AIM COLLEGE-HISAR Contact: , STATISTICAL ANALYSIS (MBA CP-205)

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1 Ques 1: - What do you understand by (Average) central tendency. Explain method of central tendency with merits & demerits. Ans:- Central Tendency:- An average is a single value, point or location which represent the whole set of figures and all other individual items concentrated around it. It is knows as central tendency. Central Tendency is known as ad verge or measures. Characteristics:- 1. It is easy to understand. 2. It should be simple to compute. 3. It should be based on all the observation. 4. It should be based on all the items. 5. It should be capable of further algebraic treatment. Method:- (A) Mathematical Averages:- - Arithmetic mean - Geometric mean - Harmonic mean - Quadratic mean (B) Positional Averages:- - Median - Mode (C) Commercial Average:- - Moving Average - Progressive Average - Composite Average 1. Mathematical Averages:- Arithmetic Mean:- It is the important popular & simplest measures of central Tendency. It is also known as Mean. AM is the number which is obtained by adding the values of all the items of a series & dividing the total by the number of items. Formula:- X = X N Two types of A.M. = (a) Simple arithmetic (b) weighted A.M. Three types of Stastical Series:- (a) Individual Series (b) Discrete frequency series (c ) Continuous series (i). It is easy to compute (ii) It is well defined. (i). It can neither be determined by inspection not by graphical location. (ii) It cannot be calculate for qualitative dates. 2. Geometrical Mean:- GM is defined as the Nth root of the product of N items. If there are 2 item, we take the square root; if three, the cube root is seen. Formula G.M. = Antilog F log M x 1 X x 2 X x x n N (i) It is capable of farther algebraic treatment. (ii) (ii) It is less affected by the extreme values. (i)it has restricted application. (ii) It is difficult to understand. (CP-205) 1/17

2 3.Harmonic Mean:- HM means is reciprocal of the arithmetic average of the reciprocal of value of various items in the variables. Formula:- N f x 1. It is rigidly defined. 2. is based on all the observations of the series. 1. It is difficult to calculate & is not understandable. 2. It is not popular. 3.Quadratic Mean:- 1. It is rigidly defined. 2. It is based on all observation of series. 2. Difficult to calculate. 2. It is not popular. (B) Positional Averages:- 1. Median:- Median of distribution is that value of the variant which divides it into the equal parts:- Median may be defined as the value of that item which divides the series into two equal parts, are half containing values greater then it & the other half containing values less than it. Formula: Median = size of n + 1 ` 2 1. It is well-defined. 2. It is easy to compute. 1. It is not based on all the items. 2. It is not capable of further algebraic treatment. 2. MODE:- It is the value which occurs the greatest number of frequency in a series. Formula:- MODE = L + 1 h L = Lower Limit = Difference of frequency of model class & preceding class. 2 = Difference of frequency of model class and succeeding class. h = width of the model class. 1. It is easy to compute. 2. It can be located graphically. 1. It is not simple to understand It is not well defined. C. Commercial Average:- (i) Moving Average (ii) Progressive Average (iii) Composite Average Merit:- 1. It is easy to compute. 2. It is easy to understand. (CP-205) 2/17

