ø©Ü1 ˜±Ò… ’±fl¬˘Ú fl¬1±º

Stratum size Sample means
[ô¶11 ’±fl¬±1] [√õ∂øÓ¬˙«1 ˜±Ò…]
150 63
250 50
100 55

14. Define and discuss the level of significance. 3
¸±Ô«fl¬Ó¬± ô¶11 ¸—:± ø˚˛± ’±1n∏ ¬ı…±‡…± fl¬1±º

15. Write down the large sample test statistic for testing the difference between
two population means. Also mention its sampling distribution. 2+1=3
≈Ȭ± ¸˜ø©Ü1 ˜±Ò…1 ¬Û±Ô«fl¬… ¬Û1œé¬± fl¬ø1¬ıÕ˘, ¬ı‘˝» √õ∂øÓ¬˙«1 ¸±Ô«fl¬Ó¬± ¬Û1œé¬±1 √õ∂øÓ¬˙«ÊÀȬ±
ø˘‡±º ˘·ÀÓ¬ Ó¬±1 √õ∂øÓ¬˙«Ê ¬ıKȬÚÀȬ±› ø‰¬Ú±Mê fl¬1±º

16. A coin is tossed 900 times and head turns up 480 times. Test the hypothesis
that ‘the coin is unbiased’. (Level of significance = 5%) 3
¤È¬± ˜≈^± 900 ¬ı±1 ›¬Û1Õ˘ øÚÀé¬¬Û fl¬1± ˝í˘ ’±1n∏ ˝◊˚˛±1 øˆ¬Ó¬1Ó¬ 480 ¬ı±1 ˜≈G Œ¬Û±ª± ·í˘º
ë˜≈^±ÀȬ± ’Úøˆ¬ÚÓ¬ ˝˚˛í √õ∂fl¬äÀȬ± ¬Û1œé¬± fl¬1±º [¸±Ô«fl¬Ó¬± ô¶1 = 5% ]

17. Write down two applications of the chi-square test. 1½+1½=3
fl¬±“˝◊ - ¬ı·«1 ¬Û1œé¬±1 ≈Ȭ± √õ∂À˚˛±ÊÚœ˚˛Ó¬± Î◊¬À~‡ fl¬1±º

29T STAT [5] Contd.

6 dx
18. What do you mean by Numerical integration. Evaluate ∫0 by
1 x
Simpson’s 13 rule and hence find log e 7 . 1+4=5
6 dx
¸—‡…±Rfl¬ ’Ú≈fl¬˘Ú ¬ı≈ø˘À˘ øfl¬ ¬ı≈ʱ∑ ø‰¬•Û‰¬Ú1 13 øÚ˚˛˜ÀȬ± ¬ı…ª˝±1 fl¬ø1 ∫0 -1 ˜±Ú
1 x
Î◊¬ø˘›ª± ’±1n∏ Ó¬±1¬Û1± log e 7 1 ˜±Ú øÚÌ«˚˛ fl¬1±º

19. State the uses of Pilot survey. Show that in S.R.S.W.R., the sample mean
is an unbiased estimator of the population mean. 3+2=5
øflƒ¬ √õ∂fl¬ä ¸˜œé¬±1 ¬ı…ª˝±1 ¸•ÛÀfl«¬ Î◊¬À~‡ fl¬1±º Œ‡≈›ª± Œ˚ ¸¬Û≈Ú–¶ö±¬ÛÚ ˚±‘ø2Âfl¬ √õ∂øÓ¬‰¬˚˛Ú1
Œé¬SÓ¬ √õ∂øÓ¬˙«1 ˜±Ò… ¸˜ø©Ü1 ˜±Ò…1 ’Úøˆ¬ÚÓ¬ ’±fl¬˘fl¬º

