Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Students can download 12th Business Maths Chapter 10 Operations Research Ex 10.4 Questions and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Choose the correct answer.

Question 1.
The transportation problem is said to be unbalanced if _______
(a) Total supply ≠ Total demand
(b) Total supply = Total demand
(c) m = n
(d) m + n – 1
Answer:
(a) Total supply ≠ Total demand

Question 2.
In a non-degenerate solution number of allocation is ________
(a) Equal to m + n – 1
(b) Equal to m + n + 1
(c) Not equal to m + n – 1
(d) Not equal to m + n + 1
Answer:
(a) Equal to m + n – 1

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Question 3.
In a degenerate solution number of allocations is ________
(a) equal to m + n – 1
(b) not equal to m + n – 1
(c) less than m + n – 1
(d) greater than m + n – 1
Answer:
(c) less than m + n – 1

Question 4.
The penalty in VAM represents the difference between the first ________
(a) Two largest costs
(b) Largest and Smallest costs
(c) Smallest two costs
(d) None of these
Answer:
(c) Smallest two costs

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Question 5.
Number of basic allocation in any row or column in an assignment problem can be ________
(a) exactly one
(b) at least one
(c) at most one
(d) none of these
Answer:
(a) exactly one

Question 6.
North-West Comer refers to _________
(a) top left corner
(b) top right comer
(c) bottom right comer
(d) bottom left comer
Answer:
(a) top left corner

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Question 7.
Solution for transportation problem using _________ method is nearer to an optimal solution.
(a) NWCM
(b) LCM
(c) VAM
(d) Row Minima
Answer:
(c) VAM

Question 8.
In an assignment problem the value of decision variable xij is _______
(a) 1
(b) 0
(c) 1 or 0
(d) none of them
Answer:
(c) 1 or 0

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Question 9.
If the number of sources is not equal to the number of destinations, the assignment problem is called _______
(a) balanced
(b) unsymmetric
(c) symmetric
(d) unbalanced
Answer:
(d) unbalanced

Question 10.
The purpose of a dummy row or column in an assignment problem is to _________
(a) prevent a solution from becoming degenerate
(b) the balance between total activities and total resources
(c) provide a means of representing a dummy problem
(d) none of the above
Answer:
(b) the balance between total activities and total resources

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Question 11.
The solution for an assignment problem is optimal if _________
(a) each row and each column has no assignment
(b) each row and each column has at least one assignment
(c) each row and each column has at most one assignment
(d) each row and each column has exactly one assignment
Answer:
(d) each row and each column has exactly one assignment

Question 12.
In an assignment problem involving four workers and three jobs, total numbers of assignments possible are ______
(a) 4
(b) 3
(c) 7
(d) 12
Answer:
(b) 3

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.4

Question 13.
Decision theory is concerned with ________
(a) analysis of information that is available
(b) decision making under certainty
(c) selecting optimal decisions in sequential problem
(d) All of the above
Answer:
(d) All of the above

Question 14.
A type of decision-making environment is _______
(a) certainty
(b) uncertainty
(c) risk
(d) all of the above
Answer:
(d) all of the above

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Students can download 12th Business Maths Chapter 10 Operations Research Additional Problems and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

I. One Mark Questions

Choose the correct Answer

Question 1.
Which of the following methods is used to verify the optimality of the current solution of the transportation problem?
(a) Least cost method
(b) Vogel’s method
(c) North-west comer rule
(d) None of these
Answer:
(a) Least cost method

Question 2.
The degeneracy’in the transportation problem indicates that _________
(a) Dummy allocations need to be added
(b) The problem has no feasible solution
(c) Multiple optimal solutions exist
(d) All of the above
Answer:
(c) Multiple optimal solutions exist

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 3.
The Hungarian method can also be used to solve ______
(a) Transportation problem
(b) Travelling salesman problem
(c) A linear programming problem
(d) All the above
Answer:
(b) Travelling salesman problem

Question 4.
An optimal solution of an assignment problem can be obtained only if, _________
(a) each row and column has only one zero element
(b) each row and column has at least one zero element
(c) The data is arranged in a square matrix
(d) None of the above
Answer:
(d) None of the above

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 5.
Say True or False.

  1. In a transportation problem, a single source may supply something to all destinations.
  2. A transportation model must have the same number of rows and columns.
  3. It is usually possible to find an optimal solution to a transportation problem that is degenerate.
  4. In a transportation problem, a dummy source is given a zero cost, while in an assignment problem, a dummy source is given a very high cost.
  5. The Hungarian method operates on the principle of matrix reduction, whereby the cost table is reduced to a set of opportunity costs.

Answer:

  1. True
  2. False
  3. True
  4. False
  5. True

Question 6.
Fill in the blanks.

  1. In a transportation problem, we must make the number of ________ and _______ equal.
  2. ______ or ______ are used to balance an assignment problem.
  3. The method of finding an initial solution based on opportunity costs is called _______
  4. ________ occurs when the number of occupied squares is less than the number of rows plus the number of columns minus one.
  5. Both transportation and assignment problems are members of a category of LP problems called ________
  6. In the case of an unbalanced problem, shipping cost coefficients of ______ are assigned to each dummy factory or warehouse.

Answer:

  1. units supplied, units demanded
  2. Dummy rows, dummy columns
  3. Vogel’s approximation method
  4. Degeneracy
  5. Network flow problems
  6. zero

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 7.
Match the following.

(a) Dummy column (i) Finding initial solution
(b) Northwest comer rule (ii) Rows ≠ Columns
(c) Hungarian method (iii) Supply ≠ Demand
(d) Feasible solution (iv) Assignment problem
(e) Unbalanced problem (v) All demand and supply constraints are met

Answer:
(a) – (iii)
(b) – (i)
(c) – (iv)
(d) – (v)
(e) – (ii)

Question 8.
The objective function of transportation problem is to ________
(a) Maximise total cost
(b) Minimise the total cost
(c) Total cost should be zero
(d) All the above
Answer:
(b) Minimise the total cost

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 9.
In transportation problem, optimal solution can be verified by using _______
(a) NWC
(b) LCM
(c) MODI method
(d) Matrix method
Answer:
(c) MODI method

Question 10.
The cells in the transportation problem can be classified as _______
(a) assigned cells and empty cells
(b) allocated cells and unallocated cells
(c) occupied and unoccupied cells
(d) assigned and unoccupied cells
Answer:
(c) occupied and unoccupied cells

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 11.
In transportation problem if total supply > total demand we add _________
(a) dummy row with cost 0
(b) dummy column with cost 0
(c) dummy row with cost 1
(d) dummy column with cost 1
Answer:
(b) dummy column with cost 0

Question 12.
In an LPP the objective function is to be ________
(a) Minimised
(b) Maximised
(c) (a) or (b)
(d) only (b)
Answer:
(c) (a) or (b)

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 13.
The method used for solving an assignment problem is called ________
(a) Reduced matrix method
(b) MODI method
(c) Hungarian method
(d) Graphical method
Answer:
(c) Hungarian method

II. 2 Mark Questions

Question 1.
Consider 3 jobs to be assigned to 3 machines. The cost for each combination is shown in the table below. Find the minimal job machine combinations.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 1
Solution:
Step 1:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 2
Step 2:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 3
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 4
Optimal assignment:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 5

