LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.COM. DEGREE EXAMINATION – COMMERCE

THIRD SEMESTER – November 2012

# ST 3202- ADVANCED STATISTICAL METHODS

Date :9/11/2012                   Dept. No.                                        Max. : 100 Marks

Time : 9.00 – 12.00

PART A                                           (10 X 2 = 20 marks)

1. Define independence of attributes.
2. What are the types of non- probability sampling?
3. State the Axioms of probability
4. State addition theorem on probability.
5. State any four properties of Poisson distribution.
6. What is meant by probable error? Mention its uses.
7. Differentiate between Small Samples and Large Samples.
8. What is meant by analysis of variance?
9. Explain the various types of control charts.
10. Distinguish between the control limits and tolerance limits.

PART  B                                              (5 X 8 = 40 Marks)

1. State and prove Baye’s theorem.

1. The result of a certain survey shows that out of 50 ordinary shop of small size 35 are managed by men

of which 17 are in cities, 12 shops in villages are run by women. Can it be inferred that shops run by

women relatively more in villages than in cities ?

1. Five men in a company of 20 are graduates, if 3 men are picked out from this 20 at random, what is the

probability that (i) all are graduates (ii) at least one is a graduate.

1. An Automatic Machine fills in tea in sealed tins with Mean Weight of tea 1 kg. and S.D. 1gm . A

random sample of 50 tins was examined and it was found that their Mean Weight was 999.50 gms. Is

the machine working properly .

1. The following data is collected on two characteristics:
 Smokers Non-Smokers Literate 83 57 Illiterate 45 68

Based on this test whether there is relation between the habit of smoking and literacy.

16 . An IQ test was administered to 5 persons before and after they were trained. The results are given

below:

 Candidates I II III IV V IQ before training 110 120 123 132 125 IQ after training 120 118 125 136 121

Test whether there is any change in IQ after the training programme. Use 5% level of  significance.

1. The following table gives the number of defective items found in 20 successive samples of 100 items

each

2    6   2   4   4   15   0   4   10   18   2   4   6   4   8   0   2   2   4   0

Comment whether the process is under control. Suggest suitable control limits for the future.

PART   C                                   (2 X 20  =  40 Marks)

19.(a) A number of school-children were examined for the presence or absence of certain

defects of which three chief descriptions were noted; A-development defects;

B-nerve signs; C low nutrition. Given the following ultimate frequencies, find the

frequencies of the classes defined by the presence of the defects.

(ABC) = 57; (aBC) = 78

(ABg) = 281; (aBg) = 670

(AbC) = 86; (abC) = 65

(Abg) = 453; (abg) = 8310                                                                                                     (10)

19 . (b)  Two boxes contain 12 white and 18 black and 15 white and 25 black balls respectively.  One  box

was taken at random and a ball was taken from the same.  It is a black ball. What is the probability

that it is from the (i) first box (ii) second box.                                                                                (10)

1. (a) If 3% of the electric bulbs manufactured by a company are defective, find the probability that in a

sample  of 100 bulbs exactly five bulbs are defective (e-3 = 0.0498).                                             (10)

1. (b) The average daily sales of 500 branch offices was Rs.150,000 and the standard deviation

Rs.15,000. Assuming the distribution to be normal, find how many branches have sales between

• 1,20,000 and Rs.1,45,000
• 1,40,000 and Rs.1,60,000                                                                                   (10)

.

21.(a) The sales manager of a large company conducted a sample survey in states A and B taking 400

Samples in each case. The results were as follow

State A               State B

Average sales             Rs.2500               Rs.2200

Standard Deviation      Rs.400                Rs.550

Test whether the average sales is the same in the two states. Test at 1% level.                                  (10)

1. (b) Value of a Variety in two samples are given below:
 Sample I 5 6 8 1 12 4 3 9 6 10 Sample II 2 3 6 8 1 10 2 8 * *

Test the significance of the difference between the two sample means.                                     (10)

1. The following table gives the fields of 15 samples of plot under three varieties of seed.
 A B C 20 18 25 21 20 28 23 17 22 16 15 28 20 25 32

(20)

Test using analysis of variance whether there is a significant difference in the average yield of seeds

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., B.Com., DEGREE EXAMINATION – ECONOMICS & COMMERCE

THIRD SEMESTER – APRIL 2011

# ST 3202/3200/4205/4200 – ADVANCED STATISTICAL METHODS

Date : 15-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A                                           (10 X 2 = 20 marks)

1. What is meant by independence of attributes?
2. What are the types of sampling?
3. Define Probability of an event.
4. Define conditional probability.
5. State any two properties of normal distribution.
6. State Central Limit Theorem.
7. State Type – I and Type – II error.
8. What is meant by analysis of variance?
9. Explain the various types of control chart.
10. What is meant by probable error? Mention its uses.

SECTION B                                              (5 X 8 = 40 Marks)

1. State and prove multiplication theorem.

1. 800 candidates of both sex appeared at an examination. The boys outnumbered the girls by 15 %

of  the total. The number of candidates who passed exceeded  the number failed by 480. Equal

number of boys and girls failed  in the examination. Prepare a 2×2 table and find the coefficient

of association and Comment.

1. Five men in a company of 20 are graduates, if 3 men are picked out from this 20 at random, what is the probability that (i) all are graduate (ii) at least one is a graduate.

1. Two random samples of sizes 400 and 500 have mean 10.9 and 11.5 respectively. Can the samples be

regarded as drawn from the same population with variance 25?  Test at 1% level.

