ASSIGNMENT BOOKLET

Bachelor’s Degree Programme

STATISTICAL TECHNIQUES

(Valid from 1st July, 2013 to 31

st March, 2014)

School of Sciences

Indira Gandhi National Open University

Maidan Garhi

New Delhi-110068

2013-2014

AST-01

It is compulsory to submit the assignment before filling in the exam form.

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Dear Student,

Please read the section on assignments in the Programme Guide for Elective courses that we sent you

after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous

evaluation, which would consist of one tutor-marked assignment for this course. The assignment is in

this booklet.

Instructions for Formatting Your Assignments

Before attempting the assignment please read the following instructions carefully.

1) On top of the first page of your answer sheet, please write the details exactly in the following format:

ROLL NO: ……………………………………………

NAME: ……………………………………………

ADDRESS: ……………………………………………

……………………………………………

……………………………………………

COURSE CODE: …………………………….

COURSE TITLE: …………………………….

ASSIGNMENT NO. ………………………….…

STUDY CENTRE: ………………………..….. DATE: ……………………….………………...

PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND

TO AVOID DELAY.

2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.

3) Leave 4 cm margin on the left, top and bottom of your answer sheet.

4) Your answers should be precise.

5) While solving problems, clearly indicate which part of which question is being solved.

6) This assignment is valid only upto March, 2014. If you have failed in this assignment or fail to

submit it by 31st March, 2014, then you need to get the assignment for the January, 2014 cycle and

submit it as per the instructions given in that assignment.

7) It is compulsory to submit the assignment before filling in the exam form.

We strongly suggest that you retain a copy of your answer sheets.

We wish you good luck.

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Assignment (To be done after studying the course material)

Course Code: AST-01

Assignment Code: AST-01/TMA/2013-14

Maximum Marks: 100

1. a) A lot contains 50 defective and 50 non defective bulbs. Two bulbs are drawn at random

one at a time with replacement. The events A, B, C are defined as follows:

A = first bulb is defective, B = second bulb is non defective, C = The two bulbs are both

defective or both non defective. Determine whether A, B and C are independent. Also

calculate P ( ) ( ) ( ).BAPandCBP,CAA ∩∩∩ (5)

b) From a population of 20,000 observations, a sample of 500 observations is selected.

Calculate the standard error of sample mean if the population standard deviation equals

20. (3)

c) A random sample of 700 units from a large consignment showed that 200 were (2)

damaged. Find 95 % confidence interval for the proportion of damaged unit in the

consignment.

2. (a) Two floppies are selected at random without replacement from a box containing (5)

7 good and 3 defective floppies. Let A be the event that the first floppy drawn is

defective, and let B be the event that the second floppy drawn is defective.

(i) Find the conditional probabilities P(B/A) and P(B/AC)

(ii) Show that P(B) = P(B/A). P(A) + P(B/AC) P(A

C) = P(A).

(b) A company produces one-kilogram sugar packets. The specifications on the net (5)

content are 1000 ± 5 grams. Assuming that the net content follows normal

distribution with mean weight as 1005 grams and the process capability equal to 30 grams, find out the proportion of packets that have weight less than lower

specification limit. What should be the mean if this proportion is to be reduced to 0.01?

3. a) Two new types of petrol, called premium and super, are introduced in the market,

and their manufactures claim that they give extra mileage. Following data were obtained on extra mileage which is defined as actual mileage minus 10. (10)

(i) Using ANOVA, test whether premium or super give an extra mileage.

(ii) What is your estimate for the error variance?

(iii) Assuming that the error variance is known and is equal to 1, obtain the

95 % confidence interval for the mean extra mileage of super.

Data on extra mileage

Ordinary Petrol 1 2 2 1

Premium Petrol 2 2 1 3

Super Petrol 4 1 2 3

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4. a) The number of total accidents during the first half of 2008 are as given in the Table below. (7)

Data on Fatal Accidents

i) Draw the graph and comment.

ii) Prepare the table of forecasts and errors by applying simple exponential

smoothing for the data given in (i) (take the exponential smoothing constant

as 0.1).

b) Among a group of 50 students of class 12, five students are 16 years old, thirty (3)

students are 17 years old, five students are 18 years old and ten students are 19 years old. Let a student be drawn from this group and let X be the age of the

selected students ( in complete years).

(i) Is the distribution of X continuous? (ii) Obtain the probability density function of X.

