LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 21

FIFTH SEMESTER – APRIL 2006

                                               ST 5503 – COMPUTATIONAL STATISTICS

(Also equivalent to STA 508)

 

 

Date & Time : 02-05-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Answer ALL the questions.  Each question carries 34 marks.

1. a) Fit a straight line trend to the following data:

Year :     1976    1977    1978    1979    1980    1981    1982    1983

Sales:      76        80        130      144      138      120      174      190

(Rs.Lakhs)

Also estimate sales for the year 1990.

1. b) Find seasonal indices using ratio to moving average method for the following data:

Quarter

Year       I          II        III        IV

1979      30        40        36        34

1980      34        52        50        44

1981      40        58        54        48

1982      57        78        68        62

1983      86        92        80        82

(or)

1. c) For the following data:

Commodity	Base year		Current year
	Price	Quantity	Price	Quantity
A	10	12	15	10
B	16	14	20	12
C	12	15	14	13
D	18	18	22	16
E	20	22	25	20

Find i) Fisher’s  ii) Walsh’s       iii) Dorbish-Bowley
iv) Marshall-Edgeworth price index numbers

1. d) Find seasonal indices using ratio-to-trend method for the following data

Quarter

Year       I          II        III        IV

1980     40.3     44.8     46.0     48.0

1981     50.1     53.1     55.3     59.5

1982     47.2     50.1     52.1     55.2

1983     55.4     59.0     61.6     65.3

2. a) In a genetical experiment the frequencies observed in four classes are 1997, 906, 904, and 32. Theory predicts that there should be a proportion .  Find the maximum likelihood estimator of the parameter and also obtain the estimate of its variance.
3. b) Five unbiased dice were thrown 96 times and the number of times 4, 5 or 6 was obtained as shown below:

No. of dice showing :   5          4          3          2          1           0

4, 5 or 6

Frequency                     8         18        35        24        10          1

Fit a binomial distribution and test the goodness of fit at 5% level of significance.

1. c) Use the following sample of size 15 to test the hypothesis of randomness against the alternative hypothesis of cyclic effect.

12.4        13.8     22.2     17.9     24.6     15.7     27.3

22.7        26        14.5     22        21.8     31.9     11.5     28.3

(or)

1. d) The following is a sequence of independent observations on the random variable x with the density function , .

The observations are 1.28          1.34     0.64     1.58     0.78     0.94     1.51     1.52     1.71     0.28     0.35   0.40.   Obtain the MLE’s of ₁ and ₂.

1. e)  Two horses A and B were tested according to the time (in seconds) to run a particular track with the following results:

Horse A:             28        30        32        33        33        29        34

Horse B:             29        30        30        24        27        29

Test whether the two horses have the same running capacity.  Use 5% significance level.

1. f) A test or rating a person’s sense of humour on the scale from 0 to 150 was given to 15 married couples.  The scores of these couples were as follows:

Husband :           56        90        38        51        85        49        55        58        68                                74        83   87        60        31        89

Wife :                 49        88        51        47        53        41        52        69        83                                89        77   62        65        44        92

Use the Wilcoxon test to test if there is a difference in the average sense of humour of husbands and wives against the two sided alternative.

3. a) In a population with N=b, the values of Y_i are 8, 3,1,11, 4 and 7 respectively. Calculate the sample mean for all simple random sample of size 2 without replacement.  Verify that the sample mean is an unbiased estimate of the population mean.  Also find the variance of the sample mean.
4. b) The following table shows the number of person (x) and the weekly expenditure on food (y) in a simple random sample of 15 families.

Family number:  1          2          3          4          5          6          7          8

x    :    2          3          3          5          4          7          2          4

y   :     14.3     20.8     22.7     30.5     41.2     28.2     24.2     30.0

 

Family number:  9          10        11        12        13        14        15

x    :    2          5          3          6          4          4          2

y   :     24.2     44.4     13.4     19.8     29.4     27.1     22.2

Estimate the mean weekly expenditure on food per person.  Also find the standard error of the estimate.

(or)

1. c) A simple random sample of 12 households was drawn from a population of 150 households.  The following table gives the number of persons in each household and whether they had seen a dentist or not.

