LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
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FIFTH SEMESTER – APRIL 2008
ST 5503 – COMPUTATIONAL STATISTICS
Date : 06/05/2008 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
Answer all questions. Each carries 34 marks.
1.(a) For the following data:
Commodity Base year Current year
Kg. Rate(RS.) Kg. Rate(RS.)
Rice 10 20 14 2
Oil 12 62 16 80
Wheat 14 10 18 19
Ghee 12 100 14 180
Tea 16 150 18 200
Find (i) Laspyre (ii) Paasche (iii)Dorbish-Bowley (iv) Marshall-Edgeworth and (v)Fisher price and quantity index numbers.(10 marks)
(b) Fit a trend line by the method of least squares for the following data:
Year : 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003
Sales(Crores): 12 14 18 20 26 25 30 32 38 42
Also estimate the trend values for the years from 2004 to 2010.Further compute 3 year
and 4 year moving averages.(14 marks)
(c) The following figures show the distribution of digits in numbers chosen at random
from a telephone directory:
Digits : 0 1 2 3 4 5 6 7 8 9
Frequency: 1026 1107 997 966 1075 933 1107 972 964 853
Test whether the digits may be taken to occur equally frequently in the directory.Use
.05 level of significance.(10 marks)
[OR]
(d) Identify the monthly seasonal indices for the 3 years of expenses for a six-unit
apartment house in southern Florida as given here.Use a 12-month moving average
calculation.
Expenses
Month Year1 Year2 Year3
January 170 180 195
February 180 205 210
March 205 215 230
April 230 245 280
May 240 265 290
June 315 330 390
July 360 400 420
August 290 335 330
September 240 260 290
October 240 270 295
November 230 255 280
December 195 220 250 (20 marks)
(e)The following table gives probabilities and observed frequencies in four classes AB,Ab,aB and ab in a genetical experiment. Estimate the parameter by the method of maximum likelihood and its standard error.
Class Probability Observed frequency
AB ¼(2+) 108
Ab ¼(1-) 27
aB ¼(1-) 30
ab ¼ () 8 (14 marks)
2(a) The National Association of Home Builders provided data on the cost of the most
popular home remodeling projects.Sample data on cost in thousands of dollars for
two types of remodeling projects are as follows.
Kitchen Master Bedroom
- 0
- 9
- 4
- 8
- 9
- 8
- 6
- 0
21.8
23.6
Develop a 95% confidence interval for the difference between the two Population means. (10 marks)
(b)Two independent samples of 8 and 7 items respectively had the following values:
Sample1: 9 11 13 11 15 9 12 14
Sample2: 10 12 10 14 9 8 10
Is the difference between the means of samples significant ? Test at 1% level of significance . (14 marks)
(c) The Dow Jones Industrial Average varies as investors buy and sell shares of the
30 stocks that make up the average.Samples of the Dow Jones Industrial Average
taken at different times during the first 5 days of November 1997 and the first 5
days of December 1997 are as follow:
November December
- 8066
- 8209
- 7842
- 7943
- 7846
- 8071
- 8055
- 8159
- 7828
- 8109
Using a .05 level of significance, test to determine whether the population variances for
the two time periods are equal. (10 marks)
[or]
(d) Fit a Poisson distribution to the following data and test the goodness of fit:
No.of accidents: 0 1 2 3 4 5 6
No. of days : 150 65 45 34 10 6 2 (14 marks)
(e) One of the questions on the Business Week 1996 Subscriber Study was ,”In the past
12 months ,when travelling for business, what type of airline ticket did you purchase
Most often ?” The data obtained are shown in the following contingency table.
Type of Flight
——————-
Type of Ticket Domestic Flights International Flights
First class 29 22
Business/executive class 95 121
Full fare economy/coach class 518 135
Using=.05 ,test for the independence of type of flight and type of ticket.(10 marks)
(f) A test was conducted of two overnight mail delivery services. Two samples of
identical deliveries were set up so that both delivery services were notified of the
need for a deliveryat the same time.The hours required to make each delivery
follow. Do the data shown suggest a difference in the delivery times for the two
services ? Use Wilcoxon signed ranks for the test at 5% significance level. The
data follows:
Service
————————–
Delivery 1 2
1 24.5 28.0
2 26.0 25.5
3 28.0 32.0
4 21.0 20.0
5 18.0 19.5
6 36.0 28.0
7 25.0 29.0
8 21.0 22.0
9 24.0 23.5
10 26.0 29.5
11 31.0 30.0 (10 marks)
3 (a) Consider a population of 6 units with values : 2, 5, 8, 11,13 and 14
[i] Write down all possible samples of size 2 without replacement from this population
[ii] Verify that the sample mean is an unbiased estimate of the population mean
[iii] Calculate the Sampling variance and verify that it agrees with the variance of
the sample mean under SRSWOR.
[iv] Also , Verify that the Sampling Variance is less than the variance of the sample
mean which is obtained from SRSWR. [14]
(b) The data given below is for a small sheep population which exhibits a steady rising trend. Each column represents a systematic sample and rows represent the strata.
[i] Calculate sampling variance under systematic sampling
[ii] Calculate sampling variance under stratified sampling
[iii] Calculate sampling variance for without stratification and without replacement
[iv] Compare the precision.
Sample Number
Stratum # |
1 | 2 | 3 | 4 |
| I | 5 | 7 | 9 | 11 |
| II | 8 | 11 | 14 | 14 |
| III | 9 | 13 | 15 | 17 |
| IV | 11 | 14 | 17 | 21 |
[20]
[OR]
[c] The table given below shows the summary of data for Paddy crop census of all the
2500 farms in a state. A sample of 125 farms is to be selected from this population
The farms were stratified according to farm size(in acres) into 4 strata as given below:
| Stratum
Number |
Farm
Size (inacres) |
No. of
Farms Ni |
Average area Under paddy crop (in acres) per farm
ŸNi |
Standard
Deviation si |
| 1 | 0-100 | 600 | 45 | 8 |
| 2 | 101-200 | 900 | 105 | 12 |
| 3 | 200-500 | 700 | 130 | 20 |
| 4 | >500 | 300 | 180 | 40 |
[i] Estimate the total area under Paddy cultivation for the state
[ii] Find the sample sizes of each stratum under proportional allocation.
[iii] Find the sample sizes of each stratum under Nayman’s Optimum allocation
[iv] Calculate the variance of the estimated total area under Proportional allocation
[v] Calculate the variance of the estimated total area under Nayman’s Optimum allocation
[vi] Calculate the variance of the estimated total area under un-stratified simple
random sampling without replacement.
[vii] Estimate gain in efficiency resulting from [iv] and [v] and compare them with [vi]. [34]
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