Stats

MBA- Semester 1

Assignment - Marks 60 (6X10=60)

MB0040- Statistics for Management- 4 credits

Subject Code - MB0040

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1. Mention the characteristics of Statistics. Explain any two applications of

Statistics.

2. Distinguish between primary and secondary data. What are the methods of

collecting primary data?

3. Find Karl Pearson’s correlation coefficient between the sales and expenses from the data

given below:

Sales

(Rs. Lakhs)

50 50 55 60 65 65 65 60 60 50

Expenses

(Rs. Lakhs)

11 13 14 16 16 15 15 14 13 13

4. The incidence of occupational disease in an industry is such that the

workers have a 20% chance of suffering from it. What is the probability that

out of six workers 4 or more will contract the disease?

5. Use chi-square test to test if the two attributes (Performance and Training) in the

following table are independent. Test at 5% level of significance.

Performance

Training

Intensive Good Average Total

Above Average 100 150 40 290

Average 100 100 100 300

Poor 50 80 150 280

Total 250 330 290 870

6. Construct index numbers of price for the following data by applying:

i) Laspeyre’s method

ii) Paasche’s method

iii) Fisher’s Ideal Index number

Commodity

Base year Current year

Price Quantity Price Quantity

A 2 8 4 6

B 5 10 6 5

C 4 14 5 10

D 2 19 2 13

Master of Business Administration- MBA Semester 1 MB0040 – Statistics for Management - 4 Credits (Book ID: B1129) Assignment Set – 1

Q1. Define “Statistics”. What are the functions of Statistics? Distinguish between Primary data and Secondary data.Ans:Statistics is usually and loosely defined as: 1. A collection of numerical data that measure something. 2. The science of recording, organising, analysing and reporting quantitative information. Professor A.L. Bowley gave several definitions of Statistics. He defined Statistics as: “i) The science of counting ii) The science of averages iii) The science of measurement of social phenomena, regarded as a whole in all its manifestations. iv) A subject not confined to any one science”1 However, none of these definitions are complete. According to Horace Secrist, “Statistics may be defined as the aggregate of facts affected to a marked extent by multiplicity of causes, numerically expressed, enumerated or estimated according to a reasonable standard of accuracy, collected in a systematic manner, for a predetermined purpose and placed in relation to each other”2. This definition is both comprehensive and exhaustive. Prof. Boddington, on the other hand, defined Statistics as „The science of estimates and probabilities‟3. This definition is also not complete. According to Croxton and Cowden, „Statistics is the science of collection, presentation, analysis and interpretation of numerical data from logical analysis‟4.Statistics is used for various purposes. It is used to simplify mass data and to make comparisons easier. It is also used to bring out trends and tendencies in the data as well as the hidden relations between variables. All this helps to make decision making much easier. Let us look at each function of Statistics in detail.

1. Statistics simplifies mass data

The use of statistical concepts helps in simplification of complex data. Using statistical concepts, the managers can make decisions more easily. The statistical methods help in reducing the complexity of the data and consequently in the understanding of any huge mass of data.

2. Statistics makes comparison easier Without using statistical methods and concepts, collection of data and comparison cannot be done easily. Statistics helps us to compare data collected from different sources. Grand totals, measures of central tendency, measures of dispersion, graphs and diagrams, coefficient of correlation all provide ample scopes for comparison. Hence, visual representation of numerical data helps you to compare the data with less effort and can make effective decisions.

3. Statistics brings out trends and tendencies in the data After data is collected, it is easy to analyse the trend and tendencies in the data by using the various concepts of Statistics.

4. Statistics brings out the hidden relations between variables Statistical analysis helps in drawing inferences on data. Statistical analysis brings out the hidden relations between variables.

5. Decision making power becomes easier With the proper application of Statistics and statistical software packages on the collected data, managers can take effective decisions, which can increase the profits in a business.

Primary data: Data collected for the first time keeping in view the objective of the survey is known as primary data. They are likely to be more reliable. However, cost of collection of such data is much higher. Primary data is collected by the census method. In other words, information with respect to each and every individual of the population is observed.

Collection of primary data can be done by any of the following methods. 1. Direct personal observation 2. Indirect oral interview 3. Information through agencies 4. Information through mailed questionnaires 5. Information through schedule filled by investigators

Secondary data Any information, that is used for the current investigation but is obtained from some data, which has been collected and used by some other agency or person in a separate investigation, or survey, is known a secondary data. They are available in published or unpublished form.

In published form, secondary data is available in research papers, news papers, magazines, government publication, international publication, and websites. Secondary data is collected for different purposes. Therefore, care should be exercised while making use of it.

The accuracy, reliability, objectives and scope of secondary data should be examined thoroughly before use. Secondary data may be collected either by census or by sampling methods.

Q2. Draw a histogram for the following distribution:

Ans:The frequency distribution is represented by a set of rectangular bars with area proportional to class frequency. If the class intervals have equal width then the

variable is taken along X-axis and frequency along Y-axis and a rectangle is constructed.