Household          Number of                     Dentist seen

Number               Persons                  Yes                  No

1                         5                         2                      3

2                         6                         1                      5

3                         3                         1                      2

4                         2                         1                      1

5                         3                         3                      0

6                         3                         0                      3

7                         4                         2                      2

8                         5                         1                      4

9                         3                         1                      2

10                        4                         1                      3

11                        2                         1                      1

12                        3                         2                      1

Estimate the proportion of people who had consulted a dentist and find the standard error of the estimate.

1. d) A population of size N=20 units is divided into 2 stratum.  A sample of size n=8 is to be drawn under proportional allocation using SRS.  Find .

Stratum 1:

Unit No.:            1          2          3          4          5          6          7          8

Value:                 10        15        13        12        8          9          6          11

Stratum 2:

Unit No.:            9          10        11        12        13        14        15        16        17        18

Value:                 15        17        13        12        20        25        21        17        19        23

Unit No.:            19        20

Value:                 17        13

 

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                        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 B.Sc. DEGREE EXAMINATION – STATISTICS

AB 16

FIFTH SEMESTER – NOV 2006

ST 5503 – COMPUTATIONAL STATISTICS

(Also equivalent to STA 508)

 

 

Date & Time : 01-11-2006/9.00-12.00   Dept. No.                                                       Max. : 100 Marks

 

 

Answer FIVE questions choosing  at least two questions from each section

SECTION A                                                  ( 5 x20 =100)

1] Consider a population of  6 units with values : 1, 2, 4, 7, 8, 9

[i] Write down all possible samples of size 3 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.

2] The table given below shows the summary of data for Paddy crop census of all the 2500

farms in a state. 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 the gain in efficiency resulting from [iv] and [v] as compared with [vi]

 

 

 

3]  Five samples were collected using systematic sampling from 4 different pools located in a

region to study the mosquito larvae  population( in ‘000/ gl) ,where the mosquito population

exhibits a fairly steady rising trend. i]  Find the average mosquito population in all four pools

and also find sample means. ii]  Compare the precision of systematic sampling , SRSWOR

and Stratified sampling.

Sample Number( mosquito nos. in ‘000/ gl)

Pool #	1	2	3	4	5
I	3	7	8	9	12
II	4	9	16	18	20
III	8	16	17	19	24
IV	14	18	23	28	32

 

4] The following data furnishes the software earnings of India (in ‘00 crore) for the time period

1996 To 2005. Fit a second degree parabola and hence predict the expected software earnings

of India for the financial year 2006.

Year	1996	1997	1998	1999	2000	2001	2002	2003	2004	2005
Earnings(‘00Cr)	4	8	15	30	90	50	110	150	300	500

 

 

SECTION B

 

1. a.) From the following data construct an index for 2001 taking 2000 as base by the

average of relatives method using i.) arithmetic mean and ii.) geometric mean for

averaged relatives:

 

Commodity                 Price in 2000               Price in 2001

( Rs )                             ( Rs )

 

A                               50                                   70

B                               40                                   60

C                               80                                   90

D                             110                                  120

E                                20                                   20

 

b.) Construct the consumer price index number for 2003 on the basis of 2002  from the following data using ’ family budget method’.

 

Items	Price in 2002 ( Rs )	Price in 2003 ( Rs )	Weights
Food	200	280	30
Rent	100	200	20
Clothing	150	120	20
Fuel	50	100	10
Miscellaneous	100	200	20

(14+6)

 

1. a.)  The life time of 10 electric bulbs selected randomly from a large consignment gave

the following data :

Life time (Hours )   4.2     4.6       3.9       4.1       5.2       3.8       3.9       4.3                                             4.4     5.6

Test at 5% level the hypothesis that the average life time of bulbs is 4.

• In a cross- breeding experiment with plants of certain species, 240 offsprings were classified into 4 classes with respect to the structure of the leaves as follows:

Class :                I                       II                     III                    IV

Frequencey:        21                    127                  40                    52

 

According to theory, the probabilities of the 4 classes should be in the ratio

1: 9: 3: 3. Are these data consistent with the theory? Use 5% level.                                                                                                                                                               (10+10)

 

1. Two samples are drawn from two normal population. From the following data, test whether the two populations have i) Equal variance ii) Equal means at 5% level.

