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M.B.A. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2010

First Semester

STATISTICS FOR MANAGEMENT)

Answer ALL questionsPART A (10 2 = 20 Marks)

1. Define probability.2. Give example for continuous and discrete variables.3. What is point estimate?4. Give the meaning of random sampling.5. Explain Type I and Type II error.6. What do you mean by one-tail test?7. Write the meaning of non-parametric test.8. How do you find the degrees of freedom in case of chi-square test?9. Specify the range of correlation.

10. What do you mean by seasonal variation?

11. What are the common types of variables used in statistics?

12. Name a few descriptive measures of data.13. What is the central limit theorem14. What are elements and variables in a data set?15. What are parametric tests?16. If a class of students is examined and the researcher wants to test the difference in performancebetween boys and girls, what test will you use?17. What is a non parametric test?18. Name four non parametric tests.

19. How will you test the accuracy of a regression equation?20. Why are index numbers used?

21Explain

a. Trial & Event b. Exhaustive case c. Favorable Case22.Define Probability.23.Write the equations for

a. Addition Theorem of Probability in case of two and three eventsb. Multiplication Theorem of Probability.

24.What is sampling? What are the different types of sampling?25.Define correlation. What are the different methods of studying Correlation? Write their

Equations.

26.Write down the formulae/ equations respectively for the following statistical operations.a. Rank correlationb. Coefficient of regressionc. Probable error & Standard errord. Binomial Distribution equation


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e. Bayes theorem equation.27. Suppose that a decision maker is faced with three decision alternative and four states of

nature. Given the following profit payoff table:

States of Nature

Acts

S1 S2 S3 S4

a1

a2

a3

16

13

11

10

12

14

12

9

15

7

9

14

Assuming that he has no knowledge of the probabilities of occurrence of the states of nature find

the decisions to be recommended under each of the following criteria

i) Minimax Regret ii) Laplace criterion

28. Define Correlation coefficient and Regression lines

29. what is t- test or T distribution.

30.The following table gives payoffs for actions A1, A2, A3 corresponding to the statesof nature S1 and S2 whose chances are 0.6 and 0.4 respectively.

Acts

States of Nature

A1 A2 A3

S1

S2

16

19

20

15

18

12

Find decision under EMV Criterion.

31.Write short Notes on:a) Central tendency b) Measure of Dispersion

32.Distinguish between Primary and Secondary Data.33.What are the different methods of collecting primary data?34.List out the steps for the computation of Median in case of Continuous frequency

distribution.

35.What do you understand by Mode? Write the interpolation formula for Mode.36.Write the formula for following:a) Range b) Mean Deviation c) Standard Deviation37.Write short notes on Lorenz Curve38.Write short notes on Skewness.


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39.Calculate the Range and the Coefficient of Range of As monthly earnings for a yearMonth 1 2 3 4 5 6 7 8 9 10 11 12

Monthlyearnings

( in00

Rs.)

139 150 151 151 157 158 160 161 162 162 173 175

40. Obtain the mean number of calls per minute

Number

of calls

0 1 2 3 4 5 6 7

Frequency 14 21 25 43 51 40 39 12

Part B ( 3*10m=30m) Answer Any 3

1. The Mean of the following frequency distribution is 50. But the frequencies 1 and 2 inthe class 20-40 and 60-80 are missing. Find the missing frequencies.

Class 0-20 20-40 40-60 60-80 80-100 Total

Frequency 17 1 32 2 19 120

2. Find theCoefficient of Mean Deviation (from Median) of the following dataMarks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90

Number of

Students

2 6 12 18 25 20 10 7

3. Calculate the standard deviation and coefficient of Variance from the following data.Age

under(inyears)

10 20 30 40 50 60 70 80

No.of

Persons

dying

15 30 53 75 100 110 115 115

4. What is positive and Negative Skewness? Calculate karl pearsons coefficient ofskewness using MODE and comment on the result.


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Wages 50-60 60-70 70-80 80-90 90-100 100-110 110-120

No.of

Persons

15 18 17 30 40 20 10

Part B. Answer all 3 ( 3* 10m= 30m)

1. a. A committee of four has to be formed from among 3 economists, 4 engineers, 2statisticians and 1 doctor.

i. What is the probability that each of the four professions is represented onthe committee?

ii. What is the probability that the committee consists of the doctor and atleast one economist?

b. Four coins are tossed simultaneously. What is the probability of getting:

i) 2 heads and 2 tails ii) at least 2 heads iii) At least 1 head

(Or)

2. a. A systematicsample of100 pages was taken from the Concise Oxford Dictionary andthe observed frequency distribution of foreign words per page was found to be as follows:

No.

of foreign words per pa

ges (X) : 0 1 2 3 4 5 6Frequency (f) : 48 27 12 7 4 1 1

Calculate the expected frequencies using Poisson distribution. (5m)

40.The average monthly slaes of5000 firm are normally distributed. Its mean mean andstandard Deviation are Rs.36,000 and Rs. 10,000 respectively. Find the number of

firms, the sales of which are overRs. 4000.

