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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