Sample 1         60        65        71        74        76        82        85        87

Sample 2         61        66        67        85        78        63        85        86        88        91

 

 

1. a.)  IQ Test on two groups of boys and girls gave the following results

	Mean	SD	Sample Size
Boys	73	15	100
Girls	78	10	50

 

Is there a significant difference in the Mean scores of boys and girls ? Test at 1% level.

 

• Let X denote the length of a fish selected at random from the lake . The

observed length of n=10 fish , were 5.0 , 3.9 ,5.2 , 5.5 , 2.8 , 6.1, 6.4 , 2.6 , 1.7

and 4.3. Test at 5 % level the hypothesis that the median length of fish in the

lake is 3.7 .                                                                                      (6+14)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

NO 27

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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

BA 13

FIFTH SEMESTER – November 2008

ST 5503 – COMPUTATIONAL STATISTICS

 

 

 

Date : 10-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL the Questions. Each question carries 20 marks.

 

1. The table given below presents the data for complete census of 2010 farms in a region.  The farms were stratified according to farm-size in acres into seven strata. The number of farms in the different strata, the strata standard deviation are given below. Find the sample sizes under each stratum under (i) Proportional allocation  and (ii) Neyman Optimum allocation.

Stratum No.	No. of farms	Stratum standard deviation
1	394	8.3
2	461	13.3
3	391	15.1
4	334	19.8
5	169	24.5
6	113	26
7	148	35.2

 

 

 

 

 

 

 

 

 

(OR)

In a population size N = 6, the values of Y_i are 24, 56, 12, 45, 25, 55. Calculate the sample mean for all possible simple random samples (without replacement) of size n = 2, also calculate s ² for all samples and verify E (s ² ) = S ².

 

2. Calculate fixed base index numbers and chain base index numbers for the

following    data.

 

Commodity	2002	2003	2004	2005	2006
I	2	3	5	7	8
II	8	10	12	4	18
III	4	5	7	9	12

 

 

 

 

 

 

 

(OR)

 

Calculate seasonal variations given the average quarterly price of a commodity for

5 years by ratio to trend method.

 

Year	I Quarter	II Quarter	III Quarter	IV Quarter
2001	28	22	22	28
2002	35	28	25	36
2003	33	34	30	35
2004	31	31	27	35
2005	37	36	31	36

 

 

 

3. (a) Glaucoma is an eye disease that is manifested by high intraocular pressure. The

distribution of intraocular pressure in the general population is known to be normal

with mean 16 mm Hg and standard deviation 3 mm Hg. Pressures in the range of 12

mm Hg to 20 mm Hg are considered safe. What percentage of the population is

unsafe?

&

(b)  The scores in a certain test from 12 men and 10 women candidates are reported below:

Men:      56, 67, 45, 78, 86, 64, 78, 88, 91, 46, 45, 84

Women:  67, 48, 91, 75, 58, 90, 46, 69, 70, 82

Test whether there is significant difference in the average scores of the two groups at 5% level of significance. (Variances are considered to be equal but are not known)

(6 + 14)

(OR)

(c) In a pediatric clinic a study is carried out to test the effectiveness of aspirin in

reducing temperatures. The temperatures of twelve five-year old children suffering

from influenza were observed before and after one hour of administering aspirin and

the paired observations are reported below:

 

Patient	Temp ( 0F) before taking  aspirin	Temp (0F) 1 hr after taking aspirin
1 2 3 4 5 6 7 8 9 10 11 12	102.4 103.2 101.9 103.0 101.2 100.7 102.5 103.1 102.8 102.3 101.9 101.4	99.6 100.1 100.2 101.1 99.8 100.2 101.0 100.1 100.7 101.1 101.3 100.2

 

Test whether aspirin is effective in reducing the temperature at 5% level of

significance.

