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Business Mathematics and Statistics. Andre Francis BSc MSc. Perinatal Institute. Birmingham. Andre Francis works as a medical statistician. He has previously. Business Mathematics and smelarpeppame.ml - Free ebook download as PDF File .pdf ), Text File .txt) or read book online for free. Business Mathematics and. CONTENTS. Section A: Fundamentals of Business Mathematics. 40%. 1. Arithmetic. 2. Algebra. 3. Calculus. Section B: Fundamentals of Business Statistics.

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Partnership Logarithm Simple Interest and Compound Interest Annuity Discounting of Bill of Exchange Stock and Shares8. Conceptual Framework of Statistics9. Statistical Survey : Meaning Its Steps Data and Collection of Data The price atwhich an article is downloadd is called cost price C. P and the price at which it is sold is calledthe selling price S. The difference between the selling price and cost price is either profit orloss.

These are also known as overheads. Anyexpenditure incurred relating to sales are to be deducted to get the net sales price. It also includesthe expenses incurred in connection with the download. Thus, cost price C. It is usually expressed as a percentage. Discounts are of two types suchas i Trade discount ii Cash discount.

P or list price or catalogue price of anarticle. Cash discount : It is allowed on selling price or on amount due from credit customer.

Find the downloader's cost price and amount of discount. Solution : Let the M. P Discount C. Find the cash selling price ifM. P of an article is A From whom the customer would prefer to downloadthe goods? Example:7 The Marked price of an article is A Acustomer pays A He was allowedtwo successive discounts. Solution : Given C. P, AAgain profit per cent is 10When C.

P is , S. P is A C. It is preparedby the supplier and sent on supply of goods to the downloadr. It contains the information regardingthe quantity, quality and price of the goods sold. The following terms are usuallyused in making invoice: a C. This means FAS along with boarding charges to ship or cargo plane but excluding the freight charges.

This means cost, insurance and freight. In this case exporter bears all expenses for sending goods to the importer's place. What percent does he gain or lose on his outlay bydefraud. His selling price of article of A at A Find his gain per cent. Solution Let C. Solution : Let the C. P is , C. P of the article is A8, Find the C.

P of the article. Solution : Let S. P Ax and C. P of the horse and thecow separately. P of house is , then C. P is xWhen Difference in S. P is 2, then C. Find the rate of discount allowed and profit. P is , Discount is 25 25When M. P is 1, Discount is 25When M. V and a Ratio. If he sells the T. Find the cost price of each.

Solution : Let the cost price of the T. If the cost price of the radio is A , then selling price is : a A b A c A Answer the following in one sentence each : a What do you mean by profit and loss? Answer the following questions within two sentences each :- i What is meant by sales price? Answer the following within six sentences each. What its contents?

Group C Long type questions 5. What percentage of profit is earned when banans are bought at A7 per dozen and sold at A65 per ? What percentage is this on selling price? A man sells two radios at A each. What is his gain or loss per cent on the whole? If C downloads it for A Find cost price to A. A man downloads two watches for A13, On the whole he neither gains nor loses.

Find the cost price of each watch. Find the cost price of table and the chair separately. By selling an article for A12,, the seller makes a profit of 3 of its cost price. Find the cost price and its gain per cent. Find the cost price of the article. If his total profit on the sales of two houses is A50, Find the cost of each house. Find the cost and selling price. What did he pay for the house? 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 downloader 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.

In other words, it asserts that there is no true difference in the sample and the population, and that the difference found is accidental and unimportant arising out of fluctua- tions of sampling. Hence the word, null means invalid, void or amounting to nothing. Decision-maker should always adopt the null attitude regarding the outcome of the sample. A hypothesis is said to be alternative hypothesis when it is complementary to the null hypothesis. The null hypothesis and alternative hypothesis are denoted by Ho and Hi respectively.

A null hypothesis consists of only a single parameter value and is simple while the alternative hypothesis is usually composite. In any statistical test, there are four possibilities which are termed as exhaustive decisions. They are: Reject H0 when it is false 2. Accept H0 when it is true 3. Reject H0 when it is true Type -I error 4. Accept H0 when it is false Type - II error The decisions are expressed in the following dichotomous table: The probability denoted by a pronounced as alpha and the probability p of type II error is denoted by a pronounced as beta.

