Statistics for management


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Statistics for management

  1. 1. Name – Vinay Aradhya M.AReg no – 1302010663Course – MBA – 1st semester. 2013Subject code – MB0040 – Statistics for Management.Q1 (a) Explain the characteristics of Statistics.Statistics is a science which deals with the method of collecting, classifying, presenting, comparing andinterpreting the numerical data to throw light on enquiry.Statistics details with an aggregate of factsA single figure cannot be analyzed. For example, the fact ‘Mr. Kiran is 170 cms tall’ cannot bestatistically analyzed. On the other hand, if we know the heights of 60 students in a class, we cancomment upon the average height and variation.Statistics gets affected to a great extent by multiplicity of causesThe statistics of the yield of a crop is the result of several factors, such as the fertility of soil,amount of rainfall, the quality of seed used, the quality and quantity of fertilizer used.Statistics are collected in a systematic mannerThe facts should be collected according to planned and scientific methods otherwise; they arelikely to be wrong and misleading.Statistics are collected for a pre-determined purposeThere must be definite purpose for collecting facts. Otherwise, indiscriminate data collectionmight take place which would lead to wrong diagnosis.Statistics are placed in relation to each otherThe facts must be placed in such a way that a comparative and analytical study becomespossible. Thus, only related facts which are arranged in a logical order can be called statistics.Statistical analysis cannot be used to compare heterogeneous data.(b) What are components of statistics? Give a brief description of each of the components.Collectionof dataPresentation of dataAnalysis ofdataInterpretationof data
  2. 2. Basis components of Statistics According to Croxton and CowdenCollection of data :Careful planning is required while collecting data. Two methods used for collecting data arecensus method and sampling method. The investigator has to take care while selecting anappropriate collection method.Presentation of data :The collection data is usually presented for further analysis in a tabular, diagrammatic or graphicfrom and it is condensed, summarized and visually represented in a tabular or graphical form.Tabulation is a systematic arrangement of classified data in rows and columns. For therepresentation of data in diagrams, we use different types of diagrams such as one-dimensional,two-dimensional and three dimensional diagrams.Analysis of dataThe data presented has to be carefully analyzed to make any inference from it. The inferencecan be various types, for example, as measure of central tendency, desperation, correlation orregression.Interpretation of dataThe final step is to draw conclusions from the analyzed data. Interpretation requires highdegree of skill and experience. We can interpret the data easily from pie-charts.Q2. Explain the objectives of statistical average. What are the requisites of a good average?The statistical average or simply an average refers to the measure of middle value of the data set. Theobjectives of statistical average are to:Present mass data in a concise form: The mass data is condensed to make the data readableand to use it for further analysis. It is very difficult for human mind to grasp a large body ofnumerical figures. A measure of average is used to summarize such data into a single figure,which makes it easier to understand.Facilities comparison: It is difficult to compare two different sets of mass data. However, we cancompare those two after computing the averages of individual data sets. While comparing thesame measure of average should be used. It leads to incorrect conclusions when the meansalary of employees is compared with the median salary of the employees.
  3. 3. Establish relationship between data sets: The average can be used to draw inferences aboutthe unknown relationships between the data sets. Computing the averages of the data sets ishelpful for establishing the average of population.Provide basis for decision making: In many fields such as business, finance, insurance and othersectors, managers compute the averages and draw useful inferences or conclusions for takingeffective decisions.Requisites of a good averageThe following are the requisites of a good average:It should be simple to calculate and easy to understand.It should be based on all the values.It should not be affected by extreme values.It should not be affected by sampling fluctuation.It should be rigidly defined, preferably by an algebraic formula, so that different persons obtainthe same value for a given set of data.Should be suitable for further mathematical treatment.Vigorously definedCapable of simple interpretationCapable of mathematical manipulation.Not unduly influenced by one or two extremely large or small values.Dependent on all the observed values.Q3. Mention the characteristics of a Chi-square test.The Chi-square test is one of the most commonly used non-parametric tests in statistical work.The following are the characteristics of Chi-Square test ( 2test).The 2test is based on frequencies and not on parametersIt is a non-parametric test where no parameters regarding the rigidity of population ofpopulations are requiredAdditive property is also found in 2testThe 2test is useful to test the hypothesis about the independence of attributesThe 2test can be used in complex contingency tablesThe 2test is very widely used for research purposes in behavioral and social sciences includingbusiness researchIt is defined as:Where, ‘O’ is the observed frequency and ‘E’ is the expected frequency.