Question 2.
Find an initial basic feasible solution by LCM.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 6
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 7
Total cost = (1 × 2) + (6 × 1) + (4 × 4) + (4 × 6)
= 2 + 6 + 16 + 24
= 48

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 3.
Find an initial basic feasible solution by the North West Corner Rule (NWC).
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 8
Solution:
Total demand = Total supply = 60
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 9
Total cost = (10 × 9) + (11 × 6) + (12 × 8) + (2 × 3) + (25 × 11)
= 90 + 66 + 96 + 6 + 275
= 533

Question 4.
Find an initial basic feasible solution using Least cost method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 10
Solution:
Total Demand = 5 + 8 + 7 + 14 = 34
Total Supply = 7 + 9 + 18 = 34
Since they are equal, problem is balanced.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 11
The minimum total transportation cost is = (7 × 10) + (2 × 70) + (7 × 40) + (3 × 40) + (8 × 8) + (7 × 20)
= 70 + 140 + 280 + 120 + 64 + 140
= Rs. 814

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 5.
Find the investment option using Maximin rule for the following:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 12
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 13
Max (5, -13, -5) = 5. Since the maximum payoff is 5, by maximin criteria, the decision is to invest in bonds.

III. 3 and 5 Marks Questions

Question 1.
Find an optimal solution to the following transportation problem by North West Corner Method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 14
Solution:
Total supply = 65 = Total demand. So the given problem is balanced.
First allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 15
Second allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 16
Third allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 17
Fourth allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 18
Total Transportation cost
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 19

Question 2.
Find an initial basic solution for the following transportation problem by Vogel’s Approximation method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 20
Solution:
Total demand = 72 + 102 + 41 = 215 and
Total supply = 76 + 82 + 77 = 235.
Total supply > Total demand. So we add a dummy constraint with 0 unit cost and with allocation 20 (235 – 215). The modified table is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 21
First allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 22
The maximum penalty is 16. Allot 20 units to cell (S2, Ddummy)
Second allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 23
Third allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 24
Fourth allocation:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 25
The final allocation table is given below.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 26
The minimum total cost = (76 × 8) + (21 × 24) + (41 × 16) + (20 × 0) + (72 × 8) + (5 × 16) = 2424

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 3.
A company has 4 men available for 4 separate jobs. Only one man can work on anyone job. The cost of assigning each man to each job is given below. Find the optimal solution by the Hungarian method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 27
Solution:
The number of rows and columns are equal. So the given problem is a balanced assignment problem and we can get an optimal solution.
Step 1:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 28
Step 2:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 29
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 30
We are not able to assign job for person B. Proceed as follows. Draw a minimum number of vertical and horizontal lines to cover all the zeros.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 31
Subtract the smallest element 1 from all the uncovered elements and add it to the elements which lie at the intersection of two lines. Thus we obtain another reduced matrix for fresh assignment.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 32
Total cost is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 33

Question 4.
There are five machines and five jobs are to be assigned and the cost matrix is given below. Find the proper assignment.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 34
Solution:
Step 1: (Row-reduction)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 35
Step 2: (Column – reduction)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 36
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 37
We are not able to assign a machine to job D. We proceed as follows.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 38
The smallest uncovered element is 2. Subtract 2 from all those elements which are not covered. Add 2 all elements which are at the intersection of two lines. Then proceed with the new matrix.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 39
The assignment is as follows
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 40

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

Question 5.
The cost of transportation from 3 sources to four destinations are given in the follow¬ing table. Obtain an initial basic feasible solution using
(i) North West Corner Rule (NWC)
(ii) Least Cost Method (LCM) and
(iii) Vogel’s Approximation Method (VAM)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 41
Solution:
(i) North West Corner Rule
We start by allotting the units to the North -West Comer cell. We show all the allocations in a single table.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 42
Total transportation cost is (200 × 4) + (50 × 2) + (350 × 7) + (100 × 5) + (200 × 3) + (1 × 300)
= 800 + 100 + 2450 + 500 + 600 + 300
= Rs. 4750

(ii) Least cost method (LCM)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 43
Transportation cost is = (250 × 2) + (200 × 3) + (150 × 7) + (100 × 5) + (200 × 3) + (300 × 1)
= 500 + 600+ 1050 + 500 + 600 + 300
= Rs. 3550

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems

(iii) Vogel Approximation Method (VAM)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 44
There are five penalties which have the maximum value 2. The cell with the least cost is row 3 and hence select cell (3, D) for allocation.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 45
There are four penalties which have maximum value 2. Select cell (1, B) which has the least cost for allocation.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 46
The largest penalty is 6. Allot units to cell (2, A)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 47
The largest penalty is 3. Allot units to cell (3, B)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 48
We first allot 50 units to cell (3, C) which has less cost. Then the balance units we allot to cell (2, C). We get the final allocation table as follows.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Additional Problems 49
Transportation cost is = (250 × 2) + (200 × 3) + (250 × 5) + (150 × 4) + (50 × 3) + (300 × 1)
= 500 + 600 + 1250 + 600 + 150 + 300
= Rs. 3400

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Students can download 12th Business Maths Chapter 8 Sampling Techniques and Statistical Inference Additional Problems and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Choose the correct answer:

Question 1.
Non sampling error is reduced by ________
(a) Increasing sample size
(b) Decreasing sample size
(c) Reducing amount of data
(d) None of these
Answer:
(d) None of these

Question 2.
Any numerical value calculated from sample data is called ______
(a) Error
(b) Statistic
(c) Bias
(d) Mean
Answer:
(b) Statistic

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Question 3.
In sampling with replacement a sampling unit can be selected _______
(a) Only once
(b) More than one time
(c) Less than one time
(d) None of above
Answer:
(b) More than one time

Question 4.
Standard deviation of sampling distribution of any statistic is called ________
(a) Sampling error
(b) Type-I error
(c) Standard error
(d) Non-sampling error
Answer:
(c) Standard error

Question 5.
The difference between statistic and parameter is called ________
(a) Random error
(b) Sampling error
(c) Standard error
(d) Bias
(e) Error
Answer:
(e) Error

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Question 6.
In random sampling, the probability of selecting an item from the population is _______
(a) unknown
(b) known
(c) undecided
(d) zero
Answer:
(b) known

Question 7.
Match the following:

(i) Type I error (a) determine whether a statistical result is significant
(ii) Type II error (b) Left-tailed test
(iii) Hypothesis testing (c) reject a true null hypothesis
(iv) H1 : µ > µ0 (d) do not reject a false null hypothesis
(v) H1 : µ < µ0 (e) Right-tailed test

Answer:
(i) – (c)
(ii) – (d)
(iii) – (a)
(iv) – (e)
(v) – (b)

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Question 8.
Fill in the blanks:

  1. Any statement whose validity is tested on the basis of a sample is called ________
  2. The alternative hypothesis is also called _______
  3. The probability of rejecting the null hypothesis when it is true is called ________
  4. The hypothesis µ ≤ 10 is a _______
  5. If a hypothesis specifies the population distribution it is called ______

Answer:

  1. Statistical hypothesis
  2. Research hypothesis
  3. Level of significance
  4. Composite hypothesis
  5. Simple hypothesis

Question 9.
Null and alternative hypothesis are statements about ________
(a) population parameters
(b) sample parameters
(c) sample statistics
(d) none of the above
Answer:
(a) population parameters