In practice, in business and social science problems, it is more risky to reject a correct hypothesis than to accept a wrong hypothesis. In other words, the consequences of Type I error are likely to be more serious than the consequences of Type II error. The quantity of risk tolerated in hypothesis testing is called the level of significance is commonly used at 5 percent respectively account for moderate and high precision.

The most commonly used test are t-test, F-ratio and Chi-square. The estimated value of the parameter which depends on the number of observations. The sample size, therefore, plays an important role in testing of hypothesis and is taken care of by degrees of freedom.

Degrees of freedom are the number of independent observations in a set. A statistical decision is a decision either to accept or to reject the null hypothesis based on the computed value in comparison with the given level of significance.

If the computed value of test statistic is less or more than the critical value, it can be said that the significant difference or insignificant difference and, the null hypothesis is rejected or accepted respectively at the given level of significance.

Test of Significance The tests of significance available to know the significance or otherwise of variables in various situations are a Test of significance for large samples and b Test of significance for small samples.

These tests aim at i comparing observation with expectation and thereon finding how far the deviation of one from the other can be attributed to variations of sampling, ii estimating from samples some characteristic of the population and iii guaging the reliability of estimates.

It is difficult to draw a line of demarcation between large and small samples; but a view among statistician is that, a sample is to be recorded as large only if its size exceeds 30 and if the sample size is less than 30, it is noted as small sample.

The tests of significance used for large samples are different from the small samples, the reason being that the assumptions made in case of large samples do not hold good for small samples.

The assumptions that made in dealing with problems relating to large samples are: In case of small samples, the above said assumptions will no longer be hold good. It should be noted that the estimates will vary from sample to sample if we work with very small samples.

We must satisfy with relatively wide confidence intervals. Of course, the wider the interval, the less is the precision. An inference drawn from the large sample is far more precise in the confidence limits it sets up than an inference based on a much smaller sample. Though, drawing a precise line of demarcation between the large sample and the small sample is not always easy, but the division of their theories is a very real one.

As a rule, the theory and methods of small samples are applicable to large samples, but the reverse is riot true. Please use headphones Large Samples a Single mean: Illustration 10 Compute the standard error of mean from the following data showing the amount mid by firms on the occasion of Deepavali. Amount paid Rs. The standard deviation of the height distribution of the population is known to be 3 inches. Test the statement that the mean height of the population is 66 inches at 0.

Also set up 0. Solution Since the difference is more than 1. Hence, the hypothesis is rejected. In other words, the mean height of the population could not be 66 inches. If the sample mean is 6. Give necessary justification for your conclusions. Solution Let us take the hypothesis that there is no difference between the sample mean a the population mean. Difference between the sample mean and population mean 6. Hence, the sample cannot be regarded as truly random sample. Illustration 13 If it costs a rupee to draw one number of a sample, how much would it cost in sampling from a universe with mean and standard deviation 9 to take sufficient number as to ensure that the mean of a sample would be within 0.

Find the extra cost necessary to double this precision. Illustration 14 The average number of defective articles in a factory is claimed to be less than for all the factories whose average is A random sample showed the following distribution. Class Limits Number 12 22 20 30 16 Calculate the mean and standard deviation of the sample and use it to test the claim that the average is less than the figure for all the factories at 0.

Solution The sample mean and population mean do not Since f Z is more than 1. Hence we reject the null hypothesis and conclude that the sample mean and population mean differ significantly. In other words, the manufacturer's claim that the average number of defectives in his product is less than the average figure for all the factories is valid.

Illustration 15 A random sample of articles selected from a batch of articles which show that the average diameter of the articles is 0. Find 95 per cent confidence interval for the average of this batch of articles. Can it reasonably be regarded as a sample from a large population with mean height of In other words, the sample of has come from the population with mean height of Illustration 17 Mrs.