3. a. The result of an investigation by an expert on a fire accident in a skyscraper are summerisedbelow:

i. Probability there could have been short circuit = 0.8ii. Probability (LPG cylinder explosion) = 0.2iii. Chance of fire accident is 30% given a short circuit and 95% given an LPG explosion.Based on these, what do you think is the most probable cause of fire? Statistically justify your

answer.


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(Or)

4. a. Fit a linear trend to the following data by the least squares method. Verify that ( y ye) = 0 , where ye is the corresponding trend value of y.

Year : 1990 1992 1994 1996 1998

Production (in 000 units): 18 21 23 27 16

Also estimate the production for the year 1999

5. a. A machine has produced washers having a thickness of0.050 inches. To determinewhether the machine is in proper working order, a sample of10 washers is chosen for

which the mean thickness is 0.053 and the standard deviation is 0.003 inches. Test the

hypothesis at = 0.05

b.A company manufactures different types of electrical appliances, it has been using

radio for advertising its products. The following table shows the amounts of radio

time (X, in minutes) and the number of electrical appliances sold (Y) over last seven

weeks.

X 25 18 32 21 35 28 30

Y 16 11 20 15 26 32 20

i. Fit regression of Y on Xii. What will be the value of Y when X is 27?

(Or)

6. Calculate the Karl Pearsons Coefficient of correlation from the data given below:Marks Age in years

18 19 20 21 22

20-25 3 2 - - -

15-20 - 5 4 - -

10-15 - - 7 10 -

5-10 - - - 3 2

0-5 - - - 3 1

PART B (5 16 = 80 Marks)

1. (a) (i) The probability of appoint of one of the 4 persons namely A, B, C and D in a Company are 1/5,


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1/4, 2/7 and 3/4 respectively. The probability that the company earns profit above Rs. 20,000 per monthdue to their appointment is 1/3, 2/3, 1/5 and 2/5 respectively. What is the probability that the companyearns about Rs. 20,000 per month? (Marks 8)(ii) In a bolt factory machines A, B, C manufacture respectively 25%, 35% and 40% of the total of theiroutput 5, 4, 2 percent are defective bolts. If A bolt is drawn at random from the product and is found to bedefective, what are the probabilities that it was manufactured by machine A, B and C? (Marks 8)

2. Fit a Poisson distribution to the following data and calculate the theoretical frequencies.

x:0 1 2 3 4

f:123 59 14 3 1 (Marks 16)

3. The age of employees in a company follows normal distribution with its mean and variance as 40 years

and 121 years respectively. If a random sample of 36 employees is taken from a finite normal population

of size 1000, what is the probability that the sample mean is

(i) less than 45

(ii) greater than 42 and

(iii) between 40 and 42? (Marks 16)

(4 ) A non-normal distribution representing the number of trips performed by Lorries per week in a coal

field has a mean of 100 trips and variance of 121 trips. A random sample of 36 Lorries is taken from the

non-normal population. What is the probability that the sample mean is (i) greater than 105 trips, (ii) less

than 102 trips and (iii) between 101 and103 trips? (Marks 16)

(5) The average number of defective articles in a certain factory is claimed to be less than the average for

all the factories. The average for all the factories is 30.5. A random sample of 100 defective articles

showed the following distribution.

Class limits : 16-20 21-25 26-30 31-35 36-40Number : 12 22 20 30 16

Calculate the mean and the standard deviation of the sample and use it to test the claim that the average is

less than the figure for all the factories at 5% level of significance. Given Z(-1.645) = 0.95 . (Marks 16)

(6) Three samples below have been obtained from normal populations with equal variance. Test the

hypothesis that the sample means are equal.

I II III

81071411

751099

129131214

(Marks 16)

(7) Two researchers adopted different sampling techniques while investigating the same group of

students to find the number of students falling in different intelligence levels. The results as follows :

No. of students

Researcher Below average Average Above average Genius


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X 80 60 44 10

Y 40 33 25 12

Would you say that the sampling techniques adopted by the two researchers are significantly different?

(Marks 16)

(8) The production volume of units assembled by three different operators during 9 shifts is summarizedin Table 9.26. Check whether there is significant difference between the production volumes of units

assembled by the three operators using Kruskal-Wallis test at a significant level of

0.05.