2 and 3 Mark Questions

Question 1.
A random sample of size 50 with mean 67.9 is drawn from a normal population. If the S.E of the sample mean is √0.7, find a 95% confidence interval for the population mean.
Solution:
n = 50, \(\bar{x}\) = 67.9
95% confidence limits for the population mean µ are
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems II Q1
Thus the 95% confidence intervals for estimating µ is given by (66.26, 69.54)

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Question 2.
A random sample of 500 apples was taken from large consignment and 45 of them were found to be bad. Find the limits at which the bad apples lie at 99% confidence level.
Solution:
Sample size n = 500
Proportion of bad apples P = \(\frac{45}{500}\) = 0.09
So proportion of good apples Q = 1 – 0.09 = 0.91
The confidence limits for population proportion are given by,
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems II Q2
= (0.09 – (2.58) (0.013), 0.09 + (2.58) (0.013)) .
= (0.09 – 0.034, 0.09 + 0.034)
= (0.056, 0.124)
Thus the bad apples lie between 5.6% and 12.4%

Question 3.
A sample of 400 students is found to have a mean height of 171.38 cms can it be regarded as a sample from a large population with mean height 171.17 cms and S.D 3.30 cms?
Solution:
Given
Sample size n = 400
Sample mean \(\bar{x}\) = 171.38 cm
Population mean µ = 171.17 cm
Population SD σ = 3.30 cm
H0 : µ = 171.17 cm
H1 : µ ≠ 171.17 cm
Test statistic:
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems II Q3
The table value of \(z_{\alpha / 2}\) at 5% level is 1.96. Since Z < \(z_{\alpha / 2}\), H0 is accepted. Therefore the given sample can be regarded as one from the population with mean 171.17 cm.

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Question 4.
An automatic machine fills tea in sealed tins with a mean weight of tea as 1 kg and S.D 1 gram. A random sample of 50 tins was examined and it was found that their mean weight was 999.5 grams. Is the machine working properly?
Solution:
Given
Sample size n = 50
Sample mean \(\bar{x}\) = 999.5 grams
Population mean µ = 1000 grams
Population SD σ = 1 gram
H0 : µ = 1 kg
H1 : µ ≠ 1 kg
Test statistic
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems II Q4
The table value of \(z_{\alpha / 2}\) at 1% level = 2.58. Since |Z| > \(z_{\alpha / 2}\), H0 is rejected. Therefore the machine is not working properly.

5 Marks Questions

Question 1.
A simple random sample of size 100 has mean
(а) 15, the population variance being 25. Find an interval estimate of the population mean with a confidence level of 95% and 99%
(b) If the population variance is not given, what should be done to find out the required estimates?
Solution:
(a) Given
Sample size n = 100
Sample mean \(\bar{x}\) = 15
Population SD σ = 5
The 95% confidence interval for the population mean is \(\bar{x} \pm \mathrm{Z}_{\alpha / 2} \frac{\sigma}{\sqrt{n}}\)
Here \(z_{\alpha / 2}\) = 1.96. So we get
= 15 ± (1.96) (\(\frac{5}{\sqrt{100}}\))
= 15 ± (1.96) (0.5)
= 15 ± 0.98
= 14.02 and 15.98
Therefore 95% confidence interval for population mean µ is (14.02, 15.98)
The 99% confidence interval is \(\bar{x} \pm 2.58 \frac{\sigma}{\sqrt{n}}\)
= 15 ± 2.58 (\(\frac{5}{\sqrt{100}}\))
= 15 ± 2.58 (0.5)
= 13.71 and 16.29
Therefore 99% confidence interval for the population mean µ is (13.71, 16.29)
(b) If population S.D a is not known, then the sample S.D can be used in the place of o in estimating the confidence interval.

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Question 2.
A factory is producing 50,000 pairs of shoes daily. From a sample of 500 pairs, 2% were found to be of sub-standard quality. Estimate the number of pairs that can be reasonably expected to be spoiled at 95% level of confidence.
Solution:
N = 50,000, n = 500, P = \(\frac{2}{100}\), Q = \(\frac{98}{100}\)
The estimated percentage of spoiled pairs in daily production = \(\frac{2}{100}\) × 50,000 = 1000
The limits for the number of spoiled pans at 95% level of confidence
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems III Q2

Question 3.
A company that packages peanuts states that at a maximum 6% of the peanut shells contain no nuts. At random, 300 peanuts were selected and 21 of them were empty. With a significance level of 1% can the statement made by the company be accepted?
Solution:
The population proportion P = 6% = 0.06
The null hypothesis: H0 : P ≤ 0.06
Alternative hypothesis: H1 : P > 0.06
For α = 1% = 0.01. Zα = 2.33
The test statistic is P + 2.33 (\(\sqrt{\frac{(0.06)(0.94)}{300}}\)) = 0.092
(where n = 300, P = 0.06, Q = 0.94)
Since the calculated value is less than the table value, 0.092 < 2.33,we accept the null hypothesis H0.
Hence the statement of the company can be accepted.

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Question 4.
A school principal claims that the students in his school are above average intelligence. A random sample of 30 students IQ scores has a mean score of 112.5. The mean population IQ is 100 with an SD of 15. Is there sufficient evidence to support the principal’s claim?
Solution:
Given
Population mean µ = 100
Population SD σ = 15
Sample size n = 30
Sample mean \(\bar{x}\) = 112.5
Null hypothesis H0 : µ = 100
(the students have average I.Q)
Alternative hypothesis H1 : µ > 100
(the students have above average I.Q scores)
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems III Q4
Let the significance level α = 0.05. The table value Zα = 1.645, since this is one-tailed test.
Test Statistic:
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems III Q4.1
Since 4.56 > 1.645 (ie) Z > Zα at 5% level, we reject the null hypothesis. Hence we conclude that the students have above average IQ scores. So the principal’s claim is right.

Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems

Question 5.
Boys of a certain age are known to have a mean weight of 85 pounds. A complaint is made that the boys living in hostels are underfed. So a sample of 25 boys are weighed and ‘ found to have a mean weight of 80.94 pounds. The population S.D is 11.6. What should be concluded about the complaint?
Solution:
Given n = 25, µ = 85, \(\bar{x}\) = 80.94, and σ = 11.6
Null hypothesis H0 : µ = 85 (the boys are not underfed)
Alternative hypothesis H1 : µ < 85 (the boys are underfed)
Test statistic:
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems III Q5.1
Let us take the significance level α = 0.05. The table value is Zα = -1.645
Samacheer Kalvi 12th Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Additional Problems III Q5
Now -1.75 < -1.645 (i.e) Z < Zα. Therefore we reject the null hypothesis. Since the calculated value falls in the rejection region. Hence we conclude that the boys are underfed. So the complaint should be addressed immediately.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Students can download 12th Business Maths Chapter 9 Applied Statistics Additional Problems and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Choose the correct answer.

Question 1.
Match the following.