                                                                   &                                                       

(d) A textile mill attempts to control the yarn defects that appear on manufactured

cloth. The occurrence of defects has been found to follow Poisson law and the

historical average number of defects per 100 m of cloth is 1.25. Recently, due to

changes implemented by the HR department, the occurrence of defects is expected

to reduce significantly. The quality control department wishes to test whether this

improvement has happened. The following numbers of defects were observed from

12 bales of cloth (each of length 100 m): 1, 2, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1. Carry out the

relevant test at 5% level of significance and report your findings to the quality

control department.                                                                                           (8 +12)

 

4. (a) The daily iron intake among children in the age group 9 – 11 has been

established to be normally distributed with average14.4 mg and standard deviation

4.75 mg. A social service organization working among ‘Below-Poverty-Line’

(BPL) families, took a sample of 50 children and reported that the average intake

among them was 12.5 mg. It is believed that the standard deviation is not different

(from the known 4.75 mg) for this group. Does this information indicate that the

iron intake is significantly lower in the BPL families?

&

 

 

(b) In a study on infarct size among people, the following data were obtained from 6

treated children and 8 untreated children:

Treated Children:       20.3, 21.0, 21.4, 18.9, 19.5, 19.7

Untreated Children:   19.6, 18.7, 19.9, 20.6, 20.1, 19.6, 19.0, 21.0

Test whether the variances differ significantly at 2% level of significance

(6 + 14)

(OR)

(c) A small factory A produces fasteners for use in machine tools and supplies it to a machine tool manufacturing company B. Factory A supplies fasteners in large lots every day and maintains a low percentage of 5% defectives. Company B carries out sampling inspection every day by taking a sample of 20 fasteners. It rejects the entire lot and sends it back to Factory A if more than one defective is observed in the sample. Find the proportion of days in which lots are sent back to factory A.

&

(d) The dispersion in the quality characteristic is an important indicator of the quality conformance of a production process. It has been historically found that a production process was operating with a variance of 18 for a normally distributed quality characteristic, But, due to certain changes made in the processes, it is believed that there could be a change in the process dispersion. The following data on the quality characteristic are available after the changes were implemented:

65, 60, 67, 70, 67, 62, 68, 63, 59, 69, 70, 58, 75, 75, 78

Test whether there is a significant change in the process variance at 1% level of

significance.                                                                                                      ( 8 +12)

 

5. The following are the weight gains (pounds) of two random samples of young Indians fed on two different diets but otherwise kept under identical conditions:

 

Diet I:  16.3     10.1     10.7     13.5     14.9     11.8     14.3     10.2     12.0     14.7     23.6     15.1     14.5     18.4     13.2            14.0

Diet II:            21.3     23.8     15.4     19.6     12.0     13.9     18.8     19.2     15.3     20.1     14.8     18.9     20.7     21.1            15.8     16.2

Use U test at 0.01 level of significance to test the null hypothesis that the two population samples are identical against the alternative hypothesis that on the average the second diet produces a  greater gain in weight.

(OR)

 

The same mathematics papers were marked by three teachers A, B and C. The final marks were recorded as follows:

Teacher A	73	89	82	43	80	73	66	60	45	93	36	77
Teacher B	88	78	48	91	51	85	74	77	31	78	62	76
Teacher C	68	79	56	91	71	71	87	41	59	68	53	79

Use Kruskal-wallis test, at the 0.05 level of significance to determine if the marks distributions given by the three teachers differ significantly.               (10 + 10)

 

 

 

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 27

FIFTH SEMESTER – April 2009

ST 5503 – COMPUTATIONAL STATISTICS

 

 

 

Date & Time: 04/05/2009 / 1:00 – 4:00       Dept. No.                                          Max. : 100 Marks

 

 

 

Answer any THREE questions. Each carries 34 marks.

1. 1. (i) Find seasonal variations by ratio-to-trend method from the data given below:

Year	1st Quarter	2nd Quarter	3rd Quarter	4th Quarter
1999	30	40	36	34
2000	34	52	50	44
2001	40	58	54	48
2002	54	76	68	62
2003	80	92	86	82

 

 

 

 

 

 

(ii) Assuming a four yearly cycle, calculate the trend by the method of moving averages from the following data relating to production of tea:

 

Year	1961	‘62	‘63	‘64	‘65	‘66	‘67	‘68	‘69	‘70
Production	464	515	518	467	502	540	557	571	586	612

 

2. In a population with N = 4, the Y_i values are 1, 2, 3, 4. Enlist all possible samples of size n = 2, with SRSWOR and verify that E (s²) = S².

