I dedicate this book to my two business math inspirations. First, to my son,. Jon, who came up with the idea of writing a book for business professionals. Section A: Fundamentals of Business Mathematics. 40%. 1. Arithmetic. 2. Algebra . 3. Calculus. Section B: Fundamentals of Business Statistics. 60%. 4. Statistical. This Fundamentals of Business Mathematics Learning System has been devised as a resource for students attempting to pass their CIMA.
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These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelor degree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix algebra, systems of linear algebraic. Business MBA students who studied business mathematics and statistics There are 4 chapters in this part of business mathematics: Algebra review, calculus. BUSINESS MATH. A Step-by-Step Handbook. An Open Text. BASE TEXTBOOK. VERSION REVISION A. ADAPTABLE | ACCESSIBLE | AFFORDABLE.
Mutually exclusive events ii. In other words, every unit or object is as likely to be considered as any other. Sometimes, this point estimate may not disclose the true parameter value. Do you agree with the claim that medicine B increases the weight significantly. Physics to count the number of disintegrating of a radioactive element per unit of time, 5. Gopi has estimated the following probability distribution for the number of copies demanded.
For example, if a coin is tossed, we can get head H or tail T. Hence exhaustive cases are 2.
Thus in a toss of two coins, exhaustive cases are 4, i. In a throw of n coins, exhaustive events are 2 n. Simple and Compound Events: For example, the probability of drawing a red ball from a bag containing 8 white and 7 red balls. The joint occurrence of two or more events are termed compound events. For example if a bag containing 9 white and 6 red balls, and if two successive draws of 3 balls are made, we are going to find out the probability of getting 3 white balls in the first draw and 3 red balls in the second draw.
Complementary Events: A is called complementary event of B and vice-versa if A and B are mutually exclusive and exhaustive events. For example, when a die is thrown, occurrence of an even number 2, 4, 6 and odd number 1, 3, 5 are complementary events.
Probability Tree: Sample Space: In other words, the sample space is the set of all exhaustive cases of the random experiment. The elements of the sample space are the outcomes. Algebra of Sets: In the figure given below, we represent the union, difference, and intersection of two events by means of Venn diagrams. The region enclosed by a rectangle is taken to represent the sample space whereas given events are represented by ovals within the rectangle.
Permutation and Combination The word permutation means arrangement and the word combination means group or selection. For example, let us take three letters A, B and C. The order of elements is immaterial in combinations, while in permutations the order of elements matters. Permutation A permutation of n different objects taken r at a time is an ordered arrangement of only r objects out of the n objects.
In other words, the number of ways of arranging n things taken r at a time. It is denoted symbolically as n Pr, where n is total number of elements in the set, r is the number of elements taken at a time, and P is the symbol for permutation. Thus, n! For example, 1. Find the number of permutations of the letters a, b, c, d, e taken 2 at a time. In how many ways can 8 differently coloured marbles be arranged in a row?
The number of combinations of n objects taken r at a time is denoted by n Cr given by n! In how many ways can a committee of 3 persons be chosen out of 8? How many different sets of 4 students can be chosen out of 20 students? In how many ways can a committee of 4 men and 3 women be selected out of 9 men and 6 women.
Probability Theorems The computation of probabilities can become easy and be facilitated to a great extent by the two fundamental theorems of probability - the Addition Theorem and the Multiplication Theorem. The probability of occurring either event A or event B where A and B are mutually exclusive events is the sum of the individual probability of A and B. If the total number of possibilities is n, then by definition the probability of either A or B event happening is: For example, if the probability of buying a pen is 0.
When the events are not mutually exclusive, the above said theorem is to be modified. The probability of occurring of at least one of the two events, A and B which are not mutually exclusive is given by: In case of three events: One ball is drawn at random.
What is the probability that it is marked with a number that is multiple of 5 or 7? Solution The sample space i. The possible sample points that are multiples of 5 are The possible sample points that are multiples of 7 are What is the probability that it is marked with a number that is a multiple of 3 or 4? Since there are some numbers which are multiples of both 3 and 4, we need to exclude them so that they don't get counted twice.
Find the probability that the number of drawn ball will be a multiple of a 5 or 9, and b 5 or 7 Solution a Probability of getting a ball that has a multiple of 5, i. What is the probability that he passes the English test?
