MB0040 – STATISTICS FOR MANAGEMENT Assignment Set- 1

Q1. Why it is necessary to summarise data? Explain the approaches available to summarize the data distributions?

Answer:

Graphical representation is a good way to represent summarised data. However, graphs provide us only an overview and thus may not be used for further analysis. Hence, we use summary statistics like computing averages. To analyse the data. Mass data, which is collected, classified, tabulated and presented systematically, is analysed further to bring its size to a single representative figure. This single figure is the measure which can be found at central part of the range of all values. It is the one which represents the entire data set. Hence, this is called the measure of central tendency.

In other words, the tendency of data to cluster around a figure which is in central location is known as central tendency. Measure of central tendency or average of first order describes the concentration of large numbers around a particular value. It is a single value which represents all units.

Statistical Averages: The commonly used statistical averages are arithmetic mean, geometric mean, harmonic mean.

Arithmetic mean is defined as the sum of all values divided by number of values and is represented by X.

Before we study how to compute arithmetic mean, we have to be familiar with the terms such as discrete data, frequency and frequency distribution, which are used in this unit.

If the number of values is finite, then the data is said to be discrete data. The number of occurrences of each value of the data set is called frequency of that value. A systematic presentation of the values taken by variable together with corresponding frequencies is called a frequency distribution of the variable.

Median: Median of a set of values is the value which is the middle most value when they are arranged in the ascending order of magnitude. Median is denoted by ‘M’.

Mode: Mode is the value which has the highest frequency and is denoted by Z.

Modal value is most useful for business people. For example, shoe and readymade garment manufacturers will like to know the modal size of the people to plan their operations. For discrete data with or without frequency, it is that value corresponding to highest frequency.

Appropriate Situations for the use of Various Averages

1. Arithmetic mean is used when:

a. In depth study of the variable is needed

b. The variable is continuous and additive in nature

c. The data are in the interval or ratio scale

d. When the distribution is symmetrical

2. Median is used when:

a. The variable is discrete

b. There exists abnormal values

c. The distribution is skewed

d. The extreme values are missing

e. The characteristics studied are qualitative

f. The data are on the ordinal scale

3. Mode is used when:

a. The variable is discrete

b. There exists an abnormal value

c. The distribution is skewed

d. The extreme values are missing

e. The characteristics studied are qualitative

4. Geometric mean is used when:

a. The rate of growth, ratios and percentages are to be studied

b. The variable is of multiplicative nature

5. Harmonic mean is used when:

a. The study is related to speed, time

b. Average of rates which produce equal effects has to be found

Positional Averages

Median is the mid-value of series of data. It divides the distribution into two equal portions. Similarly, we can divide a given distribution into four, ten or hundred or any other number of equal portions.

Q2. Explain the purpose of tabular presentation of statistical data. Draft a form of tabulation to show the distribution of population according to i) Community by age, ii) Literacy , iii) sex , and iv) marital status.

Answer:

The objectives of tabulation are to:

i. Simplify complex data

ii. Highlight important characteristics

iii. Present data in minimum space

iv. Facilitate comparison

v. Bring out trends and tendencies

vi. Facilitate further analysis

Tabulation is an orderly arrangement of data in columns and rows systematically in a tabular form. It is the logical listing of related quantitative data in vertical columns and horizontal rows. The presentation of data in tables should be simple, systematic and unambiguous.

The purpose of tabular presentation of statistical data is to:

a) Simplify complex data

Tabulation simplifies the complex data by presenting them systematically in columns and rows in a condensed form. It avoids all the unnecessary data that is found in a narrative form.

b) Highlight important characteristics

It also helps to highlight the important characteristics of the data. As it avoids all the unnecessary data that is usually found in a narrative form.

c) Present data in minimum space

Tabulation achieves economy in using the space for presenting the data. The textual matter is presented neatly in a short form without sacrificing utility of the data.

d) Facilitate comparison

The data presented in a tabular form is helpful for a comparative study. The relationship among the various items can be easily understood.

e) Bring out trends and tendencies

Tabulation depicts the data and their significance at first in the form of figures, which cannot be understood when the same data are in a narrative form.

f) Facilitate further analysis

The Tabulation is analytical in nature and hence it helps in further analysis.