(a) Seasonal variation (i) Erratic variation
(b) Secular trend (ii) Business cycle
(c) Irregular variation (iii) Long time variation
(d) Cyclical variation (iv) Short time variation

Answer:
(a) – (iv)
(b) – (iii)
(c) – (i)
(d) – (ii)

Question 2.
The secular trend can be measure by ______
(a) 4 methods
(b) 1 method
(c) 2 methods
(d) none of these
Answer:
(a) 4 methods

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Question 3.
Method of simple averages is used to measure _________
(a) Secular trend
(b) Irregular variation
(c) Seasonal variation
(d) Cyclic variation
Answer:
(c) Seasonal variation

Question 4.
Increase in the number of patients in the hospital due to heatstroke is _______
(a) Secular trend
(b) Irregular variation
(c) Seasonal variation
(d) Cyclic variation
Answer:
(c) Seasonal variation

Question 5.
In time series seasonal variations can occur within a period of ________
(a) 4 years
(b) 3 years
(c) one year
(d) 9 years
Answer:
(c) one year

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Question 6.
Fill in the blanks.

  1. The method of moving averages is used to find the _________
  2. Most frequently used mathematical model of a time series is _________
  3. The sale of air condition increases during summer is a ________
  4. The fire in a factory is an example of ________
  5. The best-fitting trend is one in which the sum of squares of residuals is _______

Answer:

  1. Secular trend
  2. Multiplicative model
  3. Seasonal variation
  4. Irregular variation
  5. Least

Question 7.
True or False

  1. An index number is used to measure changes in a variable over time.
  2. The ratio of a new price to the base year price is called the price relative.
  3. The Laspeyre’s and Paasche index numbers are examples of weighted quantity index only.
  4. \(\frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}} \times 100\) is Laspeyre’s quantity index.
  5. Laspeyre’s price index regards the base year quantities as fixed.

Answer:

  1. True
  2. True
  3. False
  4. False
  5. True

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Question 8.
Index for base period is always taken as ______
(a) 100
(b) 1
(c) 200
(d) 0
Answer:
(a) 100

Question 9.
Consumer price index indicates _______
(a) Rise
(b) Fall
(c) both (a) & (b)
(d) neither (a) & (b)
Answer:
(c) both (a) & (b)

Question 10.
The purchasing power of money can be accessed through ______
(a) simple index
(b) Fisher’s index
(c) Consumer price index (CPI)
(d) Volume index
Answer:
(c) Consumer price index (CPI)

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Question 11.
For consumer price index, price quotations are collected from _______
(a) Fair price shops
(b) Government depots
(c) Retailers
(d) Whole-sale dealers
Answer:
(c) Retailers

Question 12.
The aggregative expenditure method and family budget method always give ________
(a) Different results
(b) Approximate results
(c) Same results
(d) None of these
Answer:
(c) Same results

Question 13.
The Federal Bureau of statistics prepares ________
(a) The wholesale price index
(b) CPI
(c) Sensitive price indicator
(d) All the above
Answer:
(d) All the above

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Question 14.
Paasche’s price index number is also called _________
(a) Base year weighted
(b) Current year weighted
(c) Simple aggregative index
(d) Consumer price index
Answer:
(b) Current year weighted

Question 15.
Index number calculated by Fisher’s formula is ideal because it satisfies ________
(a) Circular test
(b) Factor reversal test
(c) Time reversal test
(d) All of the above
Answer:
(d) All of the above

2 Mark Questions

Question 1.
From the data given below calculate seasonal Indices:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems II Q1
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems II Q1.1
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems II Q1.2

Question 2.
Using 3-year moving averages, determine the trend values from the following data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems II Q2
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems II Q2.1

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Question 3.
A company estimates its average monthly sales in a particular year to be Rs.2,00,000. The seasonal indices of the sales data are given below. Draw a monthly sales budget for the company.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems II Q3
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems II Q3.1

Question 4.
Calculate the index for the data when the average percentage increases in the prices of items and weights are given. Food 15, clothing 3, Rent 4, Fuel 2, Miscellaneous 1, the percentage increases are 32, 54, 47, 78 and 58.
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems II Q4

3 and 5 Marks Questions

Question 1.
Using Fisher’s Ideal Formula, compute price and quantity index number for 1984 with 1982 as the base year, from the given information.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q1
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q1.1
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q1.2

Question 2.
Using the following data, compute Fisher’s Ideal price index number for the current year.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q2
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q2.1

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Question 3.
Calculate the cost of living Index number from the following data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q3
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q3.1
Cost of living index number = \(\frac{\sum P W}{\sum W}=\frac{1568.75}{12}\) = 130.73

Question 4.
Given below are the values of the sample mean (\(\bar{X}\)) and the range (R) for ten samples of size 5 each. Find the control charts and comment on the state of the process.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q4
Use A2 = 0.58, D3 = 0, D4 = 2.115
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q4.1
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q4.2
We observe that all the sample range values are within the control limits values of R chart, But two values of the sample \(\bar{X}\) (i.e) 37, 37 lies below the LCL and 49, 51 lie above the UCL. So the statistical process is out of control.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems

Question 5.
Fit a straight line trend equation by the method of least squares and estimate the trend values.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q5
Solution:
Let Yt = a + bx be the trend line.
Let X = \(\frac{x-1964.5}{0.5}\), x denotes year.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Additional Problems III Q5.1
When x = 1961, Yt = 91.75 – 7(1.25) = 83
When x = 1962, Yt = 91.75 – 5(1.25) = 85.5
We can find other values similarly.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems

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Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems

Question 1.
Using three yearly moving averages, Determine the trend values from the following data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 1
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 2

Question 2.
From the following data, calculate the trend values using fourly moving averages.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 3
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 4

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems

Question 3.
Fit a straight line trend by the method of least squares to the following data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 5
Solution:
Let x denote the years and y denote the sales. Since number of years is even,
we take X = \(\frac{x-1983.5}{0.5}\)
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 6
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 7
The trend values are obtained by substituting the years for x, and given in the table.
When x= 1980, Yt = 55.9875 + 0.83(-7) = 50.1775
When x = 1981, Yt = 55.9875 – 5(0.83) = 51.8375
When x = 1982, Yt = 55.9875 – 3(0.83) = 53.4975
When x= 1983, Yt = 55.9875 – (0.83) = 55.1575
When x = 1984, Yt = 55.9875 + (0.83) = 56.8175
When x = 1985, Yt = 55.9875 + 3(0.83) = 58.4775
When x = 1986, Yt = 55.9875 + 5(0.83) = 60.1375
When x = 1987, Yt = 55.9875 + 7(0.83) = 61.7975
We find that ΣYt = ΣY = 447.9

Question 4.
Calculate the Laspeyre’s, Paasche’s and Fisher’s price index number for the following data. Interpret the data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 8
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 9
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 10
From the index numbers, we conclude that for the same quantity, the price has reduced by 50.5% in the current year compared to the base year according to Laspeyre’s index. By the Paasche’s index number, we see that the price has reduced by 49.68% in the current year, and according to Fisher’s index number it has reduced by 49.9% in the current year.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems

Question 5.
Using the following data, construct Fisher’s Ideal Index Number and show that it satisfies the Factor Reversal Test and Time Reversal Test?
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 11
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 12
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 13
This shows Fisher’s ideal index number satisfies time reversal test.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 14
This shows Fisher’s ideal index number satisfies factor reversal test.