20. Derive the mean and variance of the Poisson distribution. 2+3=5
¬Û˚˛‰“¬ ¬ıKȬÚ1 ˜±Ò… ’±1n∏ √õ∂¸1Ì øÚÌ«˚˛ fl¬1±º

X   
21. (a) If X ~ N   ,   , find the distribution of Z  . 3


˚ø X ~ N   ,   , ŒÓ¬ÀôL Z   X    1 ¬ıKȬÚÀȬ± øÚÌ«˚˛ fl¬1±º


(b) If X ~ N    80,   5  , find P  75  X  85  . 2

˚ø X ~ N    80,   5  , ŒÓ¬ÀôL P  75  X  85  1 ˜±Ú øÚÌ«˚˛ fl¬1±º

22. An urn contains 4 white and 4 black balls. A second urn contains 5 white
and 4 black balls. One ball is transferred from the first urn to the second
urn and then a ball is drawn from the later. What is the probability that
it is white ? 5
¤È¬± ŒÈ¬Àfl¡ø˘Ó¬ 4 Ȭ± ¬ı·± ’±1n∏ 4 Ȭ± fl¬í˘± ¬ı˘ ’±Àº 2˚˛ÀȬ± ŒÈ¬Àfl¡ø˘Ó¬ 5 Ȭ± ¬ı·± ’±1n∏ 4 Ȭ± fl¬í˘±
¬ı˘ ’±Àº √õ∂Ô˜ ŒÈ¬Àfl¡ø˘1 ¬Û1± ¤È¬± ¬ı˘ 2˚˛ ŒÈ¬±Õ˘ ’Ú± ˝í˘ ’±1n∏ ¬Û±ÂÓ¬ 2˚˛ ŒÈ¬Àfl¡ø˘1 ¬Û1±
¤È¬± ¬ı˘ Ȭڱ ˝í˘º ¬ı˘ÀȬ± ¬ı·± Œ˝±ª±1 ¸y±øªÓ¬± øÚÌ«˚˛ fl¬1±º¬

29T STAT [6]

23. Draw all possible samples of size 2 by simple random sampling (without
replacement) from a population of 4 units (2, 4, 6, 8) and show that the
sample mean is an unbiased estimator of the population mean and find its
standard error. 5
4 Ȭ± ¬Û1 ¤È¬± ¸˜ø©Ü (2, 4, 6, 8) 1 ¬Û1± ¸1˘ ˚±‘ø2Âfl¬ √õ∂øÓ¬‰¬˚˛Ú [’¬Û≈Ú–¶ö±¬ÛÚ] ¬ÛX√√øÓ¬1 Z±1±
2 ’±fl¬±11 ¸fl¬À˘± ¸y±¬ı… √õ∂øÓ¬˙« Î◊¬ø˘›ª±º Œ‡≈›ª± Œ˚ √õ∂øÓ¬˙« ˜±Ò… ¸˜ø©Ü ˜±Ò…1 ’Úøˆ¬ÚÓ¬
’±fl¬˘fl¬ ’±1n∏ ˝◊˚˛±1 ˜±Úfl¬ Sn∏øȬ øÚÌ«˚˛ fl¬1±º

24. From a population, 10 students are chosen at random and their heights in
inches found to be 62, 60, 61, 65, 59, 71, 70, 68, 60 and 61. Test whether the
population mean height is 66'' ?

[ Given that : t 005  9 d. f .   2  26 ] 5

˚±‘ø2Âfl¬ˆ¬±Àª Œfl¬±ÀÚ± ¤È¬± ¸˜ø©Ü1 ¬Û1± øÚ¬ı«±ø‰¬Ó¬ fl¬1± 10 ÊÚ Â±S1 Î◊¬2‰¬Ó¬±1 ˜±¬Û [˝◊ø=¬
ø˝‰¬±¬ÛÓ¬] ø˚˛± ’±À 62, 60, 61, 65, 59, 71, 70, 68, 60 ’±1n∏ 61º ¸˜ø©ÜÀȬ±1 ·Î¬ˇ Î◊¬2‰¬Ó¬±
66'' ˝˚˛ÀÚ∑ ¸±Ô«fl¬Ó¬± ¬Û1œé¬± fl¬1±º
[ ø˚˛± ’±À – t 005  9 d. f .   2  26 ]

25. (a) Write a short note on statistic and standard error. 3
√õ∂øÓ¬˙«Ê ’±1n∏ ˜±Úfl¬ Sn∏øȬ1 ø¬ı¯∏À˚˛ ¤øȬ ‰¬˜≈ ŒÈ¬±fl¬± ø˘‡±º
(b) What are the different types of errors in sampling ? 2
√õ∂øÓ¬‰¬˚˛ÚÓ¬ Ô±øfl¬¬ı ¬Û1± ø¬ıøˆ¬iß õ∂fl¡±11 Sn∏øȬ¸˜”˝√ øfl¬ øfl¬∑

———— ×————

29T STAT [7]

29T STAT [8] 2·5+