Following is the form of tabulation to present the distribution of population according to Community by age, Literacy, Sex, and Marital status

Marital Status Sex Educated Non-Educated

Age:Below 20yrs

20-40 Above 40 Below 20yrs 20-40 Above 40

Married

Male

Female

Unmarried

Male

Female

Q3. Give a brief note of the measures of central tendency together with their merits & Demerits. Which is the best measure of central tendency and why?

Answer:

Condensation of data is necessary for a proper statistical analysis. A large number of big numbers are not only confusing to mind but also difficult to analyse. After a thorough

scrutiny of collected data, classification which is a process of arranging data into different homogenous classes according to resemblances and similarities is carried out first. Then of course tabulation of data is resorted to. The classification and tabulation of the collected data besides removing the complexity render condensation and comparison.

An average is defined as a value which should represent the whole mass of data. It is a typical or central value summarizing the whole data. It is also called a measure of central tendency for the reason that the individual values in the data show some tendency to centre about this average. It will be located in between the minimum and the maximum of the values in the data.

There are five types of average which are

Arithmetic Mean, Median, Mode, Geometric and Harmonic Mean

Arithmetic Mean

The Arithmetic mean or simply the mean is the best known easily understood and most frequently used average in any statistical analysis. It is defined as the sum of all the values in the data.

Median: Median is another widely known and frequently used average.It is defined as the most central or the middle most value of the data given in the form of an array. By an array, we mean an arrangement of the data either in ascending order or descending order of magnitude. In the case of ungrouped data one has to form an array first and then locate the middle most value which is the median. For ungrouped data the median is fixed by using,

Median = [n+1/2] the value in the array.

Mode: The word mode seems to have been derived French 'a la mode' which means 'that which is in fashion'. It is defined as the value in the data which occurs most frequently. In other words, it is the most frequently occurring value in the data. For ungrouped data we form the array and then fix the mode as the value which occurs most frequently. If all the values are distinct from each other, mode cannot be fixed. For a frequency distribution with just one highest frequency such data are called unimodal or two highest frequencies [such data are called bimodal],mode is found by using the formula,

Mode=l+cf2/f1+f2Where l is the lower limit of the model class, c is its class interval f1 is the frequency preceding the highest frequency and f2 is the frequency succeeding the highest frequency.

Relative merits and demerits of Mean, Median and Mode:

Mean: The mean is the most commonly and frequently used average. It is a simple average, understandable even to a layman. It is based on all the values in a given data. It is easy to calculate and is basic to the calculation of further statistical measures of dispersion, correlation etc. Of all the averages, it is the most stable one. However it has some demerits. It gives undue weightages to extreme value. In other words it is greatly influenced by extreme values. Moreover; it cannot be calculated for data with open - ended classes at the extreme. It cannot be fixed graphically unlike the median or the mode. It is the most useful average of analysis when the analysis is made with full reference to the nature of individual values of the data. In spite of a few shortcomings; it is the most satisfactory average.

Median: The median is another well-known and widely used average. It is well-defined formula and is easily understood. It is advantageously used as a representative value of such factors or qualities which cannot be measured. Unlike the mean, median can be located graphically. It is also possible to find the median for data with open ended classes at the extreme. It is amenable for further algebraic processes. However, it is an average, not based on all the values of the given data. It is not as stable as the mean. It has only a limited use in practice.

Mode: It is a useful measure of central tendency, as a representative of the majority of values in the data. It is a practical average, easily understood by even laymen. Its calculations are not difficult. It can be ascertained even for data with open-ended classes at the extreme. It can be located by graphical means using a frequency curve. The mode is not based on all the values in the data. It becomes less useful when the data distribution is not uni-model. Of all the averages, it is the most unstable average.

