LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
FIFTH SEMESTER – NOVEMBER 2003
ST-5503/STA508 - COMPUTATIONAL STATISTICS - I
10.11.2003
1.00 - 4.00
Max:100 marks
SECTION-A
Answer ALL questions.
1. a) In a survey conducted to estimate the cattle population in a district containing 120
villages, a simple random sampling of 20 villages was chosen without replacement.
The cattle population in the sampled villages is given as follows: 150, 96, 87,101, 56,
29, 120, 135, 141, 140, 125, 131, 49, 59, 105, 121, 85, 79, 141, 151. Obtain an
unbiased estimator of the total cattle population in the district and also estimate its
standard error.
b) The data given in the table represents the summary of farm wheat census of all the
2010 farms in a region. The farms were stratified according to farm size in acres into
seven strata. (i) Calculate the sampling variance of the estimated area under wheat for
the region from a sample of 150 farms case (a) If the farms are selected by the method
of SRS without stratification. Case (b): The farms are selected by the method of SRS
within each stratum and allocated in proportion to 1) number of farms in each stratum
(Ni). And 2) product of Ni Si . Also calculate gain in efficiency resulting from case (b)
1 and 2 procedures as compared with unstratified SRS.
Average Area
Standard
Stratum
Farm Size (in
No.of farms
under wheat
number
acres
(Ni)
deviations (I)
(14)
y 
Ni
1
2
3
4
5
6
7
0-40
41-80
81-120
121-160
161-200
201-240
More than 240
394
461
391
334
169
113
148
5.4
16.3
24.3
34.5
42.1
50.10
63.8
8.3
13.3
15.1
19.8
24.5
26.0
35.2
(20)
c) Consider a population of 6-units with values 1,5,8,12,15 and 19. Writ down all possible
samples of size 3 without replacements from the population and verify that the sample
mean is an unbiased estimator of the population mean. Also i) calculate the sampling
variance and verify that it agrees with the formula of variance of the sample mean. (ii)
Verify that the sampling variance is less than the variance of the sample mean from
SRSWOR.
d) Five samples were collected using systematic sampling from 4-different pools located in a
region to study the mosquito population, where the mosquito population exhibits a
fairly steady raising trend. i] Find the average mosquito population in all 4-poolss
ii] Find sample means iii] Compare the precision of systematic sampling, SRSWOR and
stratified sampling.
Pool no
Systematic Sample Number
1
2
3
4
5
I
2
5
6
8
10
II
4
8 10 11
13
III
8
10 11 13
14
IV
16 18 19 20
22
(14)
(20)
1
2. a) The following is a sequence of independent observations on the random variable X with the
density function
f(x ; 1, 2) =
1

1
(   2 x ) 2
e 2 1
, 1 , 2  0 , x  0 .
x
2x3
The observations are 1.57 0.37 0.62 1.04 0.21 1.8 1.03 0.49 0.81 0.56. Obtain the
maximum likelihood estimates of 1 and 2 .
(15)
b) Obtain a 95% confidence interval for the parameter  of the Poisson distribution based
on the following data:
No. of blood corpuscles :
0
1
2
3
4
5
No. of cells
:
142
156
96
27
5
1
(12)
c) Find a 99% confidence interval for  if the absolute values of the random sample of 8
SAT scores (scholastic Aptitude Test) in mathematics assumed to be N(, 2) are 624,
532,565,492, 407, 591, 611 and 558.
(7)
(OR)
d) The following data gives the frequency of accidents in Chennai City during 100 weeks.
No of accidents:
0
1
2
3
4
5
No. of weeks:
25
45
19
5
4
2
1  e  x e   x 
If P(X = x) = 

 ,  ,   0,
2  x!
x! 
x = 0 ,1, 2,….
estimate the parameters by the method of moments.
(12)
e) The following is a sample from a geometric distribution with the parameter p. Derive a
95% confidence interval for p.
x:
0
1
2
3
4
5
f:
143
103
90
42
8
14
(5)
2
f) An absolute sample of 11 mathematical scores are assumed to be N (,  ). The
observations are 26, 31, 27,28, 29, 28, 20, 29, 24, 31, 23.
Find a 99% confidence interval fo .
(7)
3. a) A vendor of milk products produces and sells low fat dry milk to a company that uses it to
produce baby formula. In order to determine the fat content of the milk, both the company and
the vendor take a sample from each lot and test it for fat content in percent. Ten sets of paired
test results are
Company Test
Vendor Test Results
Lot number
Results (X)
(Y)
1
0.50
0.79
2
0.58
0.71
3
0.90
0.82
4
1.17
0.82
5
1.14
0.73
6
1.25
0.77
7
0.75
0.72
8
1.22
0.79
9
0.74
0.72
10
0.80
0.91
Let D = X - Y and let mD denote the median of the differences.
Test H0 : mD = 0 against H1 : mD > 0 using the sign test. Let  = 0.05 approximately.
(14)
2
b) Freshmen in a health dynamics course have their percentage of body fat measured at the
beginning (x) and at the end (y) of the semester. These measurement are given for 26
students in Table below. Let m equal the median of the differences, x - y. Use the
Wilcoxon statistic to test the null hypothesis H0 : m = 0 against the alternative
hypothesis H1 : m > 0 at an approximate  = 0.05 significance level.
X
35.4
28.8
10.6
16.7
14.6
8.8
17.9
17.8
9.3
23.6
15.6
24.3
23.8
22.4
23.5
24.1
22.5
17.5
16.9
11.7
8.3
7.9
20.7
26.8
20.6
25.1
Y
33.6
31.9
10.5
15.6
14.0
13.9
8.7
17.6
8.9
23.6
13.7
24.7
25.3
21.0
24.5
21.9
21.7
17.9
14.9
17.5
11.7
10.2
17.7
24.1
20.4
21.9
(20)
c)
(OR)
A certain size bag is designed to hold 25 pounds of potatoes. A former fills such bags in the
field. Assume that the weight X of potatoes in a bag is N (,9). We shall test the null
hypothesis Ho :  = 25 against the alternative hypothesis H1 :  < 25. Let X1,X2 , X3, X4 be a
random sample of size 4 from this distribution, and let the critical region for this test be
defined by x  22.5 , where x is the observed value of X .
(a) What is the power function of this test?. In particular, what is the significance
level of this test? (b) If the random sample of four bags of potatoes yielded the values
x1 = 21.24, x 2 = 24.81 , x3 = 23.62, x 4 = 26.82,would you accept or reject Ho using this test?
(c) What is the p-value associated with the x in part (b) ?
(20)
3. (d) Let X equal the yield of alfalfa in tons per acre per year. Assume that X is N (1.5, 0.09).
It is hoped that new fertilizer will increase the average yield. We shall test the null
hypothesis Ho:  = 1.5 against the alternative hypothesis H1:  > 1.5. Assume that the
variance continues to equal 2 = 0.09 with the new fertilizer. Using X , the mean of a
random sample of size n, as the test statistic, reject Ho if x ≥ c. Find n and c so that
the power function  () = P ( X ≥ c) is such that
 =   (1.5) = 0.05 and   (1.7) = 0.95.
(14)
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