Question 6.
Compute the consumer price index for 2016 on the basis of 2015 from the following data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 15
Solution:
Consumer price index (CPI) is same as cost of living index. We use the aggregate expenditure method which gives CPI as CPI = \(\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times 100\)
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 16
Consumer price index for the year 2016 is = \(\frac{174}{146.50} \times 100\) = 118.77
On the basis of the year 2015, The cost of living has increased by 18.77%

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems

Question 7.
An Enquiry was made into the budgets of the middle-class families in a city gave the following information.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 17
What changes in the cost of living have taken place in the middle-class families of a city?
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 18
For the middle-class families of the city, the cost of living has increased up to 26.1 % in 2011 as compared to 2010.

Question 8.
From the following data, calculate the control limits for the mean and range chart.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 19
Solution:
Since the sample size is 5, we used A2 = 0.577, D3 = 0, D4 = 2.114, from the table given.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 20
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 21
We see that one sample \(\bar{X}\) value 47 is below the LCL of \(\bar{X}\). To infer that the process is not totally out of control since^the difference is less. Further investigation is recommended.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems

Question 9.
The following data gives the average life (in hours) and a range of 12 samples of 5 lamps each. The data are
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 22
Construct control charts for mean and range. Comment on the control limits.
Solution:
In this question the number of observations is 5 for each sample. So we use A2 = 0.577, D3 = 0, D4 = 2.114 from the table of control chart constants.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 23
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 24
We observe that the sample \(\bar{X}\) value 1080 is below the LCL. All sample range values are within the control limits for R. We say that process is out of control.

Question 10.
The following are the sample means and ranges for 10 samples, each of size 5. Calculate the control limits for the mean chart and range chart and state whether the process is in control or not.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 25
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 26
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Miscellaneous Problems 27
We observe that all sample \(\bar{X}\) values are within the control limits value of \(\bar{X}\) chart. But the sample range value 0.8 is above the UCL of the R chart. So we conclude that the process is out of control.

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3

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Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3

Question 1.
Given the following pay-off matrix (in rupees) for three strategies and two states of nature.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q1
Select a strategy using each of the following rule (i) Maximin (ii) Minimax
Solution:
(i) Maximin principle
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q1.1
Max (40, -20, -40) = 40. Since the maximum pay-off is Rs.40, the best strategy is S1 according to maximin rule.
(ii) Minimax principle
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q1.2
Min (60, 10, 150) = 10. Since the minimum pay- off is Rs. 10, the best strategy is S2 according to minimax rule.

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3

Question 2.
A farmer wants to decide which of the three crops he should plant on his 100-acre farm. The profit from each is dependent on the rainfall during the growing season. The farmer has categorized the amount of rainfall as high, medium and low. His estimated profit for each is shown in the table.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q2
If the farmer wishes to plant the only crop, decide which should be his best crop using (i) Maximin (ii) Minimax
Solution:
(i) Maximin principle
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q2.1
Max (2000, 3500, 4000) = 4000. Since the maximum profit is Rs. 4000, he must choose crop C as the best crop.
(ii) Minimax principle
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q2.2
Min (8000, 5000, 5000) = 5000. Since the minimum cost is Rs.5000, he can choose crop B and crop C as the best crop.

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3

Question 3.
The research department of Hindustan Ltd. has recommended paying the marketing department to launch a shampoo of three different types. The marketing types of shampoo to be launched under the following estimated pay-offs for various level of sales.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q2
What will be the marketing manager’s decision if (i) Maximin and (ii) Minimax principle applied?
Solution:
(i) Maximin principle
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q3.1
Max (10, 5, 3) = 10. Since the maximum pay-off is 10 units, the marketing manager has to choose Egg shampoo by Maximin rule.
(ii) Minimax principle
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q3.2
Min (30, 40, 55) = 30. Since the minimum pay-off is 30 units, the marketing manager has to choose Egg shampoo by minimax rule.

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3

Question 4.
Following pay-off matrix, which is the optimal decision under each of the following rule (i) Maximin (ii) Minimax
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q4
Solution:
(i) Maximin principle
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q4.1
Max (5, 7, 9, 8) = 9. Since the maximum pay-off is 9, the optimal decision is A3 according to maximin rule.
(ii) Minimax principle
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.3 Q4.2
Min (14, 11, 11, 13) = 11. Since the minimum pay-off is 11, the optimal decision A2 and A3 according to minimax rule.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Students can download 12th Business Maths Chapter 9 Applied Statistics Ex 9.4 Questions and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Choose the correct answer.

Question 1.
A time series is a set of data recorded _______
(a) Periodically
(b) Weekly
(c) Successive points of time
(d) all the above
Answer:
(d) all the above

Question 2.
A time series consists of ________
(a) Five components
(b) Four components
(c) Three components
(d) Two components
Answer:
(b) Four components

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 3.
The components of a time series which is attached to short term fluctuation are _______
(a) Secular trend
(b) Seasonal variations
(c) Cyclic variation
(d) Irregular variation
Answer:
(d) Irregular variation

Question 4.
Factors responsible for seasonal variations are ______
(a) Weather
(b) Festivals
(c) Social customs
(d) All the above
Answer:
(d) All the above

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 5.
The additive model of the time series with the components T, S, C and I is _______
(a) y = T + S + C × I
(b) y = T + S × C × I
(c) y = T + S + C + I
(d) y = T + S × C + I
Answer:
(c) y = T + S + C + I

Question 6.
Least square method of fitting a trend is _______
(a) Most exact
(b) Least exact
(c) Full of subjectivity
(d) Mathematically unsolved
Answer:
(a) Most exact

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 7.
The value of ‘b’ in the trend line y = a + bx is _________
(a) Always positive
(b) Always negative
(c) Either positive or negative
(d) Zero
Answer:
(c) Either positive or negative

Question 8.
The component of a time series attached to long term variation is trended as _______
(a) Cyclic variation
(b) Secular variations
(c) Irregular variation
(d) Seasonal variations
Answer:
(b) Secular variations

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 9.
The seasonal variation means the variations occurring within ________
(a) A number of years
(b) within a year
(c) within a month
(d) within a week
Answer:
(b) within a year

Question 10.
Another name of the consumer’s price index number is _______
(a) Whole-sale price index number
(b) Cost of living index
(c) Sensitive
(d) Composite
Answer:
(b) Cost of living index

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 11.
Cost of living at two different cities can be compared with the help of _______
(a) Consumer price index
(b) Value index
(c) Volume index
(d) Un-weighted index
Answer:
(a) Consumer price index

Question 12.
Laspeyre’s index = 110, Paasche’s index = 108, then Fisher’s Ideal index is equal to _______
(a) 110
(b) 108
(c) 100
(d) 109
Answer:
(d) 109
Hint:
Fisher’s Index = \(\sqrt{110 \times 108}\) = 109

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 13.
Most commonly used index number is _________
(a) Volume index number
(b) Value index number
(c) Price index number
(d) Simple index number
Answer:
(c) Price index number

Question 14.
Consumer price index are obtained by ________
(a) Paasche’s formula
(b) Fisher’s ideal formula
(c) Marshall Edgeworth formula
(d) Family budget method formula
Answer:
(d) Family budget method formula

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 15.
Which of the following Index number satisfy the time-reversal test?
(a) Laspeyre’s Index number
(b) Paasche’s Index number
(c) Fisher’s Index number
(d) All of them
Answer:
(c) Fisher’s Index number

Question 16.
While computing a weighted index, the current period quantities are used in the _______
(a) Laspeyre’s method
(b) Paasche’s method
(c) Marshall Edgeworth method
(d) Fisher’s ideal method
Answer:
(b) Paasche’s method