3 1. Time consuming 2.It is not popular. Others Partition of Positional Measures:- (1) Quartile (2) Deciles (3) Percentiles Ques 2:- Explain what is meant by dispersion. What are the methods of computing dispersion? What is the practical utility of such measures? Ans:- Meaning:- Dispersion is spread, scat redness or variation of items around an average or among themselves. Major objectives of measuring Dispersion:- 1. To test reliability of an average. 2. To compare the extent of variability in two or more distribution. 3. To facilitate the computations of other Stastical means. 4. To serve as the basis for central of variations. Method:- Measures of Dispersion:- 1. Range 2. Inter-Quartile Range 3. Mean Deviation 4. Stand & Deviation 5. Lorenz Curve 1. Range:- It is a distribution, is the difference between its two external observation, i.e. the difference between the largest and small observations. Range = L - S L + S 1. It is easy to understand & easy to calculate. 2. It gives a quick measures of variability. 1. It is not based on all observations. 2. It is very mush effected by extreme observations. (II) Inter Quartile Range:- This range is based on extreme values. It ignores the deviation in between values. I.Q. Range = Q 3 - Q 1 Quartile Deviation (QD) = Q 3 - Q It is easy to understand & easy to compute. 2. It is not influenced by the extreme values. 1. It ignores the first 25% of the item. 2. It gives only a rough measures. (III) Mean Deviation:- It is defined as the AM mean of the absolute deviations of observation from a central value like mean, median & mode It is also known as Average Deviation. Mean Deviation MD = Mean or Median & mode 1. It is easy to understand & it is easy to calculate. 2. It is rigidly6 defined. 3. It is a better measure for comparison. 4. It is not much affected by the fluctuations of sampling. 1. It is rarely used. It is not as popular as S.D. 2. It is not very accurate measures of dispersion. IV Standard Deviation:- It is the square root of the arithmetic mean of the items from their mean value. (CP-205) 3/17

4 S.D. = x 2 N 1. It is possible for further algebraic treatment. 2. It is less affected by the fluctuations of sampling. 1. It is not easy to understand & it is difficult to calculate. 2. It is affected by the value of every item in the series. (V) Lorenz Curve (LC):- It is a device used to show the measurement of economic inequalities as in the distribution of income & wealth. LC can also be used for the study of distribution of Profits, wages etc. 1. The size of items & their frequencies to be calculated. 2. It is rigidly defined. 3. Perfect measurement. 1. Time consuming 2. It is rarely used. Ques 3:- What is Statistical Quality Control? How are control limits set up? Describe the various types of control charts. Ans :- Introduction: Stastical Quality Control deals with the quality of the articles produced by a machine will be exactly similar, but this is not true as variation is inherent & unavoidable. On taking measurement pertaining to certain characteristics of articles, it is found that they do differ from one another. Stastical quality control method are applied to two phases of the manufacturing process: (i) Process Control (ii) Product Control or5 lot Control Discuss types of control charts & Control limits set up:- I. Control Chart for quality measurements (continuous Variables) II. Control Charts for attributes I. Control Chart for quality measurements:- Control Chart quality develop by Dr. Walter A. Shewhart, a physicist of Bell Telephone Laboratory in Control Chart for Process Control:- In Process, a sample of units selected at regular intervals. The quality of a large number of products is adjudged on the basis of measurement of characteristics, such as length, diameter, weight, textile strength etc. There variables are of a continuous type for continuous variables, Shewhart developed control charts known as x, Ө and R charts to keep a control on the quality of the product. These C charts in value the location parameter the mean, the scale parameters, the range the standard deviation. (A) x-chart:- Some variations do occur from unit to unit, as no two units are alike in respect of certain characteristics. To keep a control on the measurable characteristics of articles, we use x-chart, which in valves the mean x and the S.D.. X-chart is a geographical device, which depicts the n=measurements decision if all the blotted points lie on or in between upper & lower control limits, the process is under control. Setting of control limits: - x-chart Process a) Obtain the mean of each sample x1, x2, x3 etc. This is done by dividing the sum of the values divideng the sum of the valuee divided in a Sample ( x) by the number of items in the sample. X = x (CP-205) 4/17