Then, i. For what choice of P are A and B mutually exclusive? For what choice of P are A and B independent? Find the probability of the target being hit at all when they both try. A product is assembled from three components - X, Y, and Z. The probability of these components being defective is 0. What is the probability that the assembled product will not be defective? Items produced by a certain process may have one or both of the two types of defects A and B. It is known that 20 per cent of the items have type A defects and 10 per cent have type B defects.
Furthermore, 6 per cent are known to have both types of defects. What is the probability that a randomly selected item will be defective? Solution a. Let P A be the probability that an item will have A-type of defect, and P B be the probability that an item will have B-type of defect.
The behaviour of each successive customer is independent. If three customers A, B and C enter together, what is the probability that the salesman will make a sale to at least one of the customers. The probability of occurring of two independent events A and B is equal to the product of their individual probabilities. Thus, the total number of successful events in both cases is a1 x a2. Similarly, the total number of possible cases is n1 x n2.
By definition, the probability of occurrence of both events is: Illustration 10 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 Solution A pack contains 52 cards. We can draw 4 cards from 52 cards, in n Cr i.
A pack of 52 cards consisting of four cards each of King, Queen, Jack and Ace. Since drawing a King is independent of the ways of getting a Queen, a Jack and an Ace, the sample points or the favorable number of cases are 4 C1 x 4 C1 x 4 C1 x 4 C1.
A pack of 51 cards contains 13 Diamond cards. So we can draw 4 out of 13 Diamond cards in 13 C4 ways.
In a pack of 52 cards, there are 13 cards of each suit. We can draw one card of each suit in 13 C1 x 13 C1 x 13 C1 x 13 C1 ways. In a pack of 52 cards, there are 13 Club cards and 13 Diamond cards. So, we can draw two cards of Clubs and two cards of Diamonds in 13 C2 x 13 C2 ways. Two balls are drawn in succession at random. What is the probability that one of them is white and the other is red? What is the probability that 2 are red, 2 are white, and 1 is blue?
Two successive drawings of 3 balls are made. Find the probability that the first drawing will give 3 white and the second 3 red balls, if i the balls were replaced before the second trial, ii the balls were not replaced before the second trial.
In first draw, 3 balls out of 13 can be drawn in 13 C3 ways ways which is the exhaustive number of cases. Since the balls were not replaced before the second draw, events A and B are dependent events.
Find the probability that: If India and Australia play 4 test matches, what is the probability that i India will lose all the four test matches, ii India will win at least one test match. What is the probability of the University selecting a Hindi- knowing woman teacher? What is the probability of selecting a fair complexioned rich girl?
Of total , one is nominated at random to be the University Executive. Find the probability that i the teacher has only a PG degree, ii the teacher has a PhD degree and is from Commerce subject, iii the teacher has a PhD degree and if from Economics subject, iv the teacher has a PhD degree. The data is: If two events A and B are dependent, the probability of the second event occurring will be affected by the outcome of the first that has already occurred.
The term conditional probability is used to describe this situation. Robert L. Birte defined the concept of conditional probability as: In other words, probability of B, given A. The conditional probability occuring due to a particular event or reason is called its reverse or posteriori probability.
The revision of given i. The Bayes' theorem is defined as: Jolliffee in his book captioned 'Commonsense Statistics for Economists and Others' has defined the concept of Bayes theorem as: With twisting conditional probabilities the other way round, i. Probability before revision by Bayes' rule is called a priori or simply prior probability, since it is determined before the sample information is taken in account.
A probability which has undergone revision via Bayes' rule is called posterior probability because, it represents a probability computed after the sample information is taken into account.
Posterior probability is also called revised probability in the sense that it is obtained by revising the prior probability with the sample information. Posterior probability is always conditional probability, the conditional event being the sample information. Thus, a prior probability which is unconditional probability becomes a posterior probability, which is conditional probability by using Baye's rule. Please use headphones Illustration 20 Two sets of candidates are competing for positions on the Board of Directors of a company.