 

3. 3. (i) A sample of 30 students is to be drawn from a population consisting of 300 students belonging to two colleges A and B. The means and standard deviations of their marks are given below:

	Total no. of students(Ni)	Mean	Standard deviation(σi)
College A	200	30	10
College B	100	60	40

 

 

 

How would you draw the sample using proportional allocation technique? Hence obtain the variance of estimate of the population mean and compare its efficiency with simple random sampling without replacement.

 

(ii) An institution which gives coaching for students planning to appear for a competitive exam in English wishes to test the efficacy of its coaching programme in improving this course. Then pre-coaching and post-coaching tests are conducted. The following data were obtained from a sample of 15 students who were subjected to both the tests:

Pre-coaching score: 56, 47, 61, 79, 53, 68, 72, 85, 59, 61, 70, 53, 68, 77, 83

Post- coaching score: 97, 53, 63, 82, 53, 64, 83, 89, 66, 63, 70, 50, 60, 84, 89

Carry out the relevant tests and report your findings.

 

 

 

4. (i) Calculate Laspeyre’s, Paasche’s, Bowley’s and Fisher’s Ideal index from the following data.

 

Commodity	Price (2004)	Value (2004)	Price (2005)	Value (2005)
Bricks	10	100	8	96
Sand	16	96	14	98
Timber	12	36	10	40
Cement	15	60	5	25

 

 

(ii) Calculate chain based index numbers from the price of the following three commodities:

Commodity	1999	2000	2001	2002	2003
Wheat	4	6	8	10	12
Rice	16	20	24	30	36
Sugar	8	10	16	20	24

 

 

 

 

 

 

5. (i) Suppose that a group of 100 men received a flu vaccine and five of them died within the next one year. It is known that the probability of a man who did receive the vaccine will die within the next year is 0.02. Given this information, is the above mentioned event unusual or can this death rate be expected normally?

 

(ii) A new drug is to be tested for the treatment of patients with unstable angina. The effect of this drug is unknown. The following changes in the heart rate were observed among 20 patients: 4, -2, -3, 8, -3, -4, 2, 0, 1, 6,   -4, -7, 3, -3, 0, 5, 6, 3, -4, -2. The experimenter would like to know whether the drug induces a significant change in heart rate after 48 hours of administration.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2010

    ST 5507 – COMPUTATIONAL STATISTICS

 

 

 

Date : 09-11-10                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer all the questions                                                                                          (5 X 20 =100)

1 a)  Consider  the population of 7 units with values  1, 2, 3, 4, 5, 6,7  Write down all possible of sample of 2 ( without replacement) from this population and verify that this sample population mean is an unbiased estimate of the population mean.

Also calculate its sample variance and verify that

• It agrees with the formula for the variance of the sample mean, and
• This variance is less than the variance obtained from the sampling with replacement.

(Or )

3. b) The table given below presents the summary of data for complete census of all the the 2010                       farms in region. The farms were stratified according to the farm size in acres into seven strata, as shown in column 2 of the table. The number of farms in the different strata N_i are given in the column 3. The population values of the strata means (  ) and the strata standard deviation ( S_i) for the area under wheat  are given the frequency table

 

Stratum  No.	Farm size ( in acres )	No. of farms Ni	Average area under wheat per farm in acres (  )	St. Deviation of area under wheat per farm in acres  (Si)
1	0 – 40	394	5.4	8.3
2	41 – 80	461	16.3	13.3
3	81 – 120	391	24.3	15.1
4	121 – 160	334	34.5	19.8
5	161 – 200	169	42.1	24.5
6	201 – 240	113	50.1	26.0
7	More than 240	148	63.8	35.2

 

Calculate the sampling variance of the estimated area under wheat for the region from a sample of 150 farms if the farms are selected by the method of simple random sampling without stratification.

 

 

2. a) A random variable takes values 0, 1, 2 with probabilities

+  +  + where N is a known number and α andθ are unknown parameters. If 75 independent observations on X yielded the values 0, 1, 2 with frequencies 27, 38, 10 estimate α and θ by the method of moments.

 

6. b) If 6,11,4,8,7,6 is a sample from a normal population with mean 6. Find the maximum likelihood estimate for the variance .