5 N b) Obtain the mean of the Sample mean: == x x Number of Sample c) The control limit are set at UCL = x + 30x LCL = x - 30x (CP-205) 5/17 Where σ = σ AND σ = D1 R X n R is a biased estimator of σ and d1is the correction factor. The values for d1 are tabulated. There for control limits are UCL = x + A2R LCL = x - A2R Revealing the state of their scatters from the standard value. Diagram:- y Rejection Region UCL Stastical Value Acceptance Region Central Line Acceptance Region LCL rejection Region L Sample Numbers Above given figure The control charts have 3 horizontal lines; the lower line, the middle line & the upper line. The M.L., C.L. show the standard value of the qu7ality, characteristics of the manufactured units. The L.L. is the LCL of the variate value & the upper line is the upper control limit (UCL) As a tradition, the lines indicating the upper & the lower control limits are shown by dotted lines., while the center line is kept smooth. Small sample of units is drawn at regular intervals & the decision about the process, whatever it is under central not is taken on the basis of the potted points. Which in this case are3 mean values of the samples plotted against sample numbers. (B)R-chart:- Experience revels that in case of small sam0ples, the standard deviation S and the range R fluctuate simultaneously. Elaborately, if S is small, R is also small. But those relation does not hold good for large samples. In Stastical quality control, generally, small samples are drawn & hence the range can be used in place of a range are termed R-charts. R-chart is used to show the availability or dispersion of the quality produced by a given process. Setting of Chart-limits- Process r-chart 1. The range of each and ample, R 2. The mean of the sample ranges, R 3. UCL & LCL UCL = R + 30x LCL = R - 30x WHILE Σr = The error of the range standard. Therefore control limit are:- UCL R = D4R LCL R = D3R

6 UCL Quality R LCL O Sample Number (C ) Control Chart for the S.D. or σ chart. Setting of a Control Limits:- UCL = B2 σ σ = S.D. LCL = B1 σ If S.D. is not known then its estimates based on the average of sample SD s used. UCL = B4S LCL = B3S WHEN S = Sum of the Sample SD Number of Samples II. Control Charts for attributes:- A single chart is enough to decide for a number of a quality characteristic under the consideration, large samples are needed for correct answers. (A) Control Charts for Number of Defects:- In the preceding section, we considered. Only one criteria, i.e. an item is either perfect or defective But in many situations it will be advantageous to know the number of defects in a piece. For ex:- A radio set have many defects, i.e. its bulb may be out of order, circuit may be wrong, some resonance may out of order etc. If the use of the C-Chart:- is appropriate if the opportunities for a defect in each production units are infinite but the Probability of a defect at any point is very small & is constant. The number of defects in a gal vanished sheet or painted plate, aircrafts, roll of paper Sheet of photographic film Danger Sign No. of.. Defects Sample No. (C) Control Charts for Fraction Defectives:- Defective units can not be used and if not defective it is acceptable. For ex:- More defective items come in the market of sale. The reputation of the company will go down & the product may lose the market. Hence, the process is tested to see whether it is under control with regard to fraction, defectives. The p-chart is designed to control for fraction defective. P-chart major advantage easy to underst6and and more straightforward method. Steps:- (CP-205) 6/17

7 i. Compute the average fraction defective (p) by divided the number of defectives by the total number of units inspected. ii. On the chart draw a solid horizontal line to represent p. iii. Determine upper and lower control limit. UCL = p + 3 p (1-P), LCL = 3 p (1-P) N N Advantages of Stastical Quality Control:- 1) SQC involves inspection of only a fraction of items produced in a fixed period. Hence, it is very economical. 2) The inspection of each & every item has hardly been feasible. 3) Efficiency in quality 4) SQC an be carried through persons who do not posses a high degree in engineering or Stastics. 5) SQC provides protection against losses to the producer as well as to the consumer. 6) SQC provides compromise between the machine operators and engineers. 7) Good device. Disadvantage:- 1. Time consuming 2. Main responsibility of maintaining the quality 3. It is a part of production process rather than the production part of SQC. Ques 4:- What is Sampling? What are the essential of a good sample? Name the various methods of Sampling with merits & demerits? Ans:- Introduction:- The Sampling design or survey specifies the method of collecting Sample. Definition:- A Sample design is a definite plan for obtaining a sample from a given population. It refers to the technique or the procedure the researcher would adopt in selecting items for the Sample. Or the process of using a small number of items or parts of a large population to make conclusion about the whole population. Essentials of a good Sample:- 1. Truly Representative ness. 2. Independence 3. Adequacy 4. Homogeneity 5. More accurate results 6. Facilitating Timely results 7. Less cost 8. Destructive testing 9. Used-the appropriate Sampling design s 10. Destructive testing 11. The Sample is drawn in a scientific manner. 12. This method is economical in represent of money, time, and labour. 13. This method is easily used when the area covered in the population is spreads widely. 14. Perfect Information. Limitations:- 1. This method cannot be used effectively if the size of the population is very small. 2. This method may result in wrong conclusion if the Sample investigated is not representative of the population. Methods of Sampling:- I. Probability Sampling II. Non-Probability Sampling I. Probability Sampling:- A Sampling technique in which every member of the population has a known, non zero probability of selection. It is also known as chance probability. Different Probability Sampling Method: - (A) Simple Random Sampling: - A Sampling procedure that assures each element in the population on an equal chance of being included in the Sample. Lottery Method, use of random numbers. (CP-205) 7/17