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 17.
The quantities that can be numerically measured can be plotted on a ________
(a) p – chart
(b) c – chart
(c) x bar chart
(d) np – chart
Answer:
(c) x bar chart

Question 18.
How many causes of variation will affect the quality of a product?
(a) 4
(b) 3
(c) 2
(d) 1
Answer:
(c) 2

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 19.
Variations due to natural disorder is known as _______
(a) random cause
(b) non-random cause
(c) human cause
(d) all of them
Answer:
(a) random cause

Question 20.
The assignable causes can occur due to _______
(a) poor raw materials
(b) unskilled labour
(c) faulty machines
(d) all of them
Answer:
(d) all of them

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 21.
A typical control charts consists of ________
(a) CL, UCL
(b) CL, LCL
(c) CL, LCL, UCL
(d) UCL, LCL
Answer:
(c) CL, LCL, UCL

Question 22.
\(\bar{X}\) chart is a ______
(a) attribute control chart
(b) variable control chart
(c) neither Attribute nor variable control chart
(d) both Attribute and variable control chart
Answer:
(b) variable control chart

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 23.
R is calculated using ______
(a) \(x_{\max }-x_{\min }\)
(b) \(x_{\min }-x_{\max }\)
(c) \(\bar{x}_{\max }-\bar{x}_{\min }\)
(d) Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4 Q23
Answer:
(a) \(x_{\max }-x_{\min }\)

Question 24.
The upper control limit for \(\bar{X}\) chart is given by _______
(a) \(\overline{\mathrm{X}}+\mathrm{A}_{2} \overline{\mathrm{R}}\)
(b) Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4 Q24
(c) Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4 Q24.1
(d) Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4 Q24.2
Answer:
(c) Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4 Q24.1

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4

Question 25.
The LCL for R chart is given by ________
(a) \(\mathrm{D}_{2} \overline{\mathrm{R}}\)
(b) Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4 Q25
(c) Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.4 Q25.1
(d) \(\mathbf{D}_{3} \overline{\mathbf{R}}\)
Answer:
(d) \(\mathbf{D}_{3} \overline{\mathbf{R}}\)

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2

Students can download 12th Business Maths Chapter 10 Operations Research Ex 10.2 Questions and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2

Question 1.
What is the Assignment problem?
Solution:
Suppose that we have ‘m’ jobs to be performed on ‘n’ machines. The cost of assigning each job to each machine is Cij. (i = 1, 2,…, n and j = 1, 2,…. n).Our objective is to assign different jobs to different machines (one job per machine) to minimize the overall cost. This is known as the assignment problem.

Question 2.
Give the mathematical form of the assignment problem.
Solution:
The mathematical form of assignment problem is Minimize \(\mathrm{Z}=\sum_{i=1}^{n} \sum_{j=1}^{n} \mathrm{C}_{i j} x_{i j}\)
Subject to the constraints
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 1
(or) 1 for all i = 1, 2, …….. n and j = 1, 2, …….. n
where Cij is the cost of assigning ith job to jth machine and xij represents the assignment of ith job to jth machine.

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2

Question 3.
What is the difference between Assignment Problem and Transportation Problem?
Solution:
The assignment problem is a special case of the transportation problem. The differences are given below.

Transportation Problem Assignment Problem
1. This is about reducing cost of transportation merchandise 1. This is about assigning finite sources to finite destinations where only one destination is allotted for one source with minimum cost
2. Number of sources and number of demand need not be equal 2. Number of sources and the number of destinations must be equal
3. If total demand and total supply are not equal then the problem is said to be unbalanced. 3. If the number of rows are not equal to the number of columns then problems are unbalanced.
4. It requires 2 stages to solve: Getting initial basic feasible solution, by NWC, LCM, VAM and optimal solution by MODI method 4. It has only one stage. Hungarian method is sufficient for obtaining an optimal solution

Question 4.
Three jobs A, B and C one to be assigned to three machines U, V and W. The processing cost for each job machine combination is shown in the matrix given below. Determine the allocation that minimizes the overall processing cost. (cost is in ₹ per unit)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 2
Solution:
Here the number of rows and columns are equal.
the given assignment problem is balance.
Step 1: We select the smallest element from each row and subtract from other elements in its row.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 3
Column V has no zero. Go to step 2.
Step 2: Select the smallest element from each column and subtract from other elements in its column.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 4
Since each row and column contains at least one zero, assignments can be made.
Step 3: (Assignment)
Row A contains exactly one zero. We mark it by □ and other zeros in its column by x.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 5
Now proceed column wise. Column V has exactly one zero. Mark by □ and other zeros in its row by X.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 6
Now there is no zero in row B to assign the job. So proceed as follows. Draw a minimum number of lines to cover all the zeros in the reduced matrix. Subtract 5 from all the uncovered elements and add to the element at the intersection of 2 lines as shown below.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 7
Now start the whole procedure once again for assignment to get the following matrix.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 8
Thus all the 3 assignments have been made. The optimal assignment schedule and the total cost is
Job Machine Cost
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 9

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2

Question 5.
A computer centre has got three expert programmers. The centre needs three application programmes to be developed. The head of the computer centre, after studying carefully the programmes to be developed, estimates the computer time in minutes required by the experts to the application programme as follows.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 10
Assign the programmers to the programme in such a way that the total computer time is least.
Solution:
Here the number of rows equals the number of columns. So the given problem is balanced and we can find a solution.
Step 1:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 11
Step 2:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 12
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 13
Now all the 3 programmes have been assigned to the programmers. The optimal assignment schedule and the total cost is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 14
The optimal assignment (minimum) cost is ₹ 280.

Question 6.
A departmental head has four subordinates and four tasks to be performed. The subordinates differ inefficiency and the tasks differ in their intrinsic difficulty. His estimates of the time each man would take to perform each task is given below
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 15
How should the tasks be allocated to subordinates so as to minimize the total man-hours?
Solution:
A number of tasks equal the number of subordinates. So the given problem is balanced and we can get an optimal solution.
Step 1: Subtract minimum hours of each row from other elements of that row.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 16
Since column 2 has no zero, proceed further.
Step 2:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 17
We can proceed with the assignment since all the rows and columns have zeros.
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 18
Now there is no zero in row S. So we proceed as below.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 19
We have drawn the minimum number of lines to cover all the zeros in the reduced matrix obtained. The smallest element from all the uncovered elements is 1. We subtract this from all the uncovered elements and add them to the elements which lie at the intersection of two lines. Thus we obtain another reduced problem for fresh assignment.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 20
Now all the subordinates have been assigned tasks. The optimal assignment schedule and the total cost is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 21
The optimal assignment (minimum) hours = 41

Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2

Question 7.
Find the optimal solution for the assignment problem with the following cost matrix.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 22
Solution:
Number of Areas = Number of salesmen.
So the given problem is balanced and we can find an optimal solution.
Step 1:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 23
Step 2:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 24
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 25
Now all the salesmen have been assigned areas.
The optimal assignment schedule and the total cost is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 26
Thus the optimal cost is Rs. 37.