Q4. Machines are used to pack sugar into packets supposedly containing 1.20 kg each. On testing a large number of packets over a long period of time, it was found that the mean weight of the packets was 1.24 kg and the standard deviation was 0.04 Kg. A particular machine is selected to check the total weight of each of the 25 packets filled consecutively by the machine. Calculate the limits within which the weight of the packets should lie assuming that the machine is not been classified as faulty.

Answer:

Mean weight of the packets = 1.24 kg Standard Deviation,SD = 0.04kg Variance = 0.04^2 = 0.0016 Standard Error, SE = 0.04/sqrt(25)

= 0.04/5 = 0.008 Considering 99.7% confidence level

The means will lie between (1.2+3SE) and (1.2-3SE)

Upper limit is 1.224kg Lower Limit is 1.176kg

Q5. A packaging device is set to fill detergent power packets with a mean weight of 5 Kg. The standard deviation is known to be 0.01 Kg. These are known to drift upwards over a period of time due to machine fault, which is not tolerable. A random sample of 100 packets is taken and weighed. This sample has a mean weight of 5.03 Kg and a standard deviation of 0.21 Kg. Can we calculate that the mean weight produced by the machine has increased? Use 5% level of significance.

Answer:

Mean weight of packages, X1 = 5kg SD1 = 0.01kg Sample size, N= 100 Sample mean weight, X2 = 5.03kg SD2 = 0.21kg Using 95% confidence level Z = 1.96 1.96 = [(X-X2)/SD1]/sqrt N 1.96 =[(X-5.03)/0.01]/sqrt (100) Mean Weight X = 5.226kg

Q6. Find the probability that at most 5 defective bolts will be found in a box of 200 bolts. If it is known that 2 per cent of such bolts are expected to be defective. (You may take the distribution to be Poisson; e-4= 0.0183).

Answer:

Poisson distribution

A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution.

Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson probability based on the following formula:

Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e-μ) (μx) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

The Poisson distribution has the following properties:

The mean of the distribution is equal to μ.

The variance is also equal to μ .

M = 5

PX = 0.0183*4/5

=0.01464

Thus, the probability that at most 5 defective bolts will be found in a box of 200 bolts is 0.01464

MB0040 – STATISTICS FOR MANAGEMENT Assignment Set- 2

Q1. What do you mean by Statistical Survey? Differentiate between “Questionnaire” and “Schedule”.

Answer:

1. Definition of statistical survey

A Statistical survey is a scientific process of collection and analysis of numerical data. Statistical surveys are used to collect numerical information about units in a population. Surveys involve asking questions to individuals. Surveys of human populations are common in government, health, social science and marketing sectors.

Stages of Statistical Survey

Statistical surveys are categorised into two stages –

Planning and

Execution.

The figure below shows the two broad stages of Statistical survey.

Fig.1: Stages of Statistical Survey

Information is collected through mailed questionnaires

Often, information is collected through questionnaires. The questionnaires are filled with questions pertaining to the investigation. They are sent to the respondents with a covering letter

soliciting cooperation from the respondents (respondents are the people who respond to questions in the questionnaire). The respondents are asked to give correct information and to mail the questionnaire back. The objectives of investigation are explained in the covering letter together with assurance for keeping information provided by the respondents as confidential.

Good questionnaire construction is an important contributing factor to the success of a survey. When questionnaires are properly framed and constructed, they become important tools by which statements can be made about specific people or entire populations.

This method is generally adopted by research workers and other official and non-official agencies. This method is used to cover large areas of investigation. It is more economical and free from investigator’s bias. However, it results in many “non-response” situations. The respondent may be illiterate. The respondent may also provide wrong information due to wrong interpretation of questions.