Operator I : 29 34 34 20 32 45 42 24 35

Operator II : 30 21 23 25 44 37 34 19 38

Operator III : 26 36 41 48 27 39 28 46 15 (Marks 16)

(9) Obtain the two regression lines :

x : 45 48 50 55 65 70 75 72 80 85

y : 25 30 35 30 40 50 45 55 60 65 (Marks 16)

(10) Calculate seasonal index from the following data :

Year (Sales in 100 tonnes)

I quarter II quarter III quarter IV quarter

2005 30 22 15 45

2006 32 24 18 40

2007 35 29 20 37

2008 45 32 14 30

2009 50 30 12 35 (Marks 16)

11. The following data shows the yearly income distribution of a sample of 200 employees at MNM, Inc.

Yearly Income (In $1000s) Number of Employees

20 24 2

25 29 48

30 34 60

35 39 80


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40 44 10

(i) What percentage of employees has yearly income of $35,000 or more?

(ii) Is the figure (percentage) that you computed in (i) an example of statistical inference? If no, what kind

of statistics does it represent?

(iii) Based on this sample, the president of the company said that 45% of all our employees yearly

income are $35,000 or more. The presidents statement represents what kind of statistics?

(iv) With the statement made in (iii) can we assure that more than 45% of all employees yearly income

are at least $35,000? Explain.

(v) What percentage of employees of the sample has yearly income of $29,000 or less?

(vi) How many variables are presented in the above data set?

(vii) The above data set represents the results of how many observations?

(12) An experiment consists of throwing two sixsided dice and observing the number of spots on theupper faces. Determine the probability that

(i) the sum of the spots is 3

(ii) each die shows four or more spots

(iii) the sum of the spots is not 3

(iv) neither a one nor a six appear on each die

(v) a pair of sixes appear

(vi) the sum of the spots is 7.

(13) The life expectancy in the United States is 75 with a standard deviation of 7 years. A random sample

of 49 individuals is selected.

(i) What is the probability that the sample mean will be larger than 77 years?

(ii) What is the probability that the sample mean will be less than 72.7 years?

(iii) What is the probability that the sample mean will be between 73.5 and 76 years?

(iv) What is the probability that the sample mean will be between 72 and 74 years?

(v) What is the probability that the sample mean will be larger than 73.46 years?

(14) A random sample of 121 checking accounts at a bank showed an average daily balance of $280. The

standard deviation of the population is known to be $66.

(i) Is it necessary to know anything about the shape of the distribution of the account balances in order to

make an interval estimate of the mean of all the account balances? Explain.(ii) Find the standard error of the mean.

(iii) Give a point estimate of the population mean.

(iv) Construct a 80% confidence interval estimates for the mean.

(v) Construct a 95% confidence interval for the mean.

15. The Dean of Students at UTC has said that the average grade of UTC students is higher than that of


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the students at GSU. Random samples of grades from the two schools are selected, and the results are

shown below.

UTC GSU

Sample Size 14 12

Sample Mean 2.85 2.61

SampleStandardDeviation

0.40 0.35

Sample Mode 2.5 3.0

(i) Give the hypotheses.

(ii) Compute the test statistic.

(iii) At a 0.1 level of significance, test the Dean of Students statement.

(16) Random samples of employees from three different departments of NMC Corporation showed the

following yearly income (in $ 1,000).

Department A Department B Department C

40 46 46

37 41 40

43 43 41

41 33 48

35 41 39


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3842 44

At 05 . = , test to determine if there is a significant difference among the average income of the

employees from the three departments. Use both the critical and p- value approaches.

17. The sales records of two branches of a department store over the last 12 months are shown below.

(Sales figures are in thousands of dollars). We want to use the Mann-Whitney-Willcoxon test to

determine if there is a significant difference in the sales of the two branches.

Month Branch A Branch B

1 257 210

2 280 230

3 200 250

4 250 260

5 284 275

6 295 300

7 297 320

8 265 290

9 330 310

10 350 325

11 340 329

12 272 335

(i) Compute the sum of the ranks (T) for branch A.

(ii) Compute the mean T .

(iii) Compute T .

(iv) Use = 0.05 and test to determine if there is a significant difference in the populations of the sales of

the two branches.

(18) Two faculty members ranked 12 candidates for scholarships. Calculate the Spearman rank-

correlation coefficient and test it for significance. Use 0.02 level of significance.

Rank by Rank by

Candidate Professor A Professor B


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1 6 5

2 10 11

3 2 6

4 1 3

5 5 4

6 11 12

7 4 2

8 3 1

9 7 7

10 12 10

11 9 8

12 8 9

19. The following data represent the number of flash drives sold per day at a local computer shop and

their prices.

Price (x) Units Sold (y)

$34 3

36 4


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32 6

35 5

30 9

38 2

40 1

(i) Develop a least-squares regression line and explain what the slope of the line indicates.

(ii) Compute the coefficient of determination and comment on the strength of relationship between x and

y.

(iii) Compute the sample correlation coefficient between the price and the number of flash drives sold.

Use 01 . 0 = to test the relationship between x and y.

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