Q3. Find the (i)arithmetic mean (ii)median value of the followings set of values: 40,32,24,36,42,18,10.Ans:For discreet data, arithmetic mean is given by the formula;

Where n= No. of values; n=7.

Substituting we get; Therefore, the arithmetic mean of the given series is found out to be 28.85

MEDIAN VALUE

Arranging the values in the ascending order;

10, 18, 24, 32, 36, 40, 42Formula for finding the median value;

We have n=7, so substituting;

Q4. Calculate the standard deviation for the following data.

Ans:The below table represents the frequency distribution of data required for calculating the standard deviation.

Standard deviation is given by the formula;

Substituting the respective values we get;

Therefore the Standard Deviation is found to be 2.336 marks.

Q5. Explain the following terms with respect to Statistics: (i) Sample, (ii) Variable, (iii) Population.Ans:i) Sample

A sample is a part or subset of the population. By studying the sample, we can predict the characteristics of the entire population from where the sample is taken. The data that describes the characteristics of a sample is known as statistic.

If the population is large, it is hard to collect data. Hence, a part of the population is chosen to study the characteristics of the entire population. The size of the sample can never be as large as the size of the population. Proper care must be taken while choosing the samples. In the figure below illustrates the population and sample.

(ii) Population or Universe

The totality of all units or individuals in a survey is called population or universe. If the number of objects in a population is finite then it is called finite population otherwise it is known as infinite population.

The data that describes the characteristics of the population is known as parameter. In the figure below the total number of eight consumers constitutes the entire population.

(iii) Variable

In a population, some characteristics remain the same for all units and some others vary from unit to unit. The quantitative characteristic that varies from unit to unit is called a variable. The qualitative characteristic that varies from unit to unit is called an attribute.

A variable that assumes only some specified values in a given range is known as discrete variable. A variable that assumes all the values in the range is known as continuous variable.

For example, the number of children per family and number of petals in a flower are examples of discrete variables. The height and weight of persons are examples of continuous variables.

Q6. An unbiased coin is tossed six times. What is the probability that the tosses will result in: (i) at least four heads, and (ii) exactly two headsAns:

Master of Business Administration- MBA Semester 1 MB0040 – Statistics for Management (Book ID: B1129) Assignment Set - 2

Q1. Find Karl Pearson’s correlation co-efficient for the data given in the below table:

Ans:

Q2. Find the (i) arithmetic mean (ii) range and (iii) median of the following data: 15, 17, 22, 21, 19, 26, 20.Ans:i. ARITHMETIC MEAN. Here, “∑X=15+17+22+21+19+26+20=140” and n=7. Substituting we get;

Therefore the Arithmetic mean is 20.

ii. RANGE.Range (R) is given by the formula;R=Highest value – Lowest ValueHence for the above data;R=26-15=11

Therefore the Range is 11.

iii. MEDIAN.Firstly, arranging the given data in ascending order;15, 17, 19, 20, 21, 22, 26Median (M) is given by the formula;Therefore we have;

The 4th value of the ascending series is 20.Therefore, the median (M) is 20.

Q3. What is the importance of classification of data? What are the types of classification of data?Ans: The importance of classification of data are as follows:

It condenses the bulk data

It simplifies the data and makes the data more comprehensible

It facilitates comparison of characteristics

It renders the data ready for any statistical analysis

A good classification should be unambiguous, exhaustive, flexible, stable, suitable, and mutually exclusive.

The different types of classification are as follows:

Geographical classification Data classified according to region is geographical classification.

Chronological classification Data classified according to the time of its occurrence is called chronological classification.

Conditional classification Classification of data done according to certain conditions is called conditional classification.

Qualitative classification Classification of data that is immeasurable is called qualitative classification. For example, sex of a person, marital status, color and others.

Quantitative classification Classification of data that is measurable either in discrete or continuous form is called quantitative classification.

Statistical Series Data is arranged logically according to size or time of occurrence or some other measurable or non-measurable characteristics.

Ans:The below table displays the observed and expected values required to calculate χ2

We know from theory that;

Null hypothesis ‘Ho’: The week and shifts are independent.

Alternate hypothesis ‘Ha’: The week and shifts are dependent.

Level of significance is 5% and DOF(3-1)(3-1)=4

Therefore we have; χtab2=9.49

We have the χcal2=3.6459.

Conclusion: Since χcal2 (3.6459) < χtab2 (9.49), the Ho is accepted. Hence theattributes “Week” and “Shift” are independent.

Q5. What is sampling? Explain briefly the types of samplingAns:Sampling is a tool which enables us to draw conclusions about the characteristics of population.By choosing a sample technique carefully, errors can be minimised. Let us take a look at the different techniques available. The sampling techniques may be broadly classified into. i) Probability Sampling ii) Non-Probability Sampling

Probability sampling

Probability sampling provides a scientific technique of drawing samples from the population. The technique of drawing samples is according to the law in which each unit has a predetermined probability of being included in the sample. The different ways of assigning probability are: i) Each unit has the same chance of being selected. ii) Sampling units have varying probability iii) Units have probability proportional to the sample size

We will discuss here some of the important probability sampling designs.