1. The following data is collected on two characteristics:
 Smokers Non-Smokers Literate 83 57 Illiterate 45 68

Based on this test whether there is relation between the habit of smoking and literacy.

16 . A company arranged an intensive training course for its team of salesmen. A random sample of 10       salesmen was selected and the value ( in 000) of their sales made in the weeks immediately before and     after the course are shown in the following table:

 Salesman 1 2 3 4 5 6 7 8 9 10 Sales before Training 12 23 5 18 10 21 19 15 8 14 Sales after Training 18 22 15 21 13 22 17 19 12 16

Test whether there is evidence of an increase in mean sales. Test at 5% level

1. The number of defects detected in 20 items are given below

Item No       :  1   2    3    4     5    6   7    8     9    10    11     12    13    14   15   16   17   18  19    20

No. of defects         :  2    0   4   1      0     0   8     1    2     0      6        0     2      1    0      3    2      1   0    2

Test whether the process is under control. Device a suitable scheme for future

SECTION   C                                   (2 X 20  =  40 Marks)

19.(a) A number of school-children were examined for the presence or absence of certain

defects of which three chief descriptions were noted; A-development defects;

B-nerve signs; C low nutrition. Given the following ultimate frequencies, find the

frequencies of the classes defined by the presence of the defects.

(ABC) = 57; (aBC) = 78

(ABg) = 281; (aBg) = 670

(AbC) = 86; (abC) = 65

(Abg) = 453; (abg) = 8310                                                                                            (10)

19 . (b)  A factory manufacturing television has four units A, B, C and D. The units A, B, C and D manufacture 15%, 20%, 30%, and  35%, of the total output respectively. It was found that out of their outputs 1%, 2%, 2% and 3% are defective. A television is chosen at random from the output and found to be defective. What is the probability that, it came from unit D?                                                            (10)

1. (a) If 10% of the screws produced by an automatic machines are defectives, find the probability

that out of 20 screws selected at random there are (i) exactly two defectives

(ii)at the most three defectives  (iii) at least two defectives                                               (10)

1. (b) The average daily sales of 500 branch offices was Rs.150,000 and the standard deviation

Rs.15,000. Assuming the distribution to be normal, find how many branches have sales between

• 1,20,000 and Rs.1,45,000
• 1,40,000 and Rs.1,60,000                                                (10)

21.(a)Random samples of 400 men and 600 women were asked whether they would like to have a fly-over near their residence 200 men and 325 women were in favor of it. Test the equality of proportion of men and  women in the proposal? Test at 5% level.                                                                 (10)

1. (b) Value of a Variety in two samples are given below:
 Sample I 5 6 8 1 12 4 3 9 6 10 Sample II 2 3 6 8 1 10 2 8 * *

Test the significance of the difference between the two sample means.                                            (10)

1. Develop the Two- way ANOVA for the following data:

Treatment

 A B C D I 3 4 6 6 II 6 4 5 3 II 6 6 4 7

Plots of land

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.A., B.COM., DEGREE EXAMINATION – ECONOMICS & COMMERCE

THIRD SEMESTER – APRIL 2012

# ST 3202/3200/4205/4200 – ADVANCED STATISTICAL METHODS

Date : 02-05-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer ALL questions.                                                                                           (10 X 2 = 20 marks)

1. State the axioms of the Probability .

1. Write any four properties of normal distribution.
2. Define conditional probability.
3. State Type – I and Type – II error.

1. What is standard normal Variable?
2. State Central Limit Theorem.

7.What is null hypothesis?

1. What is meant by independence of attributes?
2. Distinguish between np chart and p chart.
3. Distinguish between the control limits and tolerance limits.

SECTION B

Answer any FIVE questions:                                                                              (5 X 8 = 40 Marks)

1. 800 candidates of both sex appeared in an examination. The boys outnumbered the girls by 15 %

of  the total. The number of candidates who passed exceeded  the number failed by 480. Equal

number of boys and girls failed  in the examination. Prepare a 2×2 table and find the coefficient

of association and Comment.

1. State and prove Baye’s theorem.
2. A Sub-Committee of 6 members is to be formed out of a group consisting of 7

men and 4 women. Calculate the probability that the sub-committee will consist of

(1) exactly 2 women (2) at least 2 women.

1. Two random samples of sizes 400 and 500 have mean 10.9 and 11.5 respectively. Can the samples be

regarded as drawn from the same population with variance 25?  Test at 1% level.

1. What is Sampling Technique ? Explain different types of Sampling.

1. In a survey of 200 boys, of which 75 intelligent, 40 had skilled fathers while 85 of the Unintelligent  boys has unskilled fathers. Do these figures support the hypothesis that skilled fathers have intelligent boys. Use Chi square –test of 5 % level.

1. The number of defects detected in 20 items are given below

Item No       :  1   2    3    4     5    6   7    8     9    10    11     12    13    14   15   16   17   18  19    20

No. of defects      :  2    0   4   1      0     0   8     1    2     0      6        0     2      1    0      3    2      1   0      2

Test whether the process is under control. Device a suitable scheme for future

SECTION   C

Answer any TWO questions:                                                                                   (2 X 20  =  40 Marks)

19.(a) Given (ABC) = 137;   (αBC) = 261; (ABC) = 313; (Aβg) = 284; (Abr) = 417; (αBg) = 420;

(αbC)  =  490; (abg)  =  508; Find the frequencies (AB), (A) and N.                       (10)

19.(b) Two Urns contain respectively 10 white, 6 red and 9 black and 3 white 7 red and 15 black balls.  One ball is drawn from each Urn.  Find the probability that  (i)  Both balls are red   (ii)  Both balls are of the same colour.                                                                                                                             (10)