Question 8.
Assign four trucks 1, 2, 3 and 4 to vacant spaces A, B, C, D, E and F so that distance travelled is minimized. The matrix below shows the distance.
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 27
Solution:
Here the number of trucks is 4 and vacant spaces are 6. So the given assignment problem is the unbalanced problem. So we introduce two dummy columns with all the entries zero to make is balanced. So the problem is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 28
Here only 4 vacant spaces can be assigned to four trucks
Step 1: Not necessary since all rows have zeros.
Step 2:
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 29
Step 3: (Assignment)
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 30
The optimal assignment schedule and total distance travelled is
Samacheer Kalvi 12th Business Maths Solutions Chapter 10 Operations Research Ex 10.2 31
Thus the minimum distance travelled is 12 km.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Students can download 12th Business Maths Chapter 9 Applied Statistics Ex 9.3 Questions and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 1.
Define Statistical Quality Control.
Solution:
Statistical quality control (SQC) refers to the use of statistical methods in the monitoring and maintaining of the quality of products and services. This method is used to determine the tolerance limits for accepting a production process.

Question 2.
Mention the types of causes for variation in a production process.
Solution:
There are two causes of variations between items produced under identical conditions in large production process. They are called assignable causes and non-assignable causes (chance causes).

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 3.
Define Chance Cause.
Solution:
The minor causes which do not affect the quality of the products to an extent are called as chance causes or Random causes. For example rain, floods, power cuts, etc.

Question 4.
Define Assignable Cause.
Solution:
The variations in input factors which are the causes for the variations in the output produc¬tions are called assignable causes. For example defective raw materials, fault in instruments used, fatigue of workers employed, unskilled technicians, worn out tools etc.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 5.
What do you mean by product control?
Solution:
Product control means controlling the quality of the product by a sampling technique called acceptance sampling. It aims at a certain quality level to he guaranteed to the customers. It is concerned with classification of raw materials, semi-finished goods or finished goods into acceptable or rejectable products.

Question 6.
What do you mean by process control?
Solution:
A production process is said to be under control if the products produced are according to the specifications; that is the characteristics are within the tolerance limits. This is tested through the control charts.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 7.
Define a control chart.
Solution:
Control charts are statistical tools to test whether a production process is under control. It was introduced by Watter.A.Shewhart. It is a simple technique used for detecting patterns of variations in the data. It consists of three lines namely, centre line (CL), Upper control limit (UCL) and Lower control limit (LCL)

Question 8.
Name the control charts for variables.
Solution:
A quality characteristic which can be expressed in terms of a numerical value in the production process is called as a variable. There are two types of control charts for variables.

  1. Mean chart (\(\bar{X}\) chart)
  2. Range chart (R chart).

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 9.
Define the mean chart.
Solution:
The mean chart (\(\bar{X}\) chart) is used to show the quality averages of the samples taken from the given process. The mean of the samples is first calculated. Then the mean of the sample means is found to get the control limits.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 1

Question 10.
Define R Chart.
Solution:
The R chart is used to show the variability or dispersion of the samples taken from the given process. The average range is given by \(\overline{\mathrm{R}}=\frac{\sum R}{n}\), where R = xmax – xmin for each ‘n’ samples. For samples of size less than 20, the range provides a good estimate of σ. Hence to measure the variance in the variable, range chart is used.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 11.
What are the uses of statistical quality control?
Solution:
The term Quality means a level or standard of a product which depends on Material, Manpower, Machines, and Management (4M’s). Quality Control ensures the quality specifications all along with them from the arrival of raw materials through each of their processing to the final delivery of goods. This technique is used in almost all’ production industries such as automobile, textile, electrical equipment, biscuits, bath soaps, chemicals, petroleum products etc.

Question 12.
Write the control limits for the mean chart.
Solution:
The calculation of control limits for \(\bar{X}\) chart in two different cases are
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 2

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 13.
Write the control limits for the R chart.
Solution:
The calculation of control limits for R chart in two different cases are
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 3

Question 14.
A machine is set to deliver packets of a given weight. Ten samples of size five each were recorded. Below are given relevant data:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 4
Calculate the control limits for the mean chart and the range chart and then comment on the state of control.
(conversion factors for n = 5, A2 = 0.58, D3 = 0 and D4 = 2.115)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 5
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 6
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 7
The above diagram shows all the three control lines with the data points plotted. We see that all the points of the sample mean are within the control limits.
We now draw the R chart for the given data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 8
The above diagram shows all the three control lines with the sample range points plotted. We observe that all the points are within the control limits.
Conclusion: From the above two plots of the sample mean \(\bar{X}\) and sample range R, we conclude that the process is in control.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 15.
Ten samples each of size five are drawn at regular intervals from a manufacturing process. The sample means (\(\bar{X}\)) and their ranges (R) are given below:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 9
Calculate the control limits in respect of \(\bar{X}\) chart.
(Given A2 = 0.58, D3 = 0 and D4 = 2.115) Comment on the state of control
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 10
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 11
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 12
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 13
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 14
From the \(\bar{X}\) chart, we see that 4 points are outside the control limit lines. So we say that the process is out of control.

Question 16.
Construct \(\bar{X}\) and R charts for the following data:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 15
(Given for n = 3, A2 = 1.023, D3 = 0 and D4 = 2.574)
Solution:
We first find the sample mean and range for each of the 8 given samples.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 16
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 17

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 17.
The following data show the values of the sample mean (\(\bar{X}\)) and its range (R) for the samples of Size five each. Calculate the values for control limits for mean, range chart and determine whether the process is in control.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 18
(conversion factors for n = 5, A2 = 0.58, D3 = 0 and D4 = 2.115)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 19
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 20
From the above control limits values we observe that all the sample means lie between the UCL and LCL (i.e.) 7.006 < \(\overline{\mathrm{x}}_{i}\) < 14.31 for i = 1, 2, 3,…….. 10. Also all the sample range value lie between the control limits for R (i.e) 0 < Ri < 13.32, i = 1, 2, 3,…., 10. Hence we conclude that the process is in control.

Question 18.
A quality control inspector has taken ten samples of size four packets each from a potato chips company. The contents of the sample are given below, Calculate the control limits for the mean and range chart.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 21
(Given for n = 4, A2 = 0.729, D3 = 0 and D4 = 2.282)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 22
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 23

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 19.
The following data show the values of sample means and the ranges for ten samples of size 4 each. Construct the control chart for mean and range chart and determine whether the process is in control
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 24
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 25
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 26
From the values of the control limits for \(\bar{X}\), we observe that one sample \(\bar{X}\) value (45) is above the UCL and one sample \(\bar{X}\) value (14) is below the LCL. Hence we conclude that the process is out of control.