The probability that the first and second sets will win are 0. If the first set wins, the probability of introducing a new product is 0. What is the probability that the product will be introduced. Past records show that the first machine produces 40 per cent of output and the second machine produces 60 per cent of output. Further, 4 per cent and 2 per cent of products produced by the first machine and the second machine respectively, were defectives. If a defective item is drawn at random, what is the probability that the defective item was produced by the first machine or the second machine.
And we may say that the defective item is more likely drawn from the output produced by the first machine. Illustration 22 The probability that management trainee will remain with a company is 0.
The probability that an employee earns more than Rs. The probability that an employee is a management trainee who remained with the company or who earns more than Rs. What is the probability that an employee earns more than Rs.
Illustration 23 Suppose that one of the three men, a politician, a businessman and an academician will be appointed as the Vice Chancellor of a University. Their probability of appointments respectively are 0.
The probabilities that research activities will be promoted by these people if they are appointed are 0. What is the probability that research will be promoted by the new Vice- Chancellor? Appointment of an academician as the Vice-Chancellor would certainly develop and promote education as it is known by the probability theory.
Random Variable and Probability Distribution By random variable we mean a variable value which is computed by the outcome of a random experiment. In brief, a random variable is a function which assigns a unique value to each sample point of the sample space.
A random variable is also called a chance variable or stochastic variable. A random variable may be continuous or discrete. If the random variable takes on all values within a certain interval, then it is called a continuous random variable while if the random variable takes on the integer values such as 0, 1, 2, 3, The function p x is known as the probability function of random variable of x and the set of all possible ordered pairs is called probability distribution of random variable.
The concept of probability distribution is in relation to that of frequency distribution. While the frequency distribution tells how the total frequency is distributed among different classes of the variable, the probability distribution tells how the total frequency is distributed among different classes of the variable, the probability distribution tells how the total probability of 1 is distributed among various values which random variable can take.
In brief, the word frequency is replacing by probability.
Illustration 24 A dealer of Allwyn refrigerators estimates from his past experience the probabilities of his selling refrigerators in a day. These are as follows: Getting an odd number is termed as a success. Find the probability distribution of number of success.
In two throws of a die, X denoted by S becomes a random variable and takes the values 0, 1, 2. This means, in the two throws, we can get either no odd numbers, or 1 odd number, or both odd numbers. Obtain the probability distribution of the number of bad apples in a draw of 3 apples at random. Solution Denote X as the number of bad apples drawn. Now X is a random variable which takes values of 0, 1, 2, 3. Assume that the number of deaths per one thousand is four persons in this group. What is the expected gain for the insurance company on a policy of this type?
Solution Denote premium by X and death rate by P X. Any unsold copies are, however, a dead loss. Gopi has estimated the following probability distribution for the number of copies demanded.
Also compute the variance. She has calculated that the cost of manufacturing as Rs. It is, however, perishable and any goods unsold at the end of the day are dead loss. She expects the demand to be variable and has drawn up the following probability distribution: Find the value of K.
Find an expression for her net profit or loss if she manufactures 'm' pieces and demand is 'n' pieces. Consider separately the two cases -- 'n' lesser than or equal to 'm', and 'n' greater than 'm'. Find the net profit or loss, assuming that she manufactures 12 pieces. Find the expected net profit. Calculate expected profit for different levels of production.
The probability of a distribution cannot exceed 1. If she manufactures 'm' pieces on any day, the cost is Rs. If the number of pieces demanded on any day 'n' is less than or equal to peices produced 'm', then all the pieces demanded are sold, and the sale proceeds is Rs.
But, if the number of pieces demanded on any day 'n' is greater than the pieces produced 'm', then the maximum sales is limited to 'm' and thus the sale proceeds is Rs.
Lets apply the finding in ii. Profit for m Demand Production m Probability n 10 11 12 13 14 15 10 10 8 6 4 2 0 0. Hence, the production of 12 pieces per day will optimise Ravali's food stall enterprise's expected profit. Mathematical Expectation and Variance The concept, mathematical expectation also called the expected value, occupies an important place in statistical analysis. The expected value of a random variable is the weighted arithmetic mean of the probabilities of the values that the variable can possibly assume.
Brite has defined the mathematical expectation as: It is the expected value of outcome in the long run. In other words, it is the sum of each particular value within the set X multiplied by the probability. Symbolically The concept of mathematical expectation was originally applied to games of chance and lotteries, but the notion of an expected value has become a common term in everyday parlance.