1. (a) A Company has four production sections viz. S1, S2, S3 and S4 , which contribute 30%, 20%, 28%    and  22% of the total output. It was observed that those sections  respectively produced 1%, 2%, 3% and 4% defective units. If a unit is selected at  random and found to be defective, what is the probability that   the units so selected has  come from either S1 or S4.?                                         (10)

1. (b) The customer accounts of a certain departmental store have an average balance of Rs.120 and a

standard deviation of Rs.40. Assuming that the account balances are normally distributed, find

• What proportion of accounts is over Rs.150?
• What proportion of accounts is between Rs.100 and Rs.150?
• What proportion of accounts is between Rs.60 and Rs.90?                (10)

21.(a) A random samples of 400 men and 600 women were asked whether they would like to have a fly-over near their residence. 200 men and 325 women were in favor of it. Test the equality of proportion of men and women at 5% level.                                                                                  (10)

1. (b) Value of a Variety in two samples are given below:
 Sample I 5 6 8 1 12 4 3 9 6 10 Sample II 2 3 6 8 1 10 2 8 * *

Test the significance of the difference between the two sample means.                                            (10)

1. Prepare a Two- way ANOVA on the data given below.

Treatment I

 I II III A 30 26 38 B 24 29 28 C 33 24 35 D 36 31 30 E 27 35 33

Treatment I I

Use the coding method, subtracting 30 from the given numbers.                                                                                                                                                                                                                              (20)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. B.Com., DEGREE EXAMINATION – ECO. & COMM.

FOURTH SEMESTER – APRIL 2012

# ST 4205/4200/3202/3200 – ADVANCED STATISTICAL METHODS

Date : 19-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

Answer ALL the questions:                                                                                     (10 X 2 = 20)

1. Write the formula for Yule’s Coefficient of partial association between A and B with C.
2. What are the methods of association?
3. State the addition theorem for two events.
4. Define random variable.
5. What is the difference between small sample and large sample test.
6. Define type – II error.
7. Write down the formula for F-test.
8. Explain Error sum of squares.
9. What are the various types of control charts?
10. Give the control limits for R chart.

SECTION – B

Answer any FIVE of the following:                                                                        (5 X 8 = 40)

1. Out of 5 lakh literates in a particular district of India, the number of criminals was 2000. Out       of 50 lakh illiterates in a particular in the same district, number of criminals was 80,000. On       the basis of these figures, do you find any association between illiteracy and criminality?

1. For two attributes A and B, we have:

(AB)= 16, (A) = 36, (αβ) = 10, N = 70. Calculate Yule’s coefficient of association and          Colligation

1. From the table given below, test whether the colour of son’s eyes is associated with that of father’s eyes by using chi-squares test at 5% level.

 Eyes Colour in Sons Eyes Colour in Fathers Not light Light Not light 230 148 Light 151 471

1. Explain the method of analysis of variance for One way classification.

1. The following data refers to visual defects found during the inspection of the first 10 samples of size 50

Each from a lot of two-wheelers manufactured by an automobile company:

 Sample No. 1 2 3 4 5 6 7 8 9 10 No. of Defectives 4 3 2 3 4 4 4 1 3 2

Draw the control chart for fraction defectives and state your conclusion.

1. The following table shows the distribution of number of faulty units produced in a single       shift in a factory. The data is for 400 shifts.

 No. of faults 0 1 2 3 4 No. of shifts 138 161 69 27 5

( value of e-1=0.3679)

Fit a Poisson distribution to the given data.

1. There are 4 boys and 2 girls in room-I and 5 boys and 3 girls in room-II. A girl from one of       the two rooms laughed loudly. What is the probability that the girl who laughed was from       room-I and room-II.
2. The height of ten children selected at random from a given locality had a mean 63.2 cms and variance 6.25 cms. Test at 5% level of significance the hypothesis that the children of the       given locality are on the average less than 65 cms in all.

SECTION – C

Answer any TWO of the following:                                                                     (2 X 20 = 40)

1. a) Find the value of ’ K’ and also find Mean and Variance.
 X 0 1 2 3 P(X) 1/8 3/8 K 1/8

1. b) State and prove Multiplication theorem of probability.
2. a) In a random sample of 500 persons from town A, 200 are found to be consumers of          wheat. In a sample of 400 from town B, 220 are found to be consumers of wheat. Do         these data reveal a significant difference between town A and town B as far as the          proportion of wheat consumers is concerned?
3. b) The following data show weekly sales before and after recognition of the sales organization.

 Sales before 14 18 13 19 15 14 16 18 Sales after 21 18 17 23 21 18 22 22

Test whether there is any significant difference in sales before and after recognition of the

Sample company.

1. a) The following are the number of defects noted in the final inspection of 20 bales of woolen cloth:

3, 1, 2, 4, 2, 1, 3 ,5, 2 ,1, 5 , 9, 5, 6, 7, 3, 4, 2, 1,6.

Draw C-chart and state whether the process is under control or not.