Simple random sampling Under this technique, sample units are drawn in such a way that each and every unit in the population has an equal and independent chance of being included in the sample. If a sample unit is replaced before drawing the next unit, then it is known as Simple Random Sampling With Replacement [SRSWR]. If the sample unit is not replaced before drawing the next unit, then it is called Simple Random Sampling without replacement [SRSWOR]. In first case, probability of drawing a unit is 1/N, where N is the population size. In the second case probability of drawing a unit is 1/Nn.

The selection of simple random sampling can be done by: Lottery method: In lottery method, we identify each and every unit

with distinct numbers by allotting an identical card. The cards are put in a drum and thoroughly shuffled before each unit is drawn. The figure 7.6 represents a lotto machine through which samples can be selected randomly.

The use of table of random numbers: There are several random number tables. They are Tippet‟s random number table, Fisher‟s and Yate‟s Tables, Kendall and Babington Smiths random tables,

Rand Corporation random numbers and so on. The table 7.2 represents the specimen of random numbers by Tippett‟s.

Suppose, we want to select 10 units from a population size of 100. We number the population units from 00 to 99. Then we start taking 2 digits. Suppose, we start with 41 (second row) then the other numbers selected will be 67, 95, 24, 15, 45, 13, 96, 72, 03.

Stratified random sampling This sampling design is most appropriate if the population is heterogeneous with respect to characteristic under study or the population distribution is highly skewed.We subdivide the population into several groups or strata such that : i) Units within each stratum is more homogeneous ii) Units between strata are heterogeneous iii) Strata do not overlap, in other words, every unit of population belongs to one and only one stratum

The criteria used for stratification are geographical, sociological, age, sex, income and so on. The population of size „N‟ is divided into „K‟ strata relatively homogenous of size „N1‟, „N2‟………….‟Nk‟ such that „N1 + N2 +……… + Nk = N‟. Then, we draw a simple random sample from each stratum either proportional to size of stratum or equal units from each stratum. The table 7.3 displays the merits and demerits of stratified random sampling.

Systematic sampling This design is recommended if we have a complete list of sampling units arranged in some systematic order such as geographical, chronological or alphabetical order.

Suppose the population size is „N‟. The population units are serially numbered „1‟ to „N‟ in some systematic order and we wish to draw a sample of „n‟ units. Then we divide units from „1‟ to „N‟ into „K‟ groups such that each group has „n‟ units. This implies „nK = N‟ or „K = N/n‟. From the first group, we select a unit at random. Suppose the unit selected is 6th unit, thereafter we select every 6 + Kth units. If „K‟ is 20, „n‟ is 5 and „N‟ is 100 then units selected are 6, 26, 46, 66, 86.

The table 7.5 displays the merits and demerits of systematic sampling.

Cluster sampling The total population is divided into recognisable sub-divisions, known as clusters such that within each cluster units are more heterogeneous and between clusters they are homogenous. The units are selected from each cluster by suitable sampling techniques. The figure 7.7 represents the cluster sampling where each packet of candy packet forms a cluster.

Multi-stage sampling The total population is divided into several stages. The sampling process is carried out through several stages. It is represented as in figure 7.8.

Non-probability sampling Depending upon the object of enquiry and other considerations a predetermined number of sample units is selected purposely so that they represent the true characteristics of the population. A serious drawback of this sampling design is that it is highly subjective in nature. The selection of sample units depends entirely upon the personal convenience, biases, prejudices and beliefs of the investigator. This method will be more successful if the investigator is thoroughly skilled and experienced.

Judgment Sampling The choice of sample items depends exclusively on the judgment of the investigator. The investigator‟s experience and knowledge about the population will help to select the sample units. It is the most suitable

method if the population size is less. The table 7.7 displays the merits and demerits of judgement sampling.

Convenience sampling The sample units are selected according to convenience of the investigator. It is also called “chunk” which refers to the fraction of the population being investigated which is selected neither by probability nor by judgment. Moreover, a list or framework should be available for the selection of the sample. It is used to make pilot studies. However, there is a high chance of bias being introduced.

Quota sampling It is a type of judgment sampling. Under this design, quotas are set up according to some specified characteristic such as age groups or income groups. From each group a specified number of units are sampled according to the quota allotted to the group. Within the group the selection of sample units depends on personal judgment. It has a risk of personal prejudice and bias entering the process. This method is often used in public opinion studies.

Q6. Suppose two houses in a thousand catch fire in a year and there are 2000 houses in a village.What is the probability that;i. None of the houses catch fire andii. At least one house catches fire.Ans:Given the probability if a house catching fire is:

Therefore; m=np=2000×0.002=4

The required probabilities are calculated as follows.

(i)The probability that none of the houses catch fire is given by:Therefore, the probability that none of the houses catches fire is 0.01832.

ii. The probability that at least one of the houses catch fire is given by:Therefore, the probability that at least one house catches fire is 0.98168.