P, an insurance agent in Anantapur Division has claimed that the average age of policy-holders who insure through her is less than the average of all the agents, which is A random sample of 60 policy-holders who had insured through her gave the following age distribution. Calculate the mean and standard deviation of this distribution and use these values to test her claim at the 95 per cent of level of significance.

You are given that Z at 0. Solution Since Z is less than 1. Hence, difference is insignificant. In other words, Mrs. P claim that the average age of policy-holders who insure through her is less than the average for all the agents at Illustration 18 A random sample of workers from South India shows that their mean wage of Rs.

A random sample of workers from North India gives a mean wage of Rs. Is there any significant difference between their mean wages?

Solution We are given two independent samples. Their means be denoted by x and ; sizes by m and n2 respectively. The given values are: Illustration 19 Mrs. Lahari has selected two markets, A and B at different locations of a city in order to make a survey on downloading habits of customers. Their average monthly expenditure on food is found to be Rs. The corresponding figures are Rs. Test at 1 per cent level of significance whether the average monthly food expenditure of the two populations of shoppers are equal.

Solution Since Z is 8. Hence, the data do not provide any evidence to accept the hypothesis. In other words, the monthly expenditure in two populations of shoppers in markets A and B differ significantly Illustration 20 An examination was given to 50 students at College A and to 60 students at College B.

At A, the mean grade was 75 with a standard deviation of 9, at B, the figures were 79 and 7 respectively. Is there a significant difference between the performance of the students at A and at B, given at 0.

Solution Let the hypothesis that there is no significant difference between the performance of the students at college A and B. We are given the value of Conclusion i. In other words, we conclude that mean grades of the students of college A and B are different at 0. Thus, the data consistent with the hypothesis and conclude that the mean grades of the students of college A and B is almost the same.

Illustration 21 Random samples drawn from two States give the following data relating to the heights of adult males. State A State B X Solution Let the hypothesis be there is no significant difference in mean height of adult males of two States. Since Z value is less than 1.

Illustration 22 A sample survey of boys about their intelligence gives a mean of 84, with-a standard deviation of The population standard deviation is Does the sample has come from the population? Solution Sample has come from the population Since Z is less than 1.

Illustration 23 The mean yield of two sets of plots and their variability are given below. Examine whether the difference in the variability of yields is significant. Set of 40 plots set of 60 plot X kgs kgs S2 26kgs 20kgs Solution Since Z value is much greater that the 2. We may reject the hypothesis. Illustration 24 A sample of height of Englishmen has a mean of While a sample of height of Australians has a mean Do the data indicate that Australians are, on an average, taller than Englishmen?

The greatest contribution to the theory of small samples is that of Sir William Gossett t-test , R. The t-test is to be applied to test: Single Mean We calculate the statistics for determining whether the mean of a sample drawn from a normal population deviates significantly from the stated value hypothetical population mean of the statistics is defined as: If calculated is more than the tabulated t for n-1 degrees of freedom at certain level of significance, we say it is significant and Ho is rejected.

If calculated is less than tabulated t, Ho may be accepted at the adopted level of significance. Illustration 25 A machine is designed to produce insulating washers for electrical devices of average thickness of 0.

A random sample of 10 washers was found to have an average thickness of 0. Test the significance of the deviation. Since t is 1. Hence, deviation is not significant. Ho is accepted. Illustration 26 A random sample of 16 values from a normal population showed a mean of Show that the assumption of a mean of Obtain 95 per cent confident limits for the same.

Solution Since t is more than the table value of 2. We conclude that the assumption of a mean of In the light of these data, mentioning the null hypothesis, discuss the suggestion that the mean height in the population is 66 inches.

Solution Table value for 9 d. It is not significant. Hence hypothesis at 0. We conclude that the mean height in the population may be regarded as 66 inches. Two Independent Samples Means Illustration 28 A group of 5 patients treated with medicine A weigh 42, 39, 48, 60 and 41 Kgs ; second group of 7 patients from the same hospital treated with medicine B weigh 38, 42, 56, 64, 68, 69 and 62 Kgs. Do you agree with the claim that medicine B increases the weight significantly.

Solution A and B medicines have equal effect on the increase in weight.