1. b) Draw the control chart for Mean and comment on the state of control from the given             data:
 Sample number Observations   1              2                3 1 50 55 52 2 51 50 53 3 50 53 48 4 48 53 50 5 46 50 44 6 55 51 56

1. The following table gives the yield on 20 sample plot under four varieties of seeds:

 A B C D 20 18 25 24 21 20 28 30 23 17 22 28 16 15 28 25 20 25 32 28

Perform a One-way ANOVA, using 5% level of significance.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – COMMERCE

THIRD SEMESTER – NOVEMBER 2012

# ST 3202 – ADVANCED STATISTICAL METHODS

Date : 09/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer ALL questions:                                                                                                (10 X 2 = 20 marks)

1. What is meant by independence of attributes?
2. What are the types of probability sampling?
3. Define probability and give an example.
4. Write Any four properties of normal distribution
5. Explain the term standard error.
6. State Central Limit Theorem.
7. State Type – I and Type – II error.
8. Explain the different type of errors in hypothesis testing
9. State the assumptions made in analysis of variance.
10. Distinguish between np chart and p chart.

SECTION B

Answer any FIVE questions:                                                                                        (5 X 8 = 40 Marks)

1. From the following data, prepare a 2X2 table and using Yule’s coefficient of association, discuss

Whether there is association between literacy and unemployment.

Literate unemployed 220 persons

Literate employed 20 persons

Literate employed 180 persons

Total number of persons 500.

1. State and prove multiplication theorem.
2. Student A can solve a problem in statistics in 4 out of 5 chances and B can do it in 2 out of 3 chances

If both A and B try the problem, find the probability that the problem will be solved.

1. After correcting the proofs of the first 50 pages of a book, it is found that on the average there are 3

errors per 5 pages. Use poisson probabilities and estimate the number of pages with 0,1,2,3 errors in

the whole book of 1000 pages (e-0.6=.5488)

15.What is Sampling Technique ? Explain different types of Sampling.

1. Out of 8000 graduates in a town,800 are females and out of 1600 graduate employees 120 are

females. Use  Chi-square to determine if any distinction is made in appointment on the basis of sex?

Test at 5% level.

1. Explain the various types of control charts.

1. You are given below the values of sample mean (X) and the range (R) for ten samples of size 5

Each. Draw mean and range charts and comment on the state of control of the process.

Sample No:   1     2       3     4      5      6       7        8       9         10

X:  43    49    37    44    45    37      51     46     43        47

R:    5       6      5      7      7      4        8       6      4          6

You may use the following :(for  n=5, A2=0.58, D3=0, D4=2.11)

SECTION   C

Answer any TWO questions:                                                                                    (2 X 20  =  40 Marks)

19.(a) Given    (ABC) = 137;   (αBC) = 261; (AβC) = 313; (ABg) = 284; (Abg) = 417; (αBg) = 420;

(αbC)  =  490; (abg)  =  508; Find the frequencies (AB), (A) and N.                       (10)

19.(b) )   There are 3 boxes containing respectively 1 White,2 Red, 3 block; 2 white,3 red, 1 black ball;

3 white , 1 red  and 2 black ball. A box is chosen at random and from it two balls are drawn

At random. The two balls are 1 red and 1 white. What is the probability that they come from

(i) The first box (ii) second box  (iii) third box.                                                                   (10)

1. (a) The customer accounts of a certain departmental store have an average balance of Rs.120 and a

standard deviation of Rs.40. Assuming that the account balances are normally distributed, find

• What proportion of accounts is over Rs.150?
• What proportion of accounts is between Rs.100 and Rs.150?
• What proportion of accounts is between Rs.60 and Rs.90 ?                                 (10)

1. (b) Random samples of 400 men and 600 women were asked whether they would like to have a fly-

over      near  their residence 200 men and 325 women were in favor of it. Test the equality of

proportion of men and    women in the proposal? Test at 5% level.                                       (10)

21.(a) The marks obtained by a group of 9 regular course students and  another group of 11 part- time

course students in a test are given below:

 Regular 56 62 63 54 60 51 67 69 58 Part time 62 70 71 62 60 56 75 64 72 68 66

Examine whether the marks obtained by regular students and part time students differ significantly at

5% level.                                                                                                                                           (10)

1. (b) The number of defects defected in 20 items are given below

Item No     :  1   2   3    4     5    6   7    8     9    10    11     12    13    14   15   16   17    18   19   20

No. of defects:  2    0   4   1      0     0   8     1    2     0      6        0     2      1    0      3     2       1    0     2

Test whether the process is under control. Device a suitable scheme for future.                        (10)

1. Perform two-way ANNOVA for the data given below:
 Treatment Plots of Land I II III A 38 45 40 B 40 42 38 C 41 49 42 D 39 36 42

Using coding method subtracting 40 from the given number.                                                           (20)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

# NO 14

B.Com. DEGREE EXAMINATION – COMMERCE

FOURTH SEMESTER – APRIL 2008

# ST 4205 / 4200/3202 – ADVANCED STATISTICAL METHODS

Date : 24/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

# SECTION A

Answer all questions.                                                                                      (10 x 2 = 20)

1. Check the consistency of the following data:

N = 100, (A) = 48, (AB) = 24, (β) = 35.

1. Define Partial association.
2. Consider an experiment of throwing a die once. Let A be the event of getting an odd number and B be the event of getting a prime number. Verify whether A and B are mutually exclusive and exhaustive.
3. Let X be a Poisson random variable satisfying P(X=2) = 2P(X=3). Find the mean and variance of X.
4. Let Z be a standard normal random variable. Find P(Z > 1.2) and P(0 < Z < 0.6).
5. What is meant by standard error?
6. Define Type I error and Type II error in hypothesis testing.
7. Find the missing values in the following ANOVA table:

Source             df        Sum of squares            Mean sum of squares

Treatments      4          118                              ?

Blocks             ?          201                              ?

Error                10        ?

