statistics & probability

Mathematics Assignment-II

Q1. The arithmetic mean height of 50 student of a college is 5’ 8”. The height of 30 of these is given on the frequency distribution below. Find the arithmetic mean height of the remaining 20 students.

Height(in inches) 5’ 4” 5’ 6”5’ 8” 5’ 10” 6’ 0”

Frequency 4 12 4 8 2

Q2. Comment on the performance of the students in three universities given below using simple and weighted averages:

University

CourseOf study

Bombay Calcutta Madras% of pass

No. of students (in ‘00s)

% of pass


% of pass


M.A. 71 3 82 2 81 2

M. Com. 83 4 76 3 76 3.5

B.A. 73 5 73 6 74 4.5

B. Com. 74 2 76 7 58 2

B. Sc. 65 3 65 3 70 7

M. Sc. 66 3 60 7 73 2

Q3. Eight coins were tossed together and the number of heads (x) resulting was noted. The operation was repeated 256 times and the frequency distribution of the number of heads is given below:

No. of heads (x) 0 1 2 3 4 5 6 7 8

Frequency (f) 1 9 26 59 72 52 29 7 1

Calculate median.

Q4. The following are the marks obtained by the students in statistics:

Marks No. of Students Marks No. of Students

10 or less 4 40 or less 40



Draw a curve on the graph paper and show therein:

(i) The range of marks obtained by middle 80% of the students.(ii) The median.

Also verify your result by direct formula calculation.

Q5. Find the value of mean and mode from the data given below:

Wt. in kg. No. of students Wt. in Kg. No. of students

93-97 2 113-117 14

98-102 5 118-122 6

103-107 12 123-127 3

108-112 17 128-132 1

Q6. Calculate mode from the following data:

Marks No. of students Marks No. of students

Below 10 4 below 60 86




Below 50 67

Q7. Below is given the frequency distribution of weights of a group of 60 students of a class in a school:

Wt. in kg. No. of students Wt. in Kg. No. of students

30-34 3 50-54 14

35-39 5 55-59 6

40-44 12 60-64 2

45-49 18

Q8. Calculate Karl Pearson’s coefficient of correlation between expenditure on advertising and sales from the data given below.

Advertising exp.(‘000 Rs) : 39 65 62 90 82 75 25 98 36 78

Sales (lakh Rs) 47 53 58 86 62 68 60 91 51 84

Q9. From the following table calculate the coefficient of correlation by Karl Pearson’s method.

X 6 2 10 4 8

Y 9 11 ? 8 7

Arithmetic means of X and Y series are 6 and 8 respectively.

Q10. Find Karl Pearson’s coefficient of correlation between sales and expenses of the following ten firms:

Firm 1 2 3 4 5 6 7 8 9 10

Sales(‘000 Rs) 50 50 55 60 65 65 65 60 60 50

Exp. (‘000 Rs) 11 13 14 16 16 15 15 14 13 13

Q11. Find Karl Pearson’s coefficient of correlation between the age and the playing habit of the people from the following information:

Age group (years) No. of people No. of players

15 and less than 20 200 150






Also mention what does your calculated ‘r’ indicate.

Q12. Calculate coefficient of correlation between X and Y series from the following data and calculate its probable error also :

X 78 89 96 69 59 79 68 61

Y 125 137 156 112 107 136 123 108

Take 69 as working mean for X and 112 for that for Y.

Q13. Family income and its percentage spent on food in the case of hundred families gave the following bivariate frequency distribution. Calculate the coefficient of correlation and interpret its value.

Family income (in Rs.)Food Expenditure(in %)

200-300 300-400 400-500 500-600 600-700

10-15 - - - 3 715-20 - 4 9 4 320-25 7 6 12 5 -25-30 3 10 19 8 -

Q14. Calculate Karl Pearson’s co-efficient of correlation from the data given below:

Age in years

Marks18 19 20 21 22

20-25 3 2 - - -15-20 - 5 4 - -10-15 - - 7 10 -

5-10 - - - 3 20-5 - - - 3 1

Q15. From the following data, obtain the two regression equations:

Sales :91 97 108 121 67 124 51 73 111 57

Purchase :71 75 69 97 70 91 39 61 80 47

Q16. From the data given below find:

(a) The two regression equation.(b) The coefficient of correlation between the marks in Economics and Statistics.(c) The most likely marks in Statistics when marks in Economics are 30.

Marks in Eco : 25 28 35 32 31 36 29 38 34 32

Marks in Stats : 43 46 49 41 36 32 31 30 33 39

Q17. A panel of judges A and B graded seven debators and independently awarded the following marks:

Debator Marks by A Marks by B

1 40 32

2 34 39

3 28 26

4 30 30

5 44 38

6 38 34

7 31 28

An eighth debator was awarded 36 marks by judge A while judge B was not present.

If judge B were also present, how many marks would you expect him to award to the eighth debator assuming that the same degree of relationship exists in their judgement?

Q18. Obtain the equations of the two lines of regressions for the following data:

X: 43 44 46 40 44 42 45 42 38 40 42 57

Y: 29 31 19 18 19 27 27 29 41 30 26 10

Hence obtain the value of the correlation coefficient between X and Y.