                                                                        (or)

1. c) Given below is a random sample from normal population. Determine 95% confidence interval for the population standard deviation.

160, 175, 161, 181, 158, 166, 174, 165, 172, 184, 170, 159, 169, 175, 179, 164

 

4. d) A random sample of size 17 from a normal population is found to have 7 and

find a 90% confidence interval  for the mean of the population.

 

3(a) Calculate seasonal indices by using Ratio to trend method for the following data:

 

	Quarter
Year	I	II	III	IV
2006	8	16	24	32
2007	48	36	24	12
2008	48	16	32	64
2009	72	108	114	36
2010	56	28	84	112

(OR)

 

(b) Calculate 3 yearly moving averages and also draw the graph for the following data:

 

Year	2000	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010
Sales	200	120	280	240	160	320	360	400	320	360	360

 

(c) Fit a straight line by the method of least square and also forecast the production for the year 2010 for the following data:

Year	2000	2001	2002	2003	2004
Production	10	20	30	50	40

 

4 (a) Two random samples were drawn from two normal populations and the observations are:

 

A	66	67	75	76	82	84	88	90	92	–	–
B	64	66	74	78	82	85	87	92	93	95	97

 

Test whether the  two populations have the same variance at 5% level of significance.

 

1. b) The following table show the association between the performance and training of 870persons.Is the association significant.

	Training
Performance	Intensive	Average	Normal	Total
Above average	100	150	40	290
Average	100	100	100	300
Poor	50	80	150	280
Total	250	330	290	870

 

(or)

(c)Apply the mann-whitney  -wilcoxon test  to the following data to test

X	25	30	45	52	65	75	80	42	50	60
Y	60	40	35	50	60	72	63	40	55	62

 

(d)  A group of 5 patients treated with machine A weighted 42, 39, 48, 60, 41. A second group of 5 patients treated with machine B weighted 38, 42, 48, 67, 40 kg. Do the two machine differ significantly with regard to their effect in increasing weight?

5) a) From the following data compute price index by applying weighted  average of price relatives method using:

(i) Arithmetic mean, and

(ii) Geometric mean.                                                                                                  (8)

 

Commodities	p0 Rs.	q0	p1 Rs.
Sugar	6.0	10 kg.	8.0
Rice	3.0	20 kg.	3.2
Milk	2.0	5 lt.	3.0

 

b)Construct index number of price from the following data by applying

 

• Laspeyre’s method
• Paasche’s method
• Bowleys method,
• Fisher’s ideal method,  and
• Marshall edgeworth method

 

Commodity	2007		2006
	Price Rs.	Quantity	Price	Quantity
A B C D	2 5 4 2	8 10 14 19	4 6 5 2	6 5 10 13

 

 

Or

1. c) From the following data, calculate Fisher’s ideal index and prove that it satisfies both the time reversal test and factor reversal tests. (10)

Commodity	2007		2006
	Price Rs.	Quantity	Price	Quantity
A B C D	4 5 3 8	8 10 6 5	5 6 4 10	8 12 7 4

 

d)From the following data of the wholesale prices of wheat for the ten years construct index numbers   (a) taking 1999 as base, and  (b) by chain base method.             (10)

 

Year	Price of wheat ( Rs. Per 10 kg )
1999	50
2000	60
2001	62
2002	65
2003	70
2004	78
2005	82
2006	84
2007	88
2008	90

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2012

ST 5507 – COMPUTATIONAL STATISTICS

 

 

Date : 03-05-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer any THREE of the following questions:                                      

(a) A Textile manufacturer keeps a record of the defects that occur on the material by noting down the number of defects observed per 500 meter of the cloth. The data collected from 180 such pieces of cloth are reported below.

 

No. of Defects	0	1	2	3	4	5
No. of Pieces	10	25	62	54	21	8

 

Fit a Poisson distribution to the number of defects per 500 meter length and test for

goodness of fit at 5% level of significance.

 

(b) Data on the life-time of 250 machines are given below:

Life Time (in hrs)	0-1	1-2	2-3	3-4	4-5	5-6	6-7	7-8	>8
No. of failed machines	85	51	35	24	18	15	12	7	3

Test at 5% level of significance whether the Life Time random variable follows

exponential distribution with p.d.f. f(x) = θe^{–θ x} , x > 0.