Total                20        500

1. Mention the difference between variable and attribute control charts.
2. What are control limits?

### SECTION B

Answer any FIVE questions.                                                                    (5 x 8 = 40)

1. a.) Define Yule’s coefficient of association.                                                                                 b.) A teaacher examined 280 students in Economics and Auditing and found that 160 failed in Economics, 140 failed in Auditing and 80 failed in both the subjects. Calculate Yule’s coefficient of association between failure in Economics and Auditing and interpret the result.
2. a.) A bag contains 10 white and 6 black balls. 4 balls are successively drawn out and not replaced. What is the probability that they are alternately of different colors?

b.) In a single throw of a die, what is the probability of obtaining a total of atleast 10?

1. Suppose 300 misprints are distributed randomly throughout a book of 500 pages. Find the probability that a given page contains a.) exactly 2 misprints  b.) no misprints and c.) 2 or more misprints.
2. Consider a population containing 5 values namely 12, 14, 10, 15, 12. Draw all possible random samples of size 2 from this population and obtain the sampling distribution of mean. Verify whether the sample mean is an unbiased estimator of the population mean.
3. Explain the procedure of testing the equality of proportions of two populations.
4. The number of units of a product sold in six shops before and after a promotional campaign are shown below:

Shops: A         B         C         D         E          F

Before campaign:  53        28        31        48        50        42

After campaign:  58        29        30        55        56        45

Can the campaign be judged to be a success? Test at 5% level.

1. Explain the various steps in performing a One – way Analysis of Variance.
2. 20 tape recorders were examined for quality control test. The number of defects for each tape recorder are given below:

2, 4, 3, 1, 1, 2, 5, 3, 6, 7, 3, 1, 4, 2, 3, 1, 6, 4, 1 and 1. Construct a suitable control chart and interpret it.

# SECTION C

1. a.) Given the following data, find frequencies of i.) the remaining positive classes and ii.) ultimate classes.

N = 1800, (A) = 850, (B) = 780, (C) = 326, (ABγ) = 200, (AβC) = 94,

(αBC) = 72 and (ABC) = 50.

b.) A manufacturing firm produces units of a product in four plants. Define event

Ai : a unit is produced in plant i, i = 1,2,3,4 and event B: a unit is defective. From

the past records of the proportions of defectives produced at each plant the

following conditional probabilities are set:

P(B|A1) = 0.05, P(B|A2) = 0.10, P(B|A3) = 0.15 and P(B|A4) = 0.02.

The first plant produces 30% of the units of the product, the second 25%, the third

40% and fourth 5%. A unit of the product made at one of these plants is tested

and found to be defective. What is the probability that the unit was produced either

in plant 1 or plant 3.                                                                                   (14+6)

1. a.) A fair coin is tossed four times. Let X denote the number of heads occurring. Find i.) the distribution function of X, ii.) expectation and variance of X.

b.) Suppose the weights of 2000 male students are normally distributed with mean 155 pounds and standard deviation 20 pounds. Find the number of students with weights: i.) less than 100 pounds  ii.) between 150 and 175 pounds

and iii.) more than 200 pounds.                                                            (12+8)

1. a.) Construct and R charts for the following data:

Sample:      1          2          3          4          5          6          7          8

X1:      32        28        39        50        42        50        44        22

X2:      37        32        52        42        45        29        52        35

X3:      42        40        28        31        34        21        35        44

b.) The life time (in thousand hours) of electric bulbs based on a random sample of 10 from a large consignment gave the following data:

Unit:    1          2          3          4          5          6          7          8

Life time:    4.2       4.6       3.9       4.1       5.2       3.8       3.9       4.3

Unit:    9          10

Life time:    4.4       5.6

Test at 5% level, the hypothesis that the mean life time of bulbs in the entire

consignment is 4000 hours.                                                                          (12+8)

1. Three types of indoor lighting A1, A2 and A3 were tried on three types of flowers B1, B2 and B3. The average heights (in cm’s) after 12 weeks of growth are indicated in the following table:

Flowers

Lightning        B1        B2        B3

A1        16        24        19

A2        15        25        18

A3        21        31        15

Test at 5% level whether there is significant difference in growth due to lightning and due to

flower type.

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# BA 31

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.COM. DEGREE EXAMINATION – COMMERCE

THIRD SEMESTER – November 2008

# ST 3202/ST3200/4202 – ADVANCED STATISTICAL METHODS

Date : 13-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A                                           (10 X 2 = 20 marks)

1. Explain the terms of attributes.
2. What are the differences between quota sampling and stratified sampling?
3. state the Axioms of the probability
4. Define conditional probability.
5. Define Binomial and poisson distribution.
6. Distinguish between null and alternative hypothesis
7. State central limit theorem
8. Explain the term standard error.
9. Explain the various types of control chart
10. Costruct the ANOVA table of two-way classification

SECTION B                                              (5 X 8 = 40 Marks)

1. 800 candidates of both sex appeared at an examination. The boys out numbered the girls by 15 %

of  the total. The number of candidates who passéd exceed  the number failed by 480. Equal

number of boys and girls failed  in the examination. Prepare a 2×2 table and find the coefficient

of association and Comment.

1. A can solve a problem of statistics in 4 out of 5 chances and B can do it in 2 out of 3 chances

If both A and B try the problem. Find the probability that the problem will be solved.

1. If 3% of the electric bulbs manufactured by a company are defective, find the probability that in a

Sample of 100 bulbs exactly five bulbs are defective ( e -3  = 0.0498)

1. A random sample of 200 tins of coconut oil gave an average weight of 4.95 kgs with a standard

Deviation of 0.21kg. Do we accept the hypothesis of net weight 5kg per tin at 1% level.