(iii) What is the probability that the student selected is atleast 18 years old.

5. a) An Rx − chart is being set up for monitoring the twist angle of a certain paper (4)

board. Twenty subgroups (each of size 5) of data have been collected and their

summary is furnished in the table below.

(i) Scrutinize the data (without drawing charts) and check if there are any increasing or

decreasing trends in process average.

(ii) Assuming that there are no systematic patterns in the process, compute the control

limits for Rx − charts.

(iii) Do you suspect any assignable causes? If so, at what sample numbers?

b) Define stratified sampling and explain the basic principle that forms the strata. (3)

c) Suppose three small towns are under study, having population N1 = 60000, (3)

N2 = 30000, N3 = 30000, respectively. A stratified random sample is to be taken with

a total sample size of n = 500. Using proportional allocation, determine the sample size

of each stratum.

Month January February March April May June

No. of fatal

accidents

2 4 3 4 3 2

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

X 1− 3− 1 3− 7 0 4 2− 4 1

R 5 12 10 6 6 6 9 10 13 6

Sample No 11 12 13 14 15 16 17 18 19 20

X 2− 4 2− 1− 3− 1− 1 1− 0 0

R 6 25 16 9 4 6 17 11 9 6

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6. a) The frequency distribution of the daily cost (in Rs.) of commuting back and (5) forth to work by 100 employees of a steel plant is:

Find the mean daily cost, median daily cost and standard deviation of commuting.

b) A study was conducted to see whether the natural perception having equal number (5)

of boys and girls was followed by families or not. 160 families were considered

as sample. Each family had 5 children. Frequency of families having a particular

combination was recorded as follows.

Boy 5 4 3 2 1 0

Girl 0 1 2 3 4 5

Frequency 20 25 25 20 30 40

Test whether the standard norm is followed or not at 95 % level of significance.

7. a) An economist want to estimate a multiple regression equation in which the (3) amount saved by the ith family depends on the family’s income, whether the

family is graduated or non graduate, and whether the family is headed by a male or female. Explain how a regression equation of this sort can be estimated. What

is the dependent variable ? What are the independent variables? What assumption

must be made ?

b) A population of size 20 is sampled without replacement. The standard deviation (3)

of the population is 0.35. We require the standard error of the mean to be not

more than 0.15. What is the minimum sample size required for this?

c) In a population of size N = 5, the values of an attribute are 8, 3, 1, 11 and 4 (4)

corresponding to population members 1 to 5.

i) Write all different samples of size 2 when simple random sampling (without replacement) is practiced ?

ii) Verify that y (sample mean) is an unbiased estimate of y (population

mean) , using the above data.

Cost (in Rs.) 0-50 50-100 100-150 150-200

Number of employees 29 32 29 10

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8. a) For the following series of observations, compute the moving average of length 4 and place them in line with the corresponding year. (5)

Year 1995 1996 1997 1998 1999

Annual sales

(Rs. Crores)

2 6 1 5 3

Year: 2000 2001 2002 2003 2004 2005

Annual sales

(Rs. Crores)

7 2 6 4 8 3

b) The weight of ghee obtained from a tin of milk is uniformly distributed with a mean

of 8 kg and range of 1.5 kg. Calculate (5)

i) the probability that a tin of milk will yield ghee weighing between 7 kg and

8.5 kg.

ii) the largest and the smallest weights of ghee obtained from a tin of ghee.

9. a) A sample of size of 3 is to be selected from a population of 15 households. List all the Possible samples obtained by: (4)

i) Linear systematic sampling

ii) Circular systematic sampling

b) A random sample of 10 observations is taken from a normal population having the

variance 42.5. Find the probability of obtaining a sample having standard deviation

between 2.4 and 4.9. (4)

c) Give two examples from day to day life where cluster sampling can be used. Justify your

your choice of example. (2)

10. State whether the following statements are true or false. Give brief justification for (10)

each case.

a) Pie chart is more appropriate than bar chart for graphically presenting the profits

of five plants of a manufacturing company during the financial year 2007-08.

b) If an event A implies another event B, then A contains B.

c) In a standard normal distribution, the area under the curve between ∞−

and 0 is 1.

d) A sampling plan with AQL = 0.08 and LTPD = 0.05 is a good sampling plan.

e) The number of possible samples in drawing 3 items from a population of 7 items

without replacement is equal to 21.