8 1. This method of Sampling is considered as the most scientific method. 2. The Sample is not at all affected by the bias of the investigate. 3. Method based on the Law of Statistical Regularity and the law of Inertia of Large numbers. 4. This method is respect of time, money & labour. 1. This method is not applicable when the complete list of terms of population If population are spreads not available to the investigator. 2. If population are spread very widely it would be difficult to cover & calculate. 3. This method is not suitable when the items of population are heterogeneous in nature. 4. The random sample may be badly affected by the bias of the chare. (C) Complex Random Sampling:- A probability Sampling Procedure in which Sampling Random sub samples are drawn from with in different Strata that are more or less equal on same Characteristics: Proportional & disproptional stratified Sampling. 1. In this method the investigator can exercise great central over the items of the samples by manipulating the number of strata. 2. This method is economical in respect of time, money and labour. 3. This method is easily used when the items of the population are heterogeneous in nature. Grouping the items of population on the basis of homogeneity is a very technical job. If the strata 000are not formed correctly, the specified sample would not be the representative of the population. (2) Multi-Stage Area Sampling:- It involves using a combination of other probability sample technique. 1. This method is economical in respect of money, time & labour. 2. This method is very useful in case a very wide range to be covered in the investigation. 3. This method is very useful, in case a very wide range is to be covered in the investigation. 1. Less accurate method. 2. This method is not suitable when the area to be covered is very small. (3) Systematic Random Sampling:- A Sample procedure in which an initial starting point is selected by a random process & then every nth number on the list is selected. 1. It is very simple to operate. 2. It is suitable in case a very small area is to be covered in the investigation. 3. This method is economical investigation respect of money, time & labour. 1. The sample can be badly affected by the hidden periodicities regarding some characteristics. 2. The Sample drawn may be affected by the bias of the investigator. Others --- Cluster & Area Sampling. II. NON-PROBABILITY SAMPLING:- (A) Convenience Sampling:- The Sampling procedures used to obtain in those units or people most conveniently available. 1`. This is a very easy method of Sampling. 2. This method is useful for pilot-survey. 1. This data collected may be badly affected by the bias of the investigator. (B) Judgment/ Purposive Sampling:- Judgment Sampling technique in which an experienced individual selects the Sample based upon some appropriate characteristics of the Sample members. (CP-205) 8/17