This term is popularly used in business situations which involve the consideration of expected values. Illustration 31 Mr.
Reddy, owner of petrol bunk sells an average of Rs. Statistics from the Meteorological Department show that the probability is 0. Find the expected value of petrol sale and variance. Proposal A - Profit of Rs. Solution Calculate the expected value of each proposal. Proposal A: Hence the business man should prefer proposal C. Illustration 33 The probability that there is at least one error in an accounts statement prepared by A is 0. A, B and C prepared 10, 16 and 20 statements respectively.
Find the expected number of correct statements in all and the standard deviation. In frequency distribution, measures like average, dispersion, correlation, etc. In population, the values of variable may be distributed according to some definite probability law, and the corresponding probability distribution is known as Theoretical Probability Distribu- tion. We have defined the mathematical expectation, random variable, and probability distribution function and also discussed these.
In the present section, we will cover the following univariate probability distributions: The first two distributions are discrete probability distributions and the third one is a continuous probability distribution. Binomial Distribution Binomial distribution is named after the Swiss mathematician James Bernoulli who innovated it.
The binomial distribution is used to determine the probability of success or failure of the one set in which there are only two equally likely and mutually exclusive outcomes. This distribution can be used under specific set of assumptions: The random experiment is performed under the finite and fixed number of trials.
The outcome of each trail results in success or failure. All the trails are independent in the sense the outcome of any trail is not affected by the preceding or succeeding trials. The probability of success or failure remains constant from trial to trial. The success of an event is denoted by 'p' and its failure by 'q'.
Since the binomial distribution is a set of dichotomous alternatives i. By expanding the binomial terms, we obtain probability distribution which called the binomial probability distribution or simply the binomial distribution. Rules of binomial expansion In binomial expansion, the rules should be noted. The constants of binomial distribution are: Determine the values of p and q. Multiply each term of the expanded binomial by N total frequency in order to obtain the expected frequency in each category.
Please use headphones Illustration 34 A coin is tossed six times. What is the probability of obtaining four or more heads. Solution In a toss of an unbiased coin, the probability of head as well as tail is equal, i. Solution Probability of getting head and tail are denoted by p and q respectively. The probability of r successes i. What is the probability that out of six workmen, three or more will contact the disease? Number of heads observed is recorded at each throw, and the results are given below.
Find the expected frequencies. What are the theoretical values of mean and standard deviation? Also calculate the mean and standard deviation of the observed frequencies. Arrange data: X dx Frequency F. The probability frequency is more scientific and mathematical model so that the arriving results are more accurate and precise. Illustration 39 Given data shows the number of seeds germinating out of 10 on damp filter for sets of seeds. By expanding 0. X Expected Frequencies N x n Cr q n-r p r 0 x 0.
This distribution describes the behaviour of rare events and has been known as the Law of Improbable events. Poisson distribution is a discrete probability distribution and is very popularly used in statistical inferences. The binomial distribution can be used when only the sample space number of trials n is known, while the Poisson distribution can study when we know the mean value of occurrences of an event without knowing the sample space. Such distribution is fairly common.
The standard deviation is m. Application and Uses Poisson distribution can explain the behaviour of the discrete 'random variables where the probability of occurrence of events is very small and the number of trials is sufficiently large As such, this distribution has found application in many fields like Queuing theory, Insurance, Biology, Physics, Business, Economics, Industry etc.
The practical areas where the Poisson distribution can be used is listed below. It is used in Biology to count the number of bacteria, 4. Physics to count the number of disintegrating of a radioactive element per unit of time, 5. In addition to the above, the Poisson distribution can also use in things like counting number of accidents taking place per day, in counting number of suicides in a particular day, or persons dying due to a rare disease such as heart attack or cancer or snake bite or plague, in counting number of typographical errors per page in a typed or printed material etc.
Please use headphones Illustration 40 An average number of phone calls per minute into the switch-board of Reddy Company Limited between the hours of 10 AM to 1 PM is 2. Find the probability that during one particular minute there will be i no phone calls at all, ii exactly 3 calls and iii at least 5 calls. Solution Let us denote the number of telephone calls per minute by X. The Poisson probability function is: Refer Table for e We cannot get table value for 2.