Q19. The following table shows the number of motor registrations in a certain territory for a term of 5 years and the sale of motor tyres by a firm in that territory for the same period.

Year Motor Registration No. of tyres sold

1 600 1250

2 630 1100

3 720 1300

4 750 1350

5 800 1500

Find the regression equation to estimate the sale of tyres when motor registration is known. Estimate sale of tyres when registration is 850.

Q20. Four cards are drawn at random from a pack of 52 cards. Find the probability that

(i) They are a king, a queen, a jack and an ace.(ii) Two are kings and two are aces.(iii) All are diamonds.(iv) Two are red and two are black.(v) There is one card of each suit.(vi) There are two cards of clubs and two cards of diamonds.

Q21. A committee of 4 persons is to be appointed from 3 officers of the production department, 4 officers of the purchase department, 2 officers of the sales department and 1 chartered accountant. Find the probability of forming the committee in the following manner:

(i) There must be one from each category.

(ii) It should have at least one from the purchase deptt.(iii) The chartered accountant must be in the committee.

Q22. A committee of four has to be formed from among 3 economists, 4 engineers, 2 statisticians and 1 doctor.

(i) What is the probability that each of the four professions is represented on the committee?

(ii) What is the probability that the committee consists of the doctor and at least one economist?

Q23. In a certain college, the students engage in various sports in the following proportions:

Football(F) : 60% of all students

Basketball(B) : 50% of all students

Both football and basketball : 30% of all students.

If a student is selected at random, what is the probability that he will:

(i) Play football or basketball?(ii) Play neither sports?

Q24. Probability that a man will be alive 25 years hence is 0.3 and the probability that his wife will be alive 25 years hence is 0.4. Find the probability that 25 years hence,

(i) Both will be alive.(ii) Only the man will be alive.(iii) Only the woman will be alive.(iv) At least one of them will be alive.

Q25. The probability that a management trainee will remain with a company is 0.6. The probability that an employee earns more than Rs. 10,000 per year is 0.5. The probability that an employee is a management trainee who remained with the company or who earns more than Rs. 10,000 per year is 0.7. What is the probability that an employee earns more than Rs. 10,000 per year given that he is a management trainee who stayed with the company?

Q26. A market research firm is interested in surveying certain attitudes in a small community. There are 125 households broken down according to income, ownership of a telephone or ownership of a T.V.

Household with Household with

annual income of annual income

Rs. 8000 or less above Rs. 8000

Telecom No Telecom No

Subscriber Telephone Subscriber Telephone

Only T.V. set 27 20 18 10

No T.V. set 18 10 12 10

(i) What is the probability of obtaining of a T.V. owner in drawing at random?(ii) If a household has income over Rs. 8000 and is a telephone subscriber, what is the

probability that it has a T.V. ?(iii) What is the conditional probability of drawing a household that owns a T.V., given that

the household is a telephone subscriber ?(iv) Are the events ‘ownership of a T.V.’ and ‘telephone subscriber’ statistically

independent ? Comment.

Q26. The probability that a person stopping at a petrol pump will get his tyres checked is 0.12, the probability that he will get his oil checked is 0.29, and the probability that he will get both checked is 0.07.

(i) What is the probability that a person stopping at this pump will have neither his tyres nor oil checked ?

(ii) Find the probability that a person who has his oil checked will also have his tyres checked.

Q27. The unbiased coins are tossed simultaneously. Find the probability of obtaining,

(i) Exactly 6 heads.(ii) At least 8 heads.(iii) No head(iv) At least one head.

(v) Not more than 3 heads.(vi) At least 4 heads.

Q28. A merchant’s file of 20 accounts containing 6 delinquent and 14 non-delinquent accounts. An auditor randomly selects 5 of these accounts for examination.

(i) What is the probability that the auditor finds exactly 2 delinquent accounts ?(ii) Find the expected number of delinquent accounts in the sample selected.

Q29. If the chance that the vessel arrives safely at a port is 0.9, find the chance that out of 5 vessels expected at least 4 will arrive safely.

Q30. Assume that half the population is vegetarian so that the chance of an individual being vegetarian is 0.5. Assuming that 100 investigator each take sample of 10 individual to see whether they are vegetarians, how many investigators would you expect to report that three people or less were vegetarian ?

Q31. With the usual notations, find p for a binomial random variable X if n=6 and 9P(X=4)=P(X=2).

Q32. The mean and variance of a binomial distribution are 3 and 2, respectively. Find the probability that the variate takes values

(i) Less than or equal to 2.(ii) Greater than or equal to 7.

Q33. If the probability of a defective bolt is 0.1, find

(i) The mean.(ii) Variance.(iii) Moment coefficient of skewness.(iv) Kurtosis.

For the distribution of defective bolts in a total of 400.

Q34. (a) 8 coins are tossed at a time, 256 times. Find the expected frequencies of success(getting a head) and tabulate the results obtained.

(b) Also obtain the values of the mean and standard deviation of the theoretical(fitted) distribution.

Q35. Fit a binomial distribution to the following data:

X: 0 1 2 3 4

Y: 28 62 46 10 4

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