If the questionnaire consists of invalid questions, or questions in incorrect order, or questions in inappropriate format, or questions that are biased, then the survey would be useless. An important method for checking and making sure whether a questionnaire is accurately capturing the intended information is to pre-test among a smaller subset of target respondents.

Success of questionnaire method of collection of data depends mainly on proper drafting of the questionnaire. You have to keep the following points in mind while preparing a questionnaire:

The respondent should not take much time in completing the questionnaire. It should be small and not lengthy.· The questions asked should be well structured and unambiguous.· The questions asked should be in proper logical sequence.· Questions should be unbiased. The questions in the questionnaire should not disturb the privacy of the respondents.· The task of completion of questionnaire should not have much writing work. · Necessary instructions and glossary should be given in covering letter.· Questions involving technological jargons and mathematical calculations should be avoided.· The completed questionnaire should be kept confidential and used only for the purpose of the survey as mentioned in the investigation.· There should not be any scope for misinterpretation in the questions.

There are different types of questions that can be used in the questionnaire. A questionnaire can have Contingency questions, Matrix questions, Closed ended questions and Open ended questions. Let’s have a look at each one in detail

Contingency questions are questions that are answered only if the respondent gives a particular response to a previous question. This avoids asking people questions that do not apply to them

Matrix questions are questions which are placed one under the other, forming a matrix. The response categories are placed along the top and a list of questions are placed down the side. This is used to efficiently occupy page space and respondents’ time.

Closed ended questions are those where the respondents’ answers are limited to a fixed set of responses. Usually scales are closed ended.

There are various types of closed ended questions.

Yes/no questions – here the respondents answer with “yes” or “no”. Some of the examples are:

Multiple choices – here the respondents have several options from which to choose. For example:

Scaled questions – here the responses are graded on a continuum (For example, rating the appearance of a product on a scale from 1 to 10, with 10 implying the most preferred appearance and 1 implying the least preferred appearance). Scaled questions are mostly questions related to attitudes. A Likert scale provides a number of attitude statements. The respondent has to say how much they agree or disagree with each one.

Open ended questions are those questions for which the respondent supplies their own answer without any fixed set of possible responses. Examples of types of open ended questions include:

Sentence completion – In these, respondents complete an incomplete sentence.

Story completion – In these, respondents complete an incomplete story.

Picture completion – In these, respondents fill in an empty conversation balloon.

Thematic Apperception Test – In these, respondents explain a picture or make up a story about what they think is happening in the picture.

Information through schedule filled by investigators

Information can be collected through schedules filled by investigators through personal contact. In order to get reliable information, the investigator should be well trained, tactful, unbiased and hard working.

A schedule is suitable for an extensive area of investigation through investigator’s personal contact. The problem of non-response is minimised.

There is a difference between a schedule and a questionnaire. A schedule is a form that the investigator fills himself through surveying the units or individuals. A questionnaire is a form sent (usually mailed) by an investigator to respondents. The respondent has to fill it and then send it back to the investigator.

Difference between Questionnaire and Schedule is as follows:

S.No. Basis Questionnaire Schedule

1. Liability for filling up

Informant is liable for filling it up. Enumerator fills it up after getting answers from informants.

2. Means of Information

It is sent to the informants by post. Enumerators themselves take up schedules and contact the informants.

3. Personal Relationship

Investigator does not have a personal contact with the informants.

Both investigator and informants have personal contact through schedule.

4. Nature of Information

Sometimes incomplete as there is lack of personal contact.

Complete information is received because of the personal contact between the investigator and informants.

5. Scope of The use of Questionnaire is suitable Schedule can be used for both

Enquiry where the informants are literate. literate as well as illiterate persons.

6. Economical Information by mailed questionnaire method is economical.

It is comparatively a costly method as most of the enumerators are paid.

7. Reliability The information collected through it is less reliable as informants cannot give correct answers to some of the questions.

It is reliable method as the enumerators can get correct answers after clarifying the questions to the informants.