1. In a survey of 200 boys, of which 75 intelligent, 40 had skilled fathers while 85 of the

Unintelligent  boys has unskilled fathers. Do these figures support the hypothesis that

Skilled fathers have intelligent boys. Use x2 –test of 5 % level.

1. Distinguish between np-chart and c- chart

1. You are given below the values of sample mean (X) and the range (R) for ten samples of size 5

Each. Draw mean and range charts and comment on the state of control of the process.

Sample No:   1     2       3     4      5      6       7        8       9         10

X:  43    49    37    44    45    37      51     46     43        47

R:    5       6      5      7      7      4        8       6      4          6

You may use the following control chart constraint for n = 5, A2 = 0.58, D3 =  0 ,  D4 = 2.11

1. State and prove Bolle’s inequality

SECTION   C                                   (2 X 20  =  40 Marks)

1. (a) Given (ABC) = 137; (αBC) = 261; (ABC) = 313; (Aβr) = 284; (Abr) = 417; (αBr) = 420;

(αbC)  =  490; (αbr)  =  508; Find the frequencies (AB), (A) and N.

(b)   Explain the procedure generally followed in testing of hypothesis.

1. (a) There are 3 boxes containing respectively 1 White,2 Red, 3 block; 2 white,3 red, 1 black ball;

3 white , 1 red  and 2 black ball. A box is chosen at random and from it two balls are drawn

At random. The two balls are 1 red and 1 white. What is the probability that they come from

(i) The first box (ii) second box  (iii) third box.

(b)   If 10% of the screws produced by an automatic machines are defectives, find the probability

That of 20 screws selected at random there are (i) exactly two defectives

(ii)at the most three defectives  (iii) at least two defectives.

21.(a) The lives of 12 cars manufactured by two companies A and B are given below in years

 X 14 15 18 12 18 17 19 21 19 16 12 11 Y 21 18 14 22 23 19 20 16 16 13 20 14

Which company will you choose to purchase a car? Give reason. Test at 5% level of significance.

(b) The data given below relate to two random samples of employees from the different states

Mean                Variance               Size

State I                28                        40                     16

State II                19                       42                     25

Test the hypothesis that variance of the populations are equal.

1. Prepare a Two- way ANOVA on the data given below.

Treatment I

 I II III A 30 26 38 B 24 29 28 C 33 24 35 D 36 31 30 E 27 35 33

Treatment I I

Use the coding method, subtracting 30 from the given numbers.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 YB 09

B.Com. DEGREE EXAMINATION – COMMERCE

THIRD SEMESTER – April 2009

# ST 3202 / 3200 – ADVANCED STATISTICAL METHODS

Date & Time: 27/04/2009 / 9:00 – 12:00  Dept. No.                                                  Max. : 100 Marks

PART-A

Answer all the questions                                                                               ( 10 x 2 = 20)

1) Explain the difference between an attribute and a variable with examples.

2) Name the methods used to study the association of attributes.

3) A bag contains 8 white and 4 red balls. Five balls are drawn at random. What is the

probability that 2 of them are red and 3 are white?

4) Define probability of an event.

5) For a binomial distribution mean is 20 and variance is 16. Find n, p.

6) What is the standard error of the sample mean and the sample proportion?

7) What is significance level?

8) What are the control limits for a c- chart?

9) Write the 95% confidence interval for the sample mean when n is large.

10) State any two uses of chi-square distribution.

PART-B

Answer any 5 questions                                                                         (  5 x 8 = 40 )

11) Find the association between literacy and unemployment from the following data.

Literates                                  1290

Unemployed                           1390

Literate unemployed               820

Comment on the results.

12) One bag contains 4 white, 2 black and 3 blue balls. Another contains 3 white 5 black and 2 blue

balls. One ball is drawn from each bag.

Find the probability that   a) both are white

1. b) one is white and one is black
2. c) the sample should not have any blue balls.

13) A certain automatic machine produces one defective screw out of every 100 screws.

If the screws are packed in boxes of 300, what percentage of these boxes

would you expect to have

1. i) no defective screw?
2. ii) at least one defective screw

iii) not more than 2 defectives.

14) Explain the method of analysis of variance for one way classification.

15) Explain the theory behind control charts.

16) From the following data find out whether there is any relationship between sex and                preference of color for 200 samples.

Color                           males                           females

Red                             10                                40

White                          70                                30

Green                          30                                20

17) 15,000 students appeared for an examination. The mean marks obtained are 49 and the standard

deviation is 6. Assuming normal distribution, what proportion of students scored more

than 55 marks? If grade A is given to those who scored above 70, what proportion of students

18) Ten specimens of copper wires drawn from two large lots have the following breaking strengths

(in kgs) 578, 572, 570, 568, 512, 578, 570, 575, 569, 548.

Test whether the mean breaking strength of the lot may be taken to be 578 kg.

PART-C

Answer any 2 questions                                                                         ( 2 x 20 = 40 )

19) a) A survey of male children in 128 families each having 5 children gave the following data.

 No of male children 0    1      2      3       4      5 No of families 9   17    26    39    22     12

Fit a binomial distribution to the data assuming p is not known.

1. b) In a bolt factory machines A, B, and C. produce respectively 25%, 35%, and 40%. Of the total of

their output 5, 4 and 2 percent are defective bolts. A bolt is drawn at random from the product

and is found to be defective. What is the probability that  it was manufactured by machines

A, B and C.?                                                                                                                   ( 10 + 10 )

20) a) You are working as a purchase manager for a company. The following

information has been supplied to two manufactures of electric bulbs.