First we have to find the value for e Now, multiply them to get 0. The data are: Solution First determine observed frequency. Without referring to Table, we can calculate the value of e. For example, e This can be computed as: A more suitable distribution for dealing with the variable whose magnitude is continuous is normal distribution. It is also called the normal probability distribution. Uses 1. It aids solving many business and economic problems including the problems in social and physical sciences.
Hence, it is cornerstone of modern statistics. It becomes a basis to know how far away and in what direction a variable is from its population mean. It is symmetrical. Hence mean, median and mode are identical and can be known. It has only one maximum point at the mean, and hence it is unimodel i. Definition In mathematical form, the normal probability distribution is defined by: The normal deviate at the mean will be zero viz.
This is known as changing to standardized scale. In equation the changing to standardized scale is written as The normal curve is distributed as under: Mean 1 covers Mean 2 covers Mean 3 covers Hence, in order to fit a curve we must know the ordinates i. Find the x , N and class interval, if any, of the observed distribution. Illustration 41 The customer accounts at the Departmental Store have an average balance of Rs. Assuming that the account balances are normally distributed, find i.
What proportion of the accounts is over Rs. What proportion of the accounts is between Rs. Proportion of accounts over Rs. Deduct the value of 0. Hence, Proportion of the accounts between Rs. Illustration 42 In a public examination students have appeared for statistics.
The average mark of them was 62 and standard deviation was Assuming the distribution is normal, obtain the number of students who might have obtained i 80 percent or more, ii First class i. Thus, x 0. In other words, the students who secure more than ranks fall under the area of 0. The Z value corresponding to 0. What do you mean by probability. Discuss the importance of probability in statistics? What is meant by mathematical expectation?
Explain it with the help of an example? What is Bayes' theorem? Explain it with suitable example? What is meant by the Poisson distribution? What are its uses? Explain the terms i. Mutually exclusive events ii. Independent and dependent events iii. Simple and compound events iv. Random variable v. Permutation and combination vi. Trial and event vii. Sample space 6. Find the probability that 2 are white and 1 is black A bag containing 8 white, 6 red and 4 black balls. Three balls are drawn at random.
Find the probability that they willbe white. A bag contains 4 white and 8 red balls, and a second bag 3 white and 5 black balls. One of the bags is chosen at random and a draw of 2 balls is made it. Find the probability that one is white and the other is black. A class consists of students, 25 of them are girls and the remaining are boys, 35 of them are rich and 65 poor, 20 of them are fair glamor What is the probability of selecting a fair glamor rich girl.
Three persons A, B and C are being considered for the appointment as Vice- Chancellor of a University whose chances of being selected for the post are in the proportion 5: The probability that A if selected will introduce democratisation in the University strut is 0. What is the probability that democratisation would be introduced in the University. The probability that a trainee will remain with a company is 0. The probability that an employee is a trainee who remained with the company or who earns more than Rs.
What is the probability than an employee earns more that Rs. In a bolt factory, machines A, B, C produce 30 per cent, 40 per cent and 30 per cent respectively. Of their output 3, 4, 2 per cents are defective bolts. A bolt is drawn at random from the product and is found to be defective. What are the probabilities that it was produced by machines A, B and C. A factory produces a certain type of output by two types of machines. The daily production are: Machine I - units and Machine II - units.
An item is drawn at from the day's production and is found to be defective. Dayal company estimates the net profit on a new product it is launching to be Rs.
The company assigning the probabilities to the first year prospects for the product are: Successful - 0. What are the expected profit and standard deviation of first year net profit for his product. Profit 0. A systematic sample of passes was taken from the concise Oxford Dictionary and the observed frequency distribution of foreign words per page was found to be as follows: Also calculate the variance of fitted distribution.
Income of a group of persons were found to be normally distributed with mean Rs. Of third group, about 95 per cent had income exceeding Rs.
What was the lowest income among the richest Chance, W. Gopikuttam, G. Gupta, S. Levin, R.. Inferring valid conclusions for making decision needs the study of statistics and application of statistical methods almost in every field of human activity. Statistics, therefore, is regarded as the science of decision making. The statisticians can commonly categorise the techniques of statistics which are of so diverse into a descriptive statistics and b inferential statistics or inductive statistics.