Question 20.
In a production process, eight samples of size 4 are collected and their means and ranges are given below. Construct a mean chart and range chart with control limits.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 273
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 28
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 274
From the values of the control limits for \(\bar{X}\), we observe that sample \(\bar{X}\) value 16 is above the UCL. Hence we conclude that the process is out of control.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3

Question 21.
In a certain bottling industry, the quality control inspector recorded the weight of each of the 5 bottles selected at random during each hour of four hours in the morning.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 275
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 276
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.3 29
From the above control limit values. We observe that all the sample \(\bar{X}\) values are within UCL and LCL values. Also, all the R values are also within UCL and LCL of R chart. Hence we conclude that the process is within Control.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Students can download 12th Business Maths Chapter 9 Applied Statistics Ex 9.2 Questions and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 1.
Define Index Number.
Solution:
“An Index Number is a device which shows by its variations the Changes in a magnitude which is not capable of accurate measurements in itself or of direct valuation in practice”. – Wheldon

“An Index number is a statistical measure of fluctuations in a variable arranged in the form of a series and using a base period for making comparisons” – Lawrence J Kalpan

Question 2.
State the uses of Index Number.
Solution:
The uses of Index number are as given below:

  • It is an important tool for formulating decision and management policies.
  • It helps in studying the trends and tendencies.
  • It determines the inflation and deflation in an economy

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 3.
Mention the classification of Index Number.
Solution:
Classification of Index Numbers:
Index number can be classified as follows

  1. Price Index Number: It measures the general changes in the retail or wholesale price level of a particular or group of commodities.
  2. Quantity Index Number: These are indices to measure the changes in the number of goods manufactured in a factory.
  3. Cost of living Index Number: These are intended to study the effect of change in the price level on the cost of living of different classes of people.

Question 4.
Define Laspeyre’s price index number
Solution:
The weighted aggregate index number using base period weights is called Laspeyre’s price index number.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 1
Where p1 is current year price
p0 is base year price
q0 is base year quantity

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 5.
Explain Paasche’s price index number.
Solution:
If both prices and quantities were permitted to change, then it is impossible to isolate the part of movement due to price changes alone. In this case, the current year quantities appear more realistic weights than the base year quantities. The index number based on current year quantities is called Paasche’s price index number.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 2
Where p1 is the current year price
q1 is the current year quantity
p0 is the base year price

Question 6.
Write a note on Fisher’s price index number.
Solution:
Fisher defined a weighted index number as the geometric mean of Laspeyre’s index number and Paasche’s Index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 3
The Fisher-price index number is also known as the “ideal” price index number. This requires more data than the other two index numbers and as a result, may often be impracticable. But this is a good index number because it satisfies both the time-reversal test and factor reversal test.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 4

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 7.
State the test of the adequacy of the index number.
Solution:
Index numbers are studied to know the- relative changes in price and quantity for any two years compared. There are two tests which are used to test the adequacy for an index number. The two tests are as follows

  • Time Reversal Test
  • Factor Reversal Test

The criterion for a good index number is to satisfy the above two tests.

Question 8.
Define Time Reversal Test.
Solution:
It is an important test for testing the consistency of a good index number. This test maintains time consistency by working both forward and backward with respect to time (here time refers to the base year and current year). Symbolically the following relationship should be satisfied, P01 × P10 = 1
Fisher’s index number formula satisfies the above relationship
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 5
when the base year and current year are interchanged, we get,
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 6

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 9.
Explain Factor Reversal Test.
Solution:
Factor Reversal Test:
This is another test for testing the consistency of a good index number. The product of price index number and quantity index number from the base year to the current year should be equal to the true value ratio. That is the ratio between the total value of the current period and total value pf the base period is known as the true value ratio. Factor Reversal Test is given by,
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 7
where P01 is the relative change in price.
Q01 is the relative change in quantity.

Question 10.
Define true value ratio.
Solution:
The ratio between the total value of the current period and the total value of the base period is known as the true value ratio.
(i.e) true value ratio = \(\frac{\sum p_{1} q_{1}}{\sum p_{0} q_{0}}\)

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 11.
Discuss Cost of Living Index Number.
Solution:
Cost of Living Index Number is constructed to study the effect of changes in the price of goods and services of consumers for a current period as compared with the base period. The change in the cost of living index number between any two periods means the change in income which will be necessary to maintain the same standard of living in both the periods. Therefore the cost of living index number measures the average increase in the cost to maintain the same standard of life.

Further, the consumption habits of people differ widely from class to class (rich, poor, middle class) and even with the region. The changes in the price level affect the different classes of people, consequently, the general price index numbers fail to reflect the effect of changes in their cost of living in different classes of people. Therefore, the cost of living index number measures the general price movement of the commodities consumed by different classes of people.

Question 12.
Define Family Budget Method.
Solution:
Family Budget Method:
In this method, the weights are calculated by multiplying prices and quantity of the base year.
(i.e.) V = Σp0q0. The formula is given by,
Cost of Living Index Number = \(\frac{\sum \mathrm{PV}}{\sum \mathrm{V}}\)
where P = \(\frac{p_{1}}{p_{0}} \times 100\) is the price relative
V = Σp0q0 is the value relative

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 13.
State the uses of the Cost of Living Index Number.
Solution:
Uses of Cost of Living Index Number

  • It indicates whether the real wages of workers are rising or falling for a given time.
  • It is used by the administrators for regulating dearness allowance or grant of bonus to the workers.

Question 14.
Calculate by a suitable method, the index number of price from the following data:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 8
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 9
The Laspeyres price index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 10
Paasche’s price index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 11
On an average, there is an increase of 44.8% and 44.4% in the price of the commodities by Laspeyres and Paasche’s price index number respectively for the current year 2012 as compared with the base year 2002.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 15.
Calculate price index number for 2005 by (a) Laspeyre’s (b) Paasche’s method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 12
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 13
Laspeyre’s price index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 14
Paasche’s price index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 15
On an average, there is an increase of 170.6% and 163.63% in the price of the commodities by Laspeyre’s and Paasche’s price index number respectively for the current year 2005 as compared with the base year 1995.

Question 16.
Compute (i) Laspeyre’s (ii) Paasche’s (iii) Fisher’s Index numbers for 2010 from the following data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 16
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 17
Laspeyre’s price index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 18
Paasche’s price index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 19
Fisher’s price index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 20
On an average, there is an increase of 6.6%, 6.8% and 6.7% in the price of the commodities by Laspeyre’s, Paasche’s and Fisher’s index number respectively for the current year 2010 as compared to the base year 2000.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 17.
Using the following data, construct Fisher’s Ideal index and show how it satisfies Factor Reversal Test and Time Reversal Test?
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 21
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 22
Fisher’s ideal index
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 23
Time reversal test:
To prove P01 × P10 = 1

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 24
Time reversal test is satisfied.

Factor Reversal Test:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 25
Factor Reversal Test is satisfied.

Question 18.
Using Fisher’s Ideal Formula; compute price index number for 1999 with 1996 as the base year, given the following.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 26
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 27
Fisher’s index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 28
Thus we interpret that on an average, there is a decrease of 16.41 % in the price of commodities by Fisher’s Index number for the current year 1999 as compared to the base year 1996.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 19.
Calculate Fisher’s index number to the following data. Also, show that satisfies Time Reversal Test.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 29
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 30
Fisher’s price index number
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 31
On average, there is an increase of 22.3% in the price of commodities by Fisher’s Index number for the current year 2017 as compared to the base year 2016
Time reversal test:
To prove P01 × P10 = 1
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 32
Time reversal test is satisfied.

Question 20.
The following are the group index numbers and the group weights of an average working-class family’s budget. Construct the cost of living index number:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 33
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 34

Question 21.
Construct the cost of living Index number for 2015 on the basis of 2012 from the following data using the family budget method.
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 35
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 36
Hence, the cost of living index number for a particular class of people for the year 2015 is increased by 17.31% as compared to the year 2012.

Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2

Question 22.
Calculate the cost of living index by aggregate expenditure method:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 37
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.2 38
Hence, the cost of living index number for a particular class of people for the year 2015 is increased by 30.62% as compared to the year 2010.