8. Delay There is delay in the receipt of information by this method.

The information is quickly collected by the enumerators.

Q2. The table shows the data of Expenditure of a family on food, clothing, education, rent and other items.

Items Expenditure

Food 4300

Clothing 1200

Education 700

Rent 2000

Others 600

Depict the data shown in the table using Pie chart.

Answer:

Items Food Clothing Education Others Rent

Expenditure 4300 1200 700 600 2000

Q3. Average weight of 100 screws in box „A‟ is 10.4 gms. It is mixed with 150 screws of box „B‟. Average weight of mixed screws is 10.9 gms. Find the average weight of screws of box „B‟.

Answer:

Given: Average Weight in Box ‘A’ [XA] = 10.4 gms.

Number of Screws in Box ‘A’ [NA] = 100.

Number of Screws in Box ‘B’ [NB] =150.

Average Weight of mixed Screws [XAB] =10.9 gms.

[XAB] = NA XA + NB XB

NA + NB

10.9 = (100 X 10.4) + (150 X XB)

100 + 150

XB = 11.23 gms.

Q4. (a) Discuss the rules of “Probability”. (b) What is meant by “Conditional Probability”?

Answer:

4(a)Rules of Probability (unit-5) (5.2 and 5.3)

Managers very often come across with situations where they have to take decisions about implementing either course of action A or course of action B or course of action C. Sometimes, they have to take decisions regarding the implementation of both A and B.

For Example: A Sales manager may like to know the probability that he will exceed the target for product A or product B.sometimes,he would like to know the probability that the sales of product A and B will exceed the target.the first type of probability is answered by addition rule.the second type of probability is answered by multiplication rule.

Addition rule:

The addition rule of probability states that:

i)     If ‘A’ and ‘B’ are any two events then the probability of the occurrence of either ‘A’ or ‘B’ is given by:

ii)     If ‘A’ and ‘B’ are two mutually exclusive events then the probability of occurrence of either A or B is given by:

iii)     If A, B and C are any three events then the probability of occurrence of either A or B or C is given by:

In terms of Venn diagram, we can calculate the probability of occurrence of either event ‘A’ or event ‘B’, given that event ‘A’ and event ‘B’ are dependent events. From the figure 5.5, we can calculate the probability of occurrence of either ‘A’ or ‘B’, given that, events ‘A’ and ‘B’ are independent events. From the figure 5.6, we can calculate the probability of occurrence of either ‘A’ or ‘B’ or ‘C’, given that, events ‘A’, ‘B’ and ‘C’ are dependent events.

iv)     If A1, A2, A3………, An are ‘n’ mutually exclusive and exhaustive events then the probability of occurrence of at least one of them is given by:

Multiplication rule :

If ‘A’ and ‘B’ are two independent events then the probability of occurrence of ‘A’ and ‘B’ is given by:

4(b) Conditional Probability :

Sometimes we wish to know the probability that the price of a particular petroleum product will rise, given that the finance minister has increased the petrol price. Such probabilities are known as conditional probabilities.

Thus the conditional probability of occurrence of an event ‘A’ given that the event ‘B’ has already occurred is denoted by P (A / B). Here, ‘A’ and ‘B’ are dependent events. Therefore, we have the following rules.

If ‘A’ and ‘B’ are dependent events, then the probability of occurrence of ‘A and B’ is given by:

It follows that:

 

 

For any bivariate distribution, there exists two marginal distributions and ‘m + n’ conditional distributions, where ‘m’ and ‘n’ are the number of classifications/characteristics studied on two variables.

Q5. (a) What is meant by “Hypothesis Testing”? Give Examples (b) Differentiate between “Type-I” and “Type-II” Errors

Answer:

5(a) In hypothesis testing, we must state the assumed or hypothesised value of the population parameter before we begin sampling. The assumption we wish to test is called the null hypothesis and is symbolised by ’Ho’.