9 1. This method is very useful when there is motive for adoptive others modes of sampling. 1. This method is not considered to be scientific. 2. This method of Sampling is expected to be affected by the bias of the investigator. (C) Quota Sampling:- A procedure that ensures that certain charate4ristics of a population sample will be represented to the exact extent that the investigator desires. 1. The data collected is reliable. 2. This method is economical in respect of money, time & labour. 1. The data collected is expected to be affected by the bias of the enumerators or of the investigators. Ques 5:- Explain the various methods of collecting Primary data & Secondary data print out their respective merits & demerits? Ans:- Methods of Data collection Two Types:- (I) Primary Data (II) Secondary Data I. Primary Data:- Data gathered & assembled specifically for the research Project at hand. (A) Observation Method:- This method is the most commonly used method specially in studies relating to behavioral sciences. The systematic recording non-verbal as well as verbal behaviour & communication. Different types of observation:- a. Visible observation b. Structural observation c. Hidden observation d. Unstructured observation (B) Interview Method:- of collecting data in valves presentation of oral verbal stimulate & reply in terms of oral-verbal responses. Different types of Interview:- 1. Personal Interview 2. Telephonic Interview (C ) Questionnaires:- that is filled in the respondent rather than by an interviewer. Main aspects of a Questionnaire:- 1. General Form 2. Question sequence 3. Question formulation & wording 4. Collection of Data through Schedules:- Performa containing a set of questions. 5. Some other methods of Primary Data:- a) Warranty Cards b) Store audit c) Pantry audit d) Consumer panel e) Mechanical device f) Projection techniques g) Content Analysis. II. Secondary data:- Data that have been previously collected for some purpose other than the one a5t hand. Different sources of Secondary data:- a) Magazines b) Books c) Reports, by government d) Universities, Net e) Government Publications Advantages & Disadvantages:- Advantages of Primary Data: 1) The data collected is reliable. (CP-205) 9/17

10 2) The degree of accuracy in data collected is very high. 3) Consistency & homogeneity is present in the data collected. 4) Replies can be recorded without causing embarrassment to respondents. 5) It is more flexible. Disadvantage:- 1) Time consuming 2) Expensive 3) Too vague data not collected. Easily. 4) More labour Advantages of Secondary Data:- 1) This method is economical in respect of money, labour & time. 2) This method can be very easily used if the area to be covered is widely used. 3) Secondary data can be used as a basis of comparison. Disadvantage:- 1) It is difficult to analysis 2) No choice 3) Replies respondents can not be recorded. Ques 6:- Write a short note:- (a) Skew ness (b) Kurtosis & Moments (c ) Index Numbers (d) Correlation (e) Regression Ans:- (A) Skew ness:- means lack of symmetry of a frequency distribution. It means that the highest frequency decreases on it either side at the uniform rate and form a balanced pattern. It is also known as a symmetrical or skewed. Methods of measuring Skew ness:- 1. Karl Pearson s Method = A.M. - Mode S.D. 2. Boewley s Method CO-eff. Of Skew ness = Q 3 + Q 1-2 Median Q 3 + Q 1 3. Methods of Movements Moment of skew ness = u 3 3 U 2 4. Kelley s Method Coeff of skew ness = P 90 + P 10-2 Median P 90 + P 10 (B) Kurtosis:- measures the degree of peackedness of a distribution. Another measures to test, however a particular frequency distribution conforms to the normal curve is Kurtosis. It indicates whether, a distribution is more flat-topped or more peaked than the normal distribution. According to Prof. D.N.Elhance:- Measures of Kurtosis:- Formula:- K = u 4 U 2 Type:- (i) Lepto Kurtosis (ii)platy Kurtosis (iii)meso Kurtosis (CP-205) 10/17