The former describes the characteristics of numerical data while the latter describes the judgment based on the statistical analysis. In other words, the former is process of analysis. In other words, the former is process of analysis whereas the latter is that of scientific device of inferring conclusions.
Both are the systematic methods of drawing satisfactory valid conclusions about the totality i. The process of studying the sample and then generalising the results to the population needs a scientific investigation searching for truth. Population and Sample The word population is technical term in statistics, not necessarily referring to people. It is totality of objects under consideration. In other words, it refers to a number of objects or items which are to be selected for investigation.
This term as sometimes called the universe. Figure 1. A population containing a finite number of objects say the students in a college, is called finite population. A population having an infinite number of objects say, heights or weights or ages of people in the country, stars in the sky etc. Having concrete objects say, the number of books in a library, the number of buses or scooters in a district, etc.
If the population consists of imaginary objects say, throw of die or coin in infinite numbers of times is referred to hypothetical population. For social scientist, it is often difficult, in fact impossible to collect information from all the objects or units of a population. He, therefore, interested to get sample data. Selection of a few objects or units forming true representative of the population is termed as sampling and the objects or units selected are termed as sample.
On the analysis being derived from the sample data, he generalises to the entire population from which the sample is drawn. The sampling has two objectives which are: Parameter and Statistics The statistical constants of the population such as population size N , population mean m , population variance 2 , population correlation coefficient p , etc are called parameters.
In other words, the values that are derived using population data are known as parameters. Similarly, the values that are derived using sample data are termed as statistics not to be confused with the word statistics meaning data or the science of statistics. The examples for statistics are sample mean x , sample variance S 2 , sample correlation coefficient r , sample size n , etc Obviously 1 statistics are quotients of the sample data whereas parameters are function of the 1 population data.
In brief the population constant is called parameter while the 1 sample constant is known as statistics. Random Sample Sampling refers to the method of selecting a sub-set of the population for investigation. Selection of objects or units in such a way that each and every object or unit in the population has the chance of being selected is called random sampling. The number of objects or units in the sample is termed as sample size. This size should neither be too big nor too small but should be optimum.
Over the census method, the sample method has distinct merits, which R. Fisher sums up thus: Speed, economy, adaptability and scientific. The right type of sampling plan is of paramount importance in execution of a sample survey in accordance with the objectives and scope of investigation the sampling techniques are broadly classified into a random sample, b non-random sample and c mixed sample. The term random or probability is very widely applicable technique in selecting a sample from the population.
All the objects or units in the universe will have an equal chance of being included in the random sample. In other words, every unit or object is as likely to be considered as any other. In this, the process is random in character and is usually representative. Selecting 'n' units out of N in such a way that every one of Ncn samples has an equal chance of being selected.
This is done in the ways: The former does permit replacing while the latter does not. Let x stands for the lift in hours of television produced by Konark Company under essentially identical conditions with the same set of workers working on the same machine using the same type of materials and the same technique.
If x1, X2, X3, xn are the lives of n such television, then x1, x2, X The number n is called the size of random sample. A random sample may be selected either by drawing the chits or by the use of random numbers.
The former is a random method but is subject to biases as can be identified chits. The latter is the best as numbers are drawn randomly. For example where a population consists of 15 units and a sample of size 6 is to be selected thus since 15 is a two-digit figure, units are numbered as 00, 01, 02, 03 Six random numbers are obtained from a two digit random number table They are - 69,36, 75, 91,44 and On dividing 69 by 15, the remainder is 9, hence select the unit on serial number 9.
Likewise divide 36, 75, 91 and 44 and 86 by The respective remainders are 6,0,1, 14 and Hence select units of serial numbers 09, 06, 00, 01, 14 and These selected units form the sample. Sampling Distribution A function of the random variables x1, x2, x Hence, a random variable has probability distribution. This probability distribution of a statistic is known as the sampling distribution of the statistics. This distribution describes t he way that a statistic is the function of the random variables.
In Practice the sampling distributions which commonly used are the sample mean and the sample variance.
These will give a fillip to a number of test statistics for hypothesis testing. Suppose in a simple random sample of size n picked up from a population, then the sample mean represented by x is defined as a. The Sample Variance: Suppose, the simple random sample of size n chosen from a population the sample variance is used to estimate the population variance.