(15 +18)

 

• (a) A population consists of 5 units with ‘y’ values 1, 4, 6, 9, 12. Enlist all possible simple random samples of size 3 that can be drawn without replacement and verify the results E() = and E(s²) = S².

 

(b) A population with 300 units is divided into three strata. A stratified random sample

was drawn and the observed values in the sample are reported below:

Stratum No.	Stratum Size	Sample observations
1 2 3	75 100 125	21, 26 32, 35, 37 40, 48, 49, 45

 

 

 

 

 

Obtain the estimate and get an estimate of its variance from the sample data.

(16 + 17)

 

• (a) Compute index number for the given data using the following methods (i) Laspeyre’s

method,  (ii) Passche’s method and (iii) fisher’s ideal formula                            (8)

Item (Rs.)	Base year		Current year
	Price (in Rs)	Expenditure	Price (in Rs)	Expenditure
Food	10	600	20	1000
Rent	8	400	4	480
Clothing	8	480	12	600
Fuel	25	650	24	720
Others	16	640	20	960

 

(b) Change the base year 1996 to 2000 and rewrite the series of index numbers in the

following data:

Year	2000	2001	2002	2003	2004	2005	2007	2008	2009
Index	100	112	125	160	140	165	170	175	182

(5)

 

(c) Calculate the seasonal indices by the method of least squares from the following data:

(Multiplicative model)                                                                                            (20)

	Exports of cotton textiles (million Rs.)
Year	I	II	III	IV
2001	71	65	79	71
2002	76	66	82	75
2003	74	68	84	80
2004	76	70	84	79
2005	78	72	86	85

 

 

 

 

 

 

• (a) On any given day at a warehouse, 14 trucks are loaded with a particular product.

It is claimed that the median weight m of each load of the product is 39,000 pounds.

On a particular day, the following observations were obtained:

 

41,195        39,485         41,229        36,840         38,050          40,890     35720

38,345       34,930          39,245        31,031         40,780         38,050      30,906

Test the null hypothesis H₀ : m = 39,000  against the one-sided alternative  hypothesis

H₁ : m < 39,000 using  the critical region C = { y | y ≥ 9 } where ‘y’ is the number of observations in the sample that are less than  39,000. Find the significance level α for the critical region C.  Also find the p – value of this test.              

                                                                                                                                                                                (13)                   

 

 

 

(b)  A vendor produces and sells low-fat milk powder to a company that

uses it to produce health drink formulae.  In order to determine the fat

content of the milk powder , 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 as follows:

Lot Number	Company Test Results (X)	Vendor test results( Y)                                                 Vendor Test Results (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
11	0.92	0.74
12	0.58	0.55

 

Test the hypothesis H₀ : p  =  P[X > Y]  =   against the one – sided alternative  H₁ : p >

using the critical region  C = { w | w ≥ 7 },  where ‘w’ is the number of pairs for which

X_i – Y_i  > 0. Find the significance level α  and p – value of this test.                      (20)

 

• (a) Measurements  of the fat content of two kinds of food item , Brand X  and Brand Y

yielded the  sample data :

Brand  X :   13.5      14.0    13.6   12.9     13.0    14.2     15.0    14.3     13.8

Brand Y :    12.9      13.0    12.8   143.5   12.7    15.0     18.7    11.8    14.3

Test  the null hypothesis μ₁  = μ₂   against  μ₁  μ₂  at 5% level of significance.

(9)

( b)       Two random samples drawn from two normal populations are :

Sample I     :    23      15       25       27      23     20       18      24     25

Sample II    :    27      33       45       35      32     35       33      28     41      43

Test whether  the two populations have the same  variances.  Use 5% significance

level.

(8)

(c)  Seven coins were tossed and the number of heads noted. The experiment was repeated

130 times  and the following distribution was obtained.