11 Moments:- Moments are statistical tools, used in Stastical investigations. The moments of a distribution are the arithmetic means of the various powers of the derivations of items from some numbers. Two main methods in moments:- Sheppard s Correction for Moments Alternative Method for finding moments ( c) Index Number:- Index Numbers are today one of the most-widely used statistically devices. They are used to take the pulse of the economy and they have come to be used as indicators of inflation any or deflationary tendencies. Main users of Index Numbers:- 1. Index Numbers help in framing suitable policies. 2. Index numbers help in studying tends & tendencies. 3. Index Numbers are very useful in deflating. (D) Correlation:- Correlation is a important Stastical technique. Correlation deals with the association between two or more variables. Types of Correlation:- i. Positive & Negative Correlation ii. Linear & Non-Linear Correlation iii. Simple, Partial & Multiple Correlation Methods:- I. Graphic Methods (a) Scattered Diagram (b) Correlation Graph II. Algebraic Methods: (a) Karl Pearson s Coefficient of Correlation (b) Spearmen s Rank Correlation Method (c) Concurrent Deviation Method (e) Regression Analysis:- It is an important Statistical technique. Regression is the measure of the average relationship between two or more variables. Types:- i. Simple & Multiple Regression ii. Linear & Non-Linear Regression iii. Partial & Total regression Methods:- i. Scatter diagram Method ii. Least square Method iii. Regression equations in individual series & grouped series. Important Note:- Correlation & Regression Practical Question is very important in exam s. Ques 7:- (a) Define Stastical decision Theory. Explain Bayesian Approach to Decision- Making. (b) Sampling Distribution Ans:- (a) Statistical Decision theory is a term used to apply to those methods for solving decision problems in which uncertainly plays a crucial role. Bayesian Approach to Decision-Making:- Bayesian Approach plays an very important role in decision making. Steps:- (CP-205) 11/17

12 1. Prior Probabilities are the original marginal probabilities giving the initial estimate of the statuary nature. These are first in a tree diagram. 2. Given the above additional information constitutes the conditional probability. 3. The joint probabilities are now derived by taking the 1 & 2 above. 4. The posterior Probabilities showing the state of nature are oriented by considering joint probabilities given in 3 and the managerial probabilities divided in below 5 5. The new managerial probabilities are derived by summing up appropriate point probabilities given in 3rd above. 6. Now joint probabilities of 3 rd over the managerial probabilities of 5 th will give the posterior probabilities. These probabilities started in the form of conditional probabilities show the altered state of nature in the light of information available from an experiment. Bayesians Decision Analysis:- The method of Samp0ling can be used to provide information about future events. This information can then be used for rendering the prior probabilities to from posterior probabilities. In other words, the results of Sampling can be updating our beliefs of subjectivity. Prior belief + Sample in formation (posterior beliefs) (Probabilities) (Probabilities) (B) Sampling Distribution:- It means refers to the probability distribution of all the possible means of random samples of a given size that we like from a population. Different type of Sampling Distribution:- 1. Chi-Square Distribution 2. The student Std. Distribution 3. F-Distribution. Q. 8 What is Hypothesis? Brief discussion about Testing Hypothesis. What are Parametric Tests and Non-Parametric Tests. Explain in Brief? Ans- Hypothesis- an unproven proposition or supposition that tentatively explains certain facts or phenomena; a proposition that is empirically testable. Main Characteristics- 1. It should be clear and precise. 2. It should state relationship between variables, if happens to be a relational hypothesis. 3. Hypothesis should be limited in scope & must be specific., 4. Hypothesis should be amenable testing with in a reasonable time Testing of Hypothesis:- Major two concepts in Hypothesis Testing (i) Null Hypothesis (Ho) (ii) Alternative Hypothesis (Ha) Procedure for Hypothesis Testing: Step I Making a Formal Statement, the step consists in making a formal statement of the null hypothesis and also of the alternatives Hypothesis, This means that hypothesis should be clearly stated. Step- II Selecting a significance level. The Hypothesis are tested on a pre-determinal level of significance and as such the same should be specified. (CP-205) 12/17