In an equation form. Standard Error The standard deviation measures variability variable. The standard deviation of a sampling distribution is referred to standard error S. It measures only sampling variability which occurs due to chance or random forces, in estimating a population parameter. The word error is used in place of deviation to emphasize that the variation among sample statistic is due to sampling errors.
If 0 is not known, we use the standard error given by: In drawing statistical inferences, the standard error is of great significance due to 1. That it provides an idea about the reliability of sample.
The lesser the standard error, the lesser the variation of population value from the expected sample value. Hence is greater reliability of sample. That it helps to determine the confidence limits within which the parameter value is expected to lie.
For large sample, sampling distribution tends to be close to normal distribution. In normal distribution, a range of mean one standard error, of mean3 two standard error, of mean 3 standard error will give The chance of a value lying outside 3 S. That it aids in testing hypothesis and in interval estimation. Please use headphones Estimation Theory A technique which is used for generalizing the results of the sample to the population for estimating population parameters along with the degree of confidence is provided by an important branch of statistics is called Statistical Inference.
In other words, it is the process of inferring information about a population from a sample. This statistical inference deals with two main problems namely a estimation and b testing hypothesis. The estimation of population parameters such as mean, variance, proportion, etc. The parameters estimation is very much need for making decision. For example, the manufacturer of electric tubes may be interested in knowing the average life of his product, the scientist may be eager in estimating the average life span of human being and so on.
Due to the practical and relative merits of the sample method over the census method, the scientists will prefer the former. A specific observed value of sample statistic is called estimate. A sample statistic which is used to estimate a population parameter is known as estimator. In other words, sample value is an estimate and the method of estimation statistical measure is termed as an estimator.
The theory, of Estimation was innovated by Prof. Estimation is studied under Point Estimation and Interval Estimation.
Good Estimation A good estimator is one which is as close to the true value of population parameter as possible. A good estimator possesses the features which are: An estimate is said to be unbiased if its expected value is equal to its parameter. For example, if 3c is an estimate of ft, x will be an unbiased estimate only if See Illustration 1 b Consistency: An estimator is said to be consistent if the estimate tends to approach the parameter as the example size increases.
For any distribution, i. An estimate is said to be efficient if the variance i. An estimator with less variability and the consistency more reliable than the other.
An estimator which uses all the relevant information in its estimation is said to be sufficient. If the estimator sufficiently insures all the information in the sample, then considering the other estimator is absolutely unnecessary.
Point Estimation Method A Point Estimation is a single statistic which is used to estimate a population parameter. Now, we shall discuss the sample mean and sample variance are unbiased estimate for corresponding population parameters.
Interval Estimation Method In Point Estimation, a single value of statistic is used as estimate of the population parameter. Sometimes, this point estimate may not disclose the true parameter value.
Having computed a statistic from a given random sample, can we make reasonable probability statements about the unknown parameter of the population from which the sample is drawn? The answer can provide by the technique of Interval Estimation. The Interval Estimation within which the unkown value of parameter is expected to lie is called confidence interval or Fiducial Interval which are respectively called by Neyman and Fisher.
The limits so determined are called Confidence Limits or Fiducial Limits and at required precision of estimate say 95 percent is known as Confidence Coefficient. Thus, a confidence interval indicates the probability that the population parameter lies within a specified range of values. To compute confidence interval we require: Z-Distribution Interval estimation for large samples is based on the assumption that if the size of sample is large, the sample value tends to be very close to the population value.
In other words, the size of sample is sufficiently large, the sampling distribution is approximately of normal curve shape. This is the feature of the central limit theorem. Therefore, the sample value can be used in estimation of standard error in the place of population value.
The Z-distribution is used in case of large samples to estimate confidence limits. For small sample, instead of Z-values, t-values are studied to estimate the confidence limits. One has to know the degree of confidence level before calculating confidence limits. Confidence level means the level of accuracy required. For example, the 99 per cent confidence level means, the actual population mean lies within the range of the estimated values to a tune of 99 per cent.
The risk is to a tune of one percent. To find the Z-value corresponding to 99 per cent confidence level, divide that J confidence level by 2 i. Identify this value in the Z-value. The Z-value corresponding to it can be identified in the left-most column and also in the top-most row.