 Company A Company B Mean life(hours) 1275 1248 SD 82 93 Sample size 100 100

Test whether there is any significant difference between the mean of  two products.

1. b) Before an increase in excise duty on tea 400 people out of 500 were found to be

tea drinkers. After an increase in the duty 400 persons were known to be tea drinkers

in another independent sample of 600 people. Test whether there is any

significant difference between the two cases?                                           ( 10 +10 )

21) a)100 children took three examinations. 40 passed the first, 39 passed the second

and 48 passed the third. 19 passed all three 9 passed first two but failed in the third,

19 failed in the first two and passed the third. Find how many children passed

at least two exams.

1. b) For a random sample of 10 persons fed on diet A, the increase in weights are

10, 6, 16, 17, 13, 12, 8, 14, 15, 9.For another random sample of 12 persons fed on

diet B the increase in weights are 7, 13, 22, 15, 12, 14, 18, 8, 21, 23, 10, 17.

Test whether there is any significant difference between the diets.           ( 10 + 10 )

22) a) The following data show the values of sample mean X and range R for 10 samples

of size 8 each. Calculate the control limits for mean and range

 Sample no 1 2 3 4 5 6 7 8 9 10 mean 11.2 11.8 10.8 11.6 11 9.6 10.4 9.6 10.6 10 range 7 4 8 5 7 4 8 4 7 9

Determine whether the process is in control.

1. b) A tea company appoints 4 salesmen A, B, C, and D and observes their sales in

three seasons summer, winter, monsoon. The figures are given below.

 A B C D summer 36 36 21 35 winter 28 29 31 32 monsoon 26 28 29 29

Test whether there is significant difference

1.  i) among salesmen
2. ii) among seasons.                                                                  ( 10 + 10 )

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 YB 14

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.COM. DEGREE EXAMINATION – COMMERCE

FOURTH SEMESTER – April 2009

# ST 4205/ ST 4200 – ADVANCED STATISTICAL METHODS

Date & Time: 27/04/2009 / 9:00 – 12:00          Dept. No.                                                  Max. : 100 Marks

PART-A

Answer all the questions                                                                     (10 x 2 = 20)

• When do you say that two attributes are independent?
• From the following data find out the missing frequencies. (AB) = 100, (A) = 300, N = 1000,

(B)=600.

• A bag contains 5 white and 3 black balls. Two balls are drawn one by one without replacement.

What is the probability that both are white?

• State the addition theorem for two events.
• State the properties of normal distribution
• What is standard error?
• Define type I error and type II error.
• What are the control limits for a p- chart?
• Explain briefly the term confidence interval.
• State any two uses of t- distribution.

PART-B

Answer any 5 questions                                                                                          (5 x 8 = 40)

• According to a survey the following results were obtained.
 No of students appeared 800 married 150 Married and successful 70 Unmarried and successful 550

Compute Yules coefficient of association and comment on the results.

• A committee of 4 persons is to be appointed from 3 Indians 4 Americans and 2 Japanese and

1 Pakistani.  Find the probability of forming the committee in the following manner.

1. there must be 1 from each group
2. it should have atleast 1 from Americans
• Pakistani must be there in the committee.
• The incidence of a certain disease is such that on the average 20%of the workers suffers

from it.  If 10 workers are selected at random, find the probability that are

1. exactly two workers will suffer from the disease
2. no worker will suffer from the disease
• not more than two workers will suffer from the disease
• Explain the method of analysis of variance for two way classification.
• Explain the theory of control charts.
• An experiment was conducted to study the effectiveness of a new drug. 300 patients were

treated with new drug and 200 were not treated with the drug. The results of the experiment

are given below.

 Details cured Condition worsened No effect Treated with drug 200 40 60 Not Treated with drug 120 30 50

Test the effectiveness of the drug.

• In a distribution exactly normal, 5% of the items are under 35 and 65% are under 63.

What is the mean and SD of the distribution?

• The wages of 10 workers taken at random from a factory are given below.

578, 572, 570, 568, 572, 578, 570, 572, 596, 584.

Is it possible that the mean wage of all workers of this factory is Rs 580?

PART-C

Answer any 2 questions                                                                                            (2 x 20 = 40)

• a) The following table gives the number of days in a 50-day period during which

automobile accidents occurred in a city.

 No of accidents 0 1 2 3 4 No of days 21 18 7 3 1

Fit a Poisson distribution to the data.

1. b) A company has two plants to manufacture scooters. Plant I manufactures 70% of the

scooters and plant II manufactures 30%. At plant I 80%of the scooters are rated standard

quality and at plant II 90% of the scooters are rated standard quality. A scooter is picked up

at random and is found to be of standard quality. What is the chance  that it has come from

plant I or plant II .                                                                                                      ( 12 + 8 )

• a) Intelligence test of two groups of boys and girls  gave the following results:

 Mean SD Sample size girls 75 15 150 boys 70 20 250

Test whether then mean marks of boys and girls are same.

1. b) In a  random sample of 600 men taken from a big city 400 are found to be smokers. In another

random sample of 900 men taken from another city 450 are smokers. Do the data indicate

there is a significant difference in the habit of smoking in the 2 cities?                   ( 10 + 10 )

21) a) Find all the ultimate class frequencies from the following data.

N = 800, (A) = 224,   (B) = 301, (C) = 150,   (AB) = 125, (AC) = 72, (BC) = 60, (ABC) = 32.

1. b) The following data show weekly sales before and after recognition of the sales organization.