13 Step III Deciding the distribution to use. Under III step deciding distributor method T-distribution, normal distribution Step IV Selecting Sample Selecting a random Sampled and computing an appropriate value Step-V Calculating of the probability calculating the probability that sample results would diverge as widely as it has from expectations if Null Hypothesis were true. Step VI Comparing the Probability :- In this probability equal to or smaller than null value on a case of one-tailed test and /2 in case of two tailed test. Tests of Hypothesis: Two types: (a) Parametric Test or Standard tests of hypothesis (b) Non Parametric tests or distribution free test of hypothesis. (A)Parametric Test:- Parametric test usually assume certain properties of the parent population from which we draw Samples. Assumptions like observations come from a normal population, sample size is large, variant. Important Parametric test:- (a) Z-test:- A Univariate hypothesis test using the standardized normal distribution, which is the distribution of Z. (b) T-Test:- A Univariate hypothesis test using the t-distribution rather than the Z- distribution. It is used when the population standard deviation is unknown & the Sample size is small. (c) X-test:- Chi-square test that statically determines significance in the analysis of frequency distributions. (d) F-test:- A procedure used to determine if there is more variability in the scores of one sample than in the scores of another Sample. (B) Non-Parametric Test:- NPT do not require any assumption about the parameters or about the nature of population. By non-parametric tests we mean those Stastical tests which do not depend wither upon the shape of the distribution or upon the parameters of the population on an SD variances. NPT need more observations. Important Non-Parametric Test:- a) Sign Test:- is the simplest type of all non-parametric tests. It name comes from the fact that it is based on the direction of the pluses, minus sign of observation. On a Sample and not on their numerical magnitudes. - One sample sign test - Pervade Sample Sign Test b) Wilxon s Signed Rank Test:- This test is based on the ranking of the Sample of observations like sign test. This test two type: - One sample signed rank test - Paired Sample signed rank test. (C ) Main Whitney U-Test:- Chi test is applied with two independent Samples have come from the same population. (D)K-Sample Test/ Kruskal Wallis Test:- This test is applied to test whether two or three independent Samples have come from the same population as against the alternative hypothesis that they are from population with different means. (E) Wald-Walfowilz Fun Test:- Thus test is used to test the null hypothesis that the two populations from which the two-independent samples are drawn from identical distributions. (CP-205) 13/17

14 Ques 9 Explain Stastical estimation theory? Ans:- Stastical estimation is a procedure of estimating the unknown population parameters from the corresponding sample Stastics. Some Important Term:- 1. Estimators & estimates:- Generally for the purpose of estimating a population parameter, we can use various Sample Stastics. Those Sample Stastics which are used to estimate the unknown population promoters are called estimators & the actual value taken by the estimators are called estimates. 2. Point estimate & Interval Estimate:- (a) Point Estimate:- A single value of a Stastics that is used to estimate the unknown population parameter is called a point estimate. (b) Internal Estimate:- refers to the probable range with in which the real value of a parameter is expected to lie. The two extreme limits of such a range are called fiducially or confidence limits & the range is called a confidence level. A Good Estimator:- A good estimator is one, which is as close to the true value of the parameters possible. (i) Unbiased dress (ii) efficiency (iii) Consistency (iv) Sufficiency (I) POINT ESTIMATION: - A single value of a statistics that is used to estimate the unknown population parameter is called appoint estimation. For ex:- The Sample mean X which we use for estimating the population mean u is appoint estimator of u. Similarly, the Stastics S2 is a point estimator of θ2, where the value of S2 is computed from a random sample. Applications of Print estimation:- I. Point estimation ion case of single sampling II. Point estimation in case of repeated sampling I. Point estimation in case of single sampling:- When a single independent random Sample is drawn from a unknown population parameter can be illustrated by the following examples. A sample of 10 measured n = 10 Mean X = 4.38 S =.06 (i) True mean (Population mean) X = 4.38 (ii) True variance (Populative mean) S2 = n,s2 n-1 Putting the value, we get S2 = 10 X.06 = 1.11X0.06 = Thus u = 4.38, Ə III. Point estimation in case of Repeated Sampling:- When large number of random Samples, of same size are drawn from the population with or without replacement, then the point estimates of the population parameter can be illustrated by the following examples:- (CP-205) 14/17