  4. 4. b. Answer: Let us take the hypothesis that the sampling techniques adopted by research workers aresimilar (i.e., there is no difference between the techniques adopted by research workers). This being so,the expectation of a investigator classifying the people ini. Poor income group = (200 * 300) / 500 = 120ii. Middle income group = (200 * 150) / 500 = 60iii. Rich income group = (200 * 50) / 500 = 20Similarly the expectation of B investigator classifying the people ini. Poor income group = (300 * 300) / 500 = 180ii. Middle income group = (300 * 150) / 500 = 90iii. Rich income group = (300 * 50) / 500 = 30We can now calculate as follows:Groups Observedfrequency OijExpectedfrequency EijOij– Eij (Oij– Eij)2 EijInvestigator Aclassifies people as poor 160 120 40 1600/120 =13.33classifies people asmiddle class people30 60 -30 900/60 = 15.00classifies people as rich 10 20 -10 100/20 = 5.00Investigator Bclassifies people as poor 140 180 -40 1600/180 =8.88classifies people asmiddle class people120 90 30 900/90 = 10.00classifies people as rich 40 30 10 100/30 = 3.33Hence, χ 2 =∑{ (O ij – Eij) 2 /Eij} = 55.54Degrees of freedom = (c – 1) (r – 1)= (3 – 1) (2 – 1) = 2.The table value of χ for two degrees of freedom at 5 per cent level of significance is 5.991.The calculated value of χ is much higher than this table value which means that the calculated 2valuecannot be said to have arisen just because of chance. It is significant. Hence, the hypothesis does not
  5. 5. hold good. This means that the sampling techniques adopted by two investigators differ and are notsimilar. Naturally, then the technique of one must be superior to that of the otherQ4. What do you mean by cost of living index? Discuss the methods of construction of cost of livingindex with an example for each.The ‘Cost of living index’, also known as ‘consumer price index’ or ‘cost of living price index’ is thecountry’s principal measure of price change. The consumer price index helps us in determining theeffect of rise and fall in prices on different classes of consumers living in different areas.The cost of living index does not measure the actual cost of living or the fluctuations in the cost of livingdue to causes other than the change in price level. However, its object is to find out how much theconsumers of a particular class have to pay for a certain quantity of goods and services.(i). Utility of consumer price index numbersIt is useful to measure the change in purchasing power of currency, real income.It helps the government in formulating wage policy, price policy, taxation and general economic policies.(ii). Assumptions of cost of living Index NumbersCost of living index number is based on the following assumptions.Similar needsThe needs of the people for which this index number is constructed are same.Same goodsCost of living index numbers are true on the average.(iii). Steps in construction of cost of living index numbersThere are 5 steps involved in construction of cost of living index numbers.Step 1: Select the class of peopleStep 2: Define scope of the indexStep 3: Conduct family budget inquiryStep 4: Obtain price quotationsStep 5: Prepare a frame or list of personsMethod of constructing consumer price index:There are two methods for constructing consumer price index number. They are:I. Aggregate expenditure methodII. Family budget method or method of weighted average of price relatives.Aggregate expenditure method
  6. 6. This is based on Laspeyre’s method where the base year quantities are taken as weights (W = Qo).∑ P1 Q0Po1 = -------------------- x 100∑ P0 Q0Family budget methodFamily budget method or the method of weighted relatives is the method where weights relatives is themethod where weight are the value (Po Qo) in the base year often denoted by W.∑ PW P1Po1 = --------------------, where P= ------- X 100 for each item and∑ W P0W = value weight, i.e. PoQoExampleCalculate the cost of living index for the current year on the basis of the base year from the followingdata, usingSolution: - Aggregate expenditure MethodThe formula of aggregate expenditure method is giving by:
  7. 7. ∑ P1 Q0 315.6Po1 = -------------------- x 100 = -------------- x 100 = 106.87∑ P0 Q0 295.3Therefore the cost of living index number is 106.87Q5. Define trend. Enumerate the methods of determining trend in time series.The trend is a pattern of data. The trend shows how the series has been moving in the past and what itsfuture course is likely to be over a long period of time.To measure the secular trend, the short-term variations should be removed and irregularities should besmoothed out. The following are the methods of measuring trend.Graphic methodThe values of the time series are plotted on a graph paper with the time (t) along x-axis and the valuesof the variable (y) along y-axis. A freehand curve is drawn through these points in such a manner that itmay show a general trend. A free hand curve removes the short-term variations and irregularmovements.It is the simplest method, Time and labor is saved. It is very flexible method as it represents both linearand non-linear trends.The main drawback of this method is that it is highly subjective as different persons will draw differentfree hand curves. Because of its subjective nature it is useless in forecasting.Semi-Average MethodThis method is sometimes used when a straight line appears to be an adequate expression of trend. Inthis method, the original data are divided into two equal parts. The averages of each part are thencalculated. The average of each part is centered in the period of the time of the part from which it hasbeen computed and then plotted on the graph paper. In this way, a line may be drawn to pass throughthe plotted points which give the trend line. In case of odd number of years, the mid-year is eliminatedwhile dividing the data into two equal parts.This method is not subjective and ·everyone gets the same trend line. It is possible to extend the trendline both the ways to estimate future or past values. But the method assumes the presence of lineartrend which may not exist.Moving Average MethodMoving averages method is used for smoothing the time series. It smoothens the fluctuations of thedata by the moving averages method.Least squares method
  8. 8. The method of least squares is a standard approach to the approximate solution of over determinedsystems, i.e., sets of equations in which there are more equations than unknowns. "Least squares"means that the overall solution minimizes the sum of the squares of the errors made in the results ofevery single equation.Q6. The following data represent the number of units of production per day turned out by 5 differentworkmen using different types of machines.Workmen Machine typeA B C D1 44 38 47 362 46 40 52 433 34 36 44 324 43 38 46 335 38 42 49 39i) Test whether the mean productivity is the same for the four different machine types.ii) Test whether 5 men differ with respect to mean productivity.Let H0: (a) Mean Productivity is same for all machines(b) Men do not differ with respect to mean productivity decoding the data by subtracting 40 from eachfigure.Source ofVariationSum ofSquaresDegrees ofFreedomMean Square Variance RatioBetweenmachine type338.8 3 112.933 F1 = 112.933 / 6.142= 18.387Betweenworkers161.5 4 40.375 F2 = 40.375 / 6.574= 6.574Residual error 73.7 12 6.142Total 574 19(a) F0.05 = 3.49 at df1 = 3 and df2 = 12. Since the calculated value F1 = 18.387 is greater than thetable value, the null hypothesis is rejected.(b) F0.05 = = 3.26 at df1 = 4 and df2 = 12. Since the calculated value F2 = 6.574 is greater than thetable value, the null hypothesis is rejected.