LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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

Answer any TWO questions.

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

A_i : 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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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)

     Answer ALL questions.         

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)

     Answer any FIVE questions

11. 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.

12. 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.

13. 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)

14. 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.

15. 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.

16. Distinguish between np-chart and c- chart

17. 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

18. State and prove Bolle’s inequality

SECTION   C                                   (2 X 20  =  40 Marks)

Answer any TWO questions

19. (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.

20. (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

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.

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

                                                         Treatment I

Treatment I I 

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

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       LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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.

            Total adults                             10,000

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

will receive grade A?

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.

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.

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

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.

Test whether there is significant difference

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

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          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.

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.

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.

Fit a Poisson distribution to the data.

19. 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:

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.

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.

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

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

machines.                                                                                                                    ( 10 + 10 )

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