 

No. of heads :   0          1          2             3            4             5              6             7

Frequency    :   7           6          19           35        30           23             9             1

Fit a binomial distribution to the given data and test the goodness of fit at 1% level of

significance.                                                                                                       (16marks)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5507/5503 – COMPUTATIONAL STATISTICS

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer any THREE of the following questions:                                             

 

• (a) A study of randomly selected motor-cycle accidents and drivers who use cellular phones provided the following data. Based on the following data, does it appear that use of cellular phones affects driving safety? (15)

 

Had Accidents               Had no Accidents

 

Cell phones used                                                    23                                   282

 

Cell phones not used                                              46                                   407

 

(b)  Find an α level Likelihood Ratio Test of     against based on a sample of size 10 from, where both µ and σ²  are unknown. If the observed value of sample mean and variance are 0.6 and 0.36 respectively, should the hypothesis H₀  be accepted or rejected?       (18)

 

2) a) From the following informations, Compare the precision of Systematic Sample, Simple Random Sampling and Stratified sampling.

Strata	1	2	3	4	5	6
I	28	32	33	35	37	39
II	15	16	17	21	22	25
III	2	3	4	7	9	9
IV	5	7	9	12	14	15
V	25	22	21	17	17	23

 

b).        A sample of 40 students is to be drawn from a population of two hundred students belonging to A&B localities. The mean & standard deviation and their heights are given below

Locality	Total No.Of People	Mean (Inches)	S.D(Inches)
A	150	53.5	5.4
B	50	62.5	6.2

 

1. Draw a sample for each locality using proportional allocation
2. Obtain the variance of the estimate of the population mean under proportional allocation.

 

(16+ 17)

• (a) Compute index number for the given data using the following methods (i)

Laspeyre’s  method,  (ii) Passche’s method and (iii) fisher’s ideal formula                                   (8)

 

 

Item (Rs.)	Base year		Current year
	Price (in Rs)	Quantity	Price (in Rs)	Quantity
Food	12	20	20	22
Rent	40	10	42	12
Clothing	8	50	12	50
Fuel	20	20	24	22
Others	16	20	25	20

 

(b) Change the base year 2000 to 2003 and rewrite the series of index numbers in the

following data:

 

Year	2000	2001	2002	2003	2004	2005	2006	2007	2008
Index	100	115	120	122	125	128	130	135	140

(5)

 

(c) Calculate the seasonal indices by the method of least squares from the following data:

(Multiplicative model)                                                                                          (20)

	Exports of cotton textiles (million Rs.)
Year	I	II	III	IV
2001	71	68	79	71
2002	76	69	82	74
2003	74	66	84	80
2004	76	73	84	78
2005	78	74	86	82

 

 

 

 

 

 

 

 

 

• (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. 10 sets of paired results are :

 

 

Lot no.	1	2	3	4	5	6	7	8	9	10
Company test results(X)	0.5	0.58	0.9	1.17	1.14	1.25	0.75	1.22	0.74	0.80
Vendor test result (Y)	0.79	0.71	0.82	0.82	0.73	0.77	0.72	0.79	0.72	0.91

 

Test  against, using a paired t test with the differences. Let.  (D=X-Y)   (20)

 

(b)  Let  be a random sample from. Test  against.   Find the Uniformly Most Powerful Test.                      (13)

 

 

 

 

 

 

 

 

 

• (a)  The following  are the weight gains (in pounds) of two random samples of young Indians fed on two different diets but otherwise kept under identical conditions:

 

Diet I	16.3	10.1	10.7	13.5	14.9	11.8	14.3	10.2
Diet II	21.3	23.8	15.4	19.6	12	13.9	18.8	19.2
Diet I	12	14.7	23.6	15.1	14.5	18.4	13.2	14
Diet II	15.3	20.1	14.8	18.9	20.7	21.1	15.8	16.2

 

Use U test at 0.01 level of significance to test the null hypothesis that the two population samples are identical against the alternative hypothesis that on the average the second diet produces a greater gain in weight.                                                         (16)                   

 

(b) The following are the speeds at which every fifth passenger car was timed at a certain

checkpoint: 46, 58, 60, 56, 70, 66, 48, 54, 62, 41, 39, 52, 45, 62, 53, 69, 65, 67, 76,

52, 52, 59, 59, 67, 51, 46, 61, 40, 43, 42, 77, 67, 63, 59, 63, 63, 72, 57, 59, 42, 56, 47,

62, 67, 70, 63, 66, 69 and 73. Test the null hypothesis of randomness at the 0.05 level

of significance.                                                                                                      (17)

 

 

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