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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

U.G DEGREE EXAMINATION – ECO., COMMERCE, BUS. ADMIN., CORP. & SEC. SHIP

FOURTH SEMESTER – APRIL 2006

# ST 4200 – ADVANCED STATISTICAL METHODS

(Also equivalent to ST 3200/STA 200)

Date & Time : 22-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

PART – A

Answer all the questions.                                                           (10 x 2 = 20 Marks)

1. Define dichotomous classification and give an example.
2. If A and B are two attributes with N = 200, (A) = 120, (B) = 180 and

(AB) = 100, find  (aB), (Ab),  (ab)   and   (a).

1. If A,B and C are three independent events write the expressions for the
probability of getting

(i)  at least one event                       (ii)  exactly one event

1. For any two events with P (A|B) = 3/5 and P (B) = ¼, find P(AB) and P(ACB).
2. Define Poisson distribution and write its mean and variance.
3. Write the moment generating function of Poisson distribution.
4. Write a short note on (i)  Level of Significance  (ii)  Test statistic
5. State the formula for testing the equality of two population means when the sample is small.
6. Write any two uses of statistical quality control.
7. Write the control limits for c chart and give any one application.

PART – B

Answer any Five questions.                                                       (5 x 8 = 40 Marks)

1. For n attributes show that (A1 . A2 …….. An) ³ (A1+ A2 +……+ An ) – (n – 1) N.
2. Check whether the following data is consistent:

N = 900,   (A) = 210,     (B) = 300,        (C) = 420,        (AB) = 150,     (AC) = 175

(BC) = 230  and (ABC) = 100.

1. Consider 3 urns. Urn I contains 5W, 6 R.  Urn II contains 7W, 5 R and Urn III contains 8W, 6R marbles.  One marble is chosen from each urn.  Find the probability of having

(i)  2W, 1R         (ii) 1W, 2R      (iii)  3W           (iv)  3R.

1. If P (A) = ¼ , P (B) = ½ and P (AB) = 1/8, find

(i)  P (A | B)     (ii)  P (B | A)      (iii)  P (Ac | B) and     (iv)  P (A | Bc)

1. If X has the p.d.f.    f (x)  =     3 (1 – x)2 , 0 < x <1, 0 elsewhere ,

find E (X) and V (X).

1. (a) Write any two applications of Poisson distribution.
• If P(X = 2) = 3 P (X = 4) + 2 P (X = 6), find mean, standard deviation and variance of X.                                                                                   (2+6)
1. In a large city A, 20% of a random sample of 900 school children had defective eye sight. In other large city B, 15% of a random sample of 1600 children had the same effect.  Is the difference between the two proportions significant?  Test at 5% significance level.
2. A certain stimulus administered to each of the 12 patients resulted in the following increase of blood pressure:

5,  2,  8,  -1,  3,  0,  -2,  1,  5,  0,  4 and 6

Can it be concluded that the stimulus will, in general, be accompanied by an increase in blood pressure?  Use 1% significance level.   (t – table value is 1.80)

PART – C

Answer any Two questions.                                                       (2 x 20 = 40 Marks)

1. (a) When two attributes A and B are said to be independent?

(b)  Given the following data, find the remaining class frequencies:

N   = 25713  ,  (A)   = 1618   (B)  = 2015

(C) = 770      , (AB) = 587,   (AC)  =  428

(BC) = 335   ,  (ABC) =  156                                                  (2 + 18)

1. (a) If X is a normal variate with mean 30 and standard deviation 5,  find the probability that (i) 26 < X < 40,           (ii)  X > 45      (iii)    |X – 30|    > 5

(b)    In a distribution exactly normal, 10.03% of the items are under 25 kilogram weight and 89.97% of the items are under 70 kilogram weight.  What are the mean and standard deviation of the distribution?   (10+10)

1. The following table gives quality rating of service stations by five professional raters:

SERVICE STATION

RATER           1         2         3         4         5         6         7         8         9         10

A                        99        70        90        99        65        85        75        70        85        92

B                        96        65        80        95        70        88        70        51        84        91

C                        95        60        48        87        48        75        71        93        80        93

D                        98        65        70        95        67        82        73        94        86        80

E             97        62        62        99        60        80        76        92        90        89

Analyse the data and discuss whether there is any significant difference between rating or between service stations.   Use 5% significance level.

1. (a) The following are the figures of defectives in 22 lots each containing 2,000 rubber belts:

425      430      216      341      225      322      280      306      337      305                              356      402      216      264   126      409      193      326      280      389                              451      420

Draw control chart for fraction defective and comment on the state of control of the process.

(b)   In welding of seams defects included pinholes, cracks, cold laps, etc.  A record was made of the number of defects found in one seam each hour and is given below.

1.12.83                 8 A.M.                       2                                   12 NOON      6

9 A.M.                       4                                    1 P.M.           4

10 A.M.                       7                                    2 P.M.           9

11 A.M.                       3                                    3 P.M.           9

12 NOON                   1          3.12.83              8 A.M.           6

1  P.M.                      4                        .           9 A.M            4.

2  P.M.                      8                                    10 A.M.         3

3  P.M.                      9                                    11 A.M.         9

2.12.83                 8  A.M.                      5                                    12 NOON     7

9  A.M.                      3                                      1 P.M.         4

10 A.M.                     7                                      2 P.M.         7

11 A.M.                     11                                    3 P.M.         12

Draw the control chart for number of defects and give your comments.

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