The confidence coefficient for 99 per cent confidence level is 2. The 99 per cent of items or cases falls within x 2. Superiority of Interval Estimate In estimating the value by the Point Estimate Method and the Interval Estimate Method, the former provides only a point in the sample with no tolerance or confidence level attached to it.
The latter provides accuracy of the estimate at a confidence level. Further it helps in hypothesis testing and becomes a basis for decision-making under the conditions of uncertainly or probability. The interval estimate, therefore, has a superiority or practical application over the point estimate.
Illustration 1 A Universe consists of four numbers 3, 5, 7 and 9. Consider all possible samples of size two which can be drawn with replacement from the universe. Calculate the mean and variance. Further, examine whether the statistics are unbiased for corresponding parameters.
What is the sampling mean and sample variance? Calculation of sample mean and sample variance Any one of the four numbers, 3, 5, 7 and 9 drawn in the first draw can be associated with any one of these four numbers drawn at random with replacement in the subsequent draw i. Illustration 2 Consider a hypothetical three numbers 2, 5 and 8. Draw all possible samples of size 2 and examine the statistics are unbiased for corresponding parameters.
Solution The given universe consists of three values namely 2,5,8. Thus there are 9 samples of size 2. Illustration 3 Consider the population of 5 units with values 1,2,3,4 and 5. Write down all possible samples of size 2 without replacement and verify that sample mean is an unbiased estimate of the population mean.
Also calculate sampling variance and verify that i it agrees with the formula for variance of the sample mean and ii this variance is less than the variance obtained from the sampling with replacement iii and find the standard error.
Solution Thus, the variance of sample mean distribution is agreed with the formula for the variance of the sample without replacement. Illustration 4 The Golden Cigarettes Company has developed a new blended tobacco product. The marketing, department has yet to determine the factory price, A sample of wholesalers were selected and were asked about price. Determine the sample mean for the following prices supplied by the wholesalers. Both the buyer as well as seller accept the use of this point estimate as a basis for fixing the price.
The point estimate can save time and expense to the producer of cigarettes. Illustration 5 Sensing the downward in demand for a product, the financial manager was con- sidering shifting his company's resources to a new product area.
Find point estimate of the mean and variance of the population from data given below. Illustration 6 A random sample of appeals was taken from a large consignment and 66 of them were found to be bad. Find the limits at which the bad appeals lie at 99 per cent confidence level. Solution Calculation of confidence limits for the proportion of bad appeals. Illustration 7 Out of 20, customer's ledger accounts, a sample of accounts was taken to accuracy of posting and balancing wherein 40 mistakes were found.
Assign limits Within which the number of defective cases can be expected to lie at 95 per cent confidence. Calculation of confidence limits for defective cases.
Find 99 per cent confidence limits for TV viewers who watch this programme. Assign limits within which the number of students who done the problem wrongly in whole universe of students at 99 per cent confidence level.
Such point of view or proposition is termed as hypothesis. Hypothesis is a proportion which can be put to test to determine validity. A hypothesis, in statistical parlance is a statement about the nature of a population which is to be tested on the basis of outcome of a random sample. Testing Hypothesis The testing hypothesis involves five steps which are as: The formulation of a hypothesis about population parameter is the first step in testing hypothesis.
The process of accepting or rejecting a null hypothesis on the basis of sample results is called testing of hypothesis. The two hypothesis in a statistical test are normally referred to: Null hypothesis ii. Alternative hypothesis. A reasoning for possible rejection of proposition is called null hypothesis. Search for books, journals or webpages All Webpages Books Journals. View on ScienceDirect. Michael Sentlowitz.
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Institutional Subscription. Free Shipping Free global shipping No minimum order. Preface Part 1 Arithmetic: Finding the Percentage 3. Finding the Rate 3. Finding the Base 3. Miscellaneous Problems 3. Manufacturers and Wholesalers 4.
Retailers 5. Quantity on Hand 6. An Introduction 7. Straight-Line Method 7. Units-of-Production Method 7. Declining-Balance Method 7.
Sum-of-the-Years'-Digits Method 7. Salaried Employees 8. Base Salary Plus Commission 8. Hourly Wages 8. Piece Rate 8.