 Week no 1 2 3 4 5 6 7 8 9 10 Sales before 15 17 12 18 16 13 15 17 19 18 Sales after 20 19 18 22 20 19 21 23 24 24

Test whether there is any significant difference in sales before and after recognition of the

sample company.                                                                                                     ( 10 + 10 )

22) a) Assume that 15 litre milk bottles are selected at random from a process. The number

of air bubbles (defects) observed from the bottles is given below.

Draw a suitable control chart.

 Bottle Number Number of defects 1 4 2 5 3 7 4 3 5 3 6 5 7 6 8 2 9 4 10 8 11 3 12 5 13 4 14 3 15 4

1. b) A manufacturing company has purchased 3 new machines of different makes and wishes

to determine whether one of them is faster than the other in producing a certain output.

Five hourly production figures are observed at random from each machine and the results

are given below.

Machines

 Makes A1 A2 A3 1 25 31 24 2 30 39 30 3 36 38 28 4 38 42 25 5 31 35 28

Use analysis of variance and test whether there is any significant difference among the

machines.                                                                                                                    ( 10 + 10 )

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 AC 10

B.A.  DEGREE EXAMINATION – ECONOMICS

FOURTH SEMESTER – APRIL 2007

ST 4200 / 3200 – ADVANCED STATISTICAL METHODS

Date & Time: 25/04/2007 / 9:00 – 12:00        Dept. No.                                                     Max. : 100 Marks

PART-A

Answer all the questions.                                                                            10×2=20 marks

1. Define dichotomous and manifold classification.
2. For two attributes A and B if (AB)=20, (Aβ)=25 ,(αB)=32 and N=100 ,find

(A) and (B)

1. Write the conditions for the consistency of data for three attributes.
2. State the axiomatic definition of probability.
3. When do you say that three events A,B and C are independent?
4. Define binomial distribution.
5. What are TypeI and TypeII errors in testing of hypothesis.
6. Write any two applications of chi-square distribution.
7. Write a note on chance and assignable causes in quality control.
8. State the control limits for p chart.

PART-B

Answer any five questions.                                                                           5×8=40 marks

1. Can vaccination be regarded as a preventive measure for small-pox from the

data given below?

“Of 1482 persons in a locality exposed to small-pox ,368 in all were attacked.”

“Of 1482 persons ,343 had been  vaccinated and of these only 35 were attacked.”

1. For n attributes A1,A2,…An, show that

(A1 A2,…An)(A1) + (A2) +…. +(An)-(n-1)N, where N is the total number of

observations.

1. If 10 fair coins were tossed simultaneously, find the probability of getting

1. If X is Poisson variate such that

P(X=2)=9 P(X=4) + 90 P(X=6)  find mean and variance.

15 (a) If X follows normal distribution with mean show that   and variance

then show that (X-)/ follows standard normal distribution.

(b) Write any four characteristics of normal distribution.

1. A random sample of 10 boys had the following

I.Q’s:70,120,110,101,88,83,95,107,100.

Do these data support the assumption of a population mean I.Q.of 100?

Use 5% level of significance.

1. The mean height of 50 male students who showed above average participation in

college athletics was 68.2 inches with a standard deviation of 2.5 inches ;while 50

male students who showed no interest in such participation had a mean height of

67.5 inches with a   standard deviation of 2.8 inches.Test the hypothesis that male

students who participate in   college athletics are taller than other male students. Use

1% significance level.

18.Draw a c chart for the following number of defects found in welding of seams:

2  4  7  3  1  4  8  9  5  3  7  11  6  4  9  9  6  4  3  9  7  4  7  12. Check whether the

process is in control .

PART-C

Answer  any two questions.                                                                      2×20 = 40 marks

1. (a) Establish the relationship between Yule’s coefficient of association and

coefficient of colligation.

(b) Given the following data find the postive classes:

(ABC) = 148, (AB)=738  (AC)=225  (A)=1196  (BC)=204

(B)=1762   (C)=171  and ()=21842.

(c) Among the adult population of a certain town 50 % are males ,60%are

wage earners  and 50% are 45 years of age  or over,10%of the males are not

wage earners and 40% of the males are under 45.Make the best possible

inference about the limits within which the percentage of persons(male or

female) of 45 years or over are wage earners .

20.(a). Fit a Poisson distribution to the following data and test for the goodness of fit:

No.of mistakes/ page:  0           1                 2                3             4

No. of pages:                109       65               22             3              1

Use 1% significance level.

(b). If X is a normal variate with mean 30 and S.D. 5,find the probability of

(i) 26 ≤ X ≤ 40     (ii) X  45    (iii)|X-30| >5.

1. Analyze the following data at 1% significance level:

Treatments

1              2              3               4                  5                6

Blocks

1                    24.7          20.6        27.7           16.2            16.2           24.9

2                     27.3          28.8       22.9           15.0            17.0           22.5

3                     38.5          39.5        36.5           19.6           15.4           26.3

4                     28.5          31.0         34.9           14.1           17.7          22.6

22.(a).Consruct a control chart for mean and the range for the following data on the

basis of fuses ,samples of 5 being taken every hour . Comment on whether the

production seems to be under control ,assuming that these are the primary data.

42   42   19   36   42   51   60   18   15   69   64   61

65   45   24   54   51   74   60   20   30   109  90  78

75   68   80   69   57   75   72   27   39   113   93   94

78   72   81   77   59    78   95   42   62   118   109  109

87   90   81   84   78   132   138   60   84  153  112  136

(b).The following are the figures of defectives in 22 lots each containing 2000 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.

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