The term ‘null hypothesis’ arises from earlier agricultural and medical applications of statistics. In order to test the effectiveness of a new fertilizer or drug, the tested hypothesis (the null hypothesis) was that it had no effect, that is, there was no difference between treated and untreated samples. If we use a hypothesised value of a population mean in a problem, we would represent it symbolically as ‘mH0’. This is read – ‘The hypothesised value of the population mean’.

If our sample results fail to support the null hypothesis, we must conclude that something else is true. Whenever we reject the hypothesis, the conclusion we do accept is called the alternative hypothesis and is symbolised H1 (“H sub-one”).

For the null hypothesis H0: m = 200, we will consider three alternative hypothesis as:

H1: m ¹ 200 (population mean is not equal to 200)

H1: m > 200 (population mean greater than 200)

H1: m < 200 (population mean less than 200)

Example

We want to test the hypothesis that the population mean is equal to 500. We would symbolise it as follows and read it as,

The null hypothesis is that the population mean = 500 written as,

The purpose of hypothesis testing is not to question the computed value of the sample statistic but to make a judgment about the difference between that sample statistic and a hypothesised population parameter. The next step after stating the null and alternative hypotheses is to decide what criterion to be used for deciding whether to accept or reject the null hypothesis. If we assume the hypothesis is

correct, then the significance level will indicate the percentage of sample means that is outside certain limits (In estimation, the confidence level indicates the percentage of sample means that falls within the defined confidence limits).

 5(b) Type I error:

Suppose that making a Type I error (rejecting a null hypothesis when it is true) involves the time and trouble of reworking a batch of chemicals that should have been accepted. At the same time, making a Type II error (accepting a null hypothesis when it is false) means taking a chance that an entire group of users of this chemical compound will be poisoned. Obviously, the management of this company will prefer a Type I error to a Type II error and, as a result, will set very high levels of significance in its testing to get low b’s.

Type II error:

Suppose, on the other hand, that making a Type I error involves disassembling an entire engine at the factory, but making a Type II error involves relatively inexpensive warranty repairs by the dealers. Then the manufacturer is more likely to prefer a Type II error and will set lower significance levels in its testing.

Q6. From the following table, calculate Laspyres Index Number, Paasches Index Number, Fisher‟s Price Index Number and Dorbish & Bowley‟s Index Number taking 2008 as the base year?

Commodity 2008 2009

Price (Rs) per Kg Quantity in Kg Price (Rs) per Kg

Quantity in Kg

A 6 50 10 56

B 2 100 2 120

C 4 60 6 60

D 10 30 12 24

E 8 40 12 36

Answer:

Commodity

2008 2009

P0q0 P1q1 P0q1 P1q0Price (Rs.) per Kg (P0)

Qty in Kg

(q0)

Price (Rs.) per Kg (P1)

Qty in Kg

(q1)

A 6 50 10 56 300 560 336 500

B 2 100 2 120 200 240 240 200

C 4 60 6 60 240 360 240 360

D 10 30 12 24 300 288 240 360

E 8 40 12 36 320 432 288 480

Total (Σ) 1360 1880 1344 1900

a) Laspeyer’s Method= P01= Σ P1q0 X 100

Σ P0q0

Laspeyer’s Index Number= P01= 1900 X 100 = 139.71

1360

b) Paasche’s Method= P01= Σ P1q1 X 100

Σ P0q1

Paasche’s Index Number= P01= 1880 X 100 = 139.88

1344

c) Dorbish and Bowley’s Method= P01= Σ P1q0 + Σ P1q1

Σ P0q0 Σ P0q1 X 100

2

Or

L + P

2

Dorbish & Bowley’s Index Number= P01= 139.71+139.88 = 139.80

2

d) Fisher’s Method= P01= √Σ P1q0 X Σ P1q1 X 100

Σ P0q0 Σ P0q1

Fisher’s Price Index Number= P01= √ 1900 X 1880 X 100 = 139.79

1360 1344