LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

## FIFTH SEMESTER – NOVEMBER 2003

**ST-5503/STA508 – COMPUTATIONAL STATISTICS – I**

10.11.2003 Max:100 marks

1.00 – 4.00

__SECTION-A__

__Answer ALL questions.__* *

- 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. (14)

- 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

(N_{i}). And 2) product of N_{i} S_{i }. Also calculate gain in efficiency resulting from case (b)

1 and 2 procedures as compared with unstratified SRS.

Stratum number |
Farm Size (in acres |
No.of farms (N_{i}) |
Average Area under wheat |
Standard deviations (s_{I}) |

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. (14)

- 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
II
III
IV |
2 5 6 8 10
4 8 10 11 13
8 10 11 13 14
16 18 19 20 22 |

(20)

- a) The following is a sequence of independent observations on the random variable X with the

density function

f(x ; q_{1}, q_{2}) = .

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 q_{1} and q_{2} . (15)

- b) Obtain a 95% confidence interval for the parameter l 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 m if the absolute values of the random sample of 8

SAT scores (scholastic Aptitude Test) in mathematics assumed to be N(m, s^{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

If P(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)

- f) An absolute sample of 11 mathematical scores are assumed to be N (m, s
^{2}). The

observations are 26, 31, 27,28, 29, 28, 20, 29, 24, 31, 23.

Find a 99% confidence interval fo s. (7)

- 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

Lot number |
Company Test Results (X) |
Vendor Test Results (Y) |

1
2
3
4
5
6
7
8
9
10 |
0.50
0.58
0.90
1.17
1.14
1.25
0.75
1.22
0.74
0.80 |
0.79
0.71
0.82
0.82
0.73
0.77
0.72
0.79
0.72
0.91 |

Let D = X – Y and let m_{D} denote the median of the differences.

Test H_{0} : m_{D} = 0 against H_{1} : m_{D} > 0 using the **sign test**. Let a = 0.05 approximately. (14)

- 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 H_{0} : m = 0 against the alternative

hypothesis H_{1} : m > 0 at an approximate a = 0.05 significance level.

X |
Y |

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 |
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)

**(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 (m,9). We shall test the null hypothesis H
_{o} : m = 25 against the alternative hypothesis H_{1} : m < 25. Let X_{1},X_{2} , X_{3,} X_{4} be a random sample of size 4 from this distribution, and let the critical region for this test be defined by , where is the observed value of .

(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

= 21.24, = 24.81 , = 23.62, = 26.82,would you accept or reject H_{o }using this test? (c) What is the p-value associated with the in part (b) ? (20)

- (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 H_{o}: m = 1.5 against the alternative hypothesis H_{1}: m > 1.5. Assume that the

variance continues to equal s^{2} = 0.09 with the new fertilizer. Using , the mean of a

random sample of size n, as the test statistic, reject H_{o} if ≥ c. Find n and c so that

the power function b_{f}(m) = P ( ≥ c) is such that

a = b_{f} (1.5) = 0.05 and b_{f} (1.7) = 0.95. (14)

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