15 Five Values = 3,4,5,6,7,10 The total number of possible Samples of size 3 without replacement are 5c3 = 10. Sample NO. 1 Sample Value II Sample Means (X) III 1 (3,4,5) 1/3 (3+4+5) = 12/3 = 4 2 (3,4,6) 1/3 (3+4+6) = 13/3 = (3,4,7) 1/3 (3+4+7) = 14/3 = (3,5,6) 1/3 (3+5+6) = 14/3 = (3,5,7) 1/3 (3+5+1) = 15/3 = (3,6,7) 1/3 (3+6+7) = 16/3 = (4,5,6) 1/3 (4+5+6) = 15/3 = (4,5,7) 1/3 (4+5+7) = 16/3 = (4,6,7) 1/3 (4+6+7) = 17/3 = (5.6.7) 1/3 (5+6+7) = 18/3 = 6.0 Total K = 9 x = 50 Mean of Sampling distribution of Means:- = ux = x = 50 = 5 k 10 Population Mean = u = = 5 5 Since ux = u, Sample mean x is an unbiased estimate pf the population u. II. Interval Estimatiion:-0 refers to the probable range written, which the real value of a parameter, is expected to lie. The two px time limits as such a range are called fiducial or confidence limits and in a range is called a confidence level. - Internal estimation for Large Sample (n> 30) - Interval estimation for small sample (n< 30) i. Interval estimation for Large Sample:- Z = t-(ct) N (0,1) or Z = t E (t) N (0,1) S.E. (+) S.E. (t) ii. Confidence limits for mean u or Z = X u N (0,1) Z = x -u N (0,1) /n /n iii. Confidence limit proportions Z = p - (cp0) or Z = P.E (p) S1 E CP) S.E (p) Iv Confidence limits for S.D> or S.D. = V(s) = 2 Var (s) = 2 2n 2n v Confidence limits for difference of Means S.e> X1 - X2 = N1 + N2 Ques 10. What is meant by probability? Explain conditional and Baye s theorem? Discuss in brief binomial, Normal and Poisson distributions. (CP-205) 15/17

16 Ans:- Probability:- it is a very important in Stastics or chance is very common in day to day life of human being. Probability(A)= Number of favorable cases =m Total no of equally likely cases =n Importance of Probability- 1. It is used in making economic decision in situation & uncertainty by sales manager4 & production manager etc. 2. It is used in theory of games which is further used in managerial decision. 3. Probability is the back bone of insurance company because life table are based on the theory of Probability. In probability conditional Theorem & Byes their is very important. Conditional Theorem:- If sub-event are not independent and nature of dependence is known, we have the theorem of conditional probabilities. This theorem is more or less corollary of the multiplication theorem. The theorem is that the probability that both of two dependent sub-events can occur is the product of the probability of the firm sub-event and the probability of the second after the first sub-event has occurred. In notation (A and B) = (A) x p(b/a) is the conditional probability of B when A has ready happened. For example, if out of a pack of cards shuffled or each time ability of King turning up again 4/52 X 451 = 12/2652, since there are 4 kings at the first shuffle of 52 cards and 3 kings only at the second shuffle of 51 cards. The term condition probabilities is often known as probabilities due to partial exhaustion of a sample space. Bays Theorem:- Probabilities can be ravished when new information pertaining to a random experiment is obtained. The notion of ravishing probabilities is a familiar one., for all of us, even to those with no previous experience in calculating probabilities have lived in an environment ruled by whims of chance and have made informal probability judgments. We do also intuitively revise these probabilities upon observing certain facts are change our actions accordingly. Our concern for revising probabilities arises from a need make better use of experimental information. This is referred to as Bays thermo after Reverend Thomas Bayes, who proposed in the eighteenth century, that probabilities be revised accordance with empirical findings. Quite often the businessman has the extra information on a particular event or propositions, either through a personal belief or from the past history of the event. Probabilities assigned on the base of personal experience., before observing the outcomes of the experiment are called prior proved abilities. For example:- probabilities assigned to past sales records, to past number of defective produced by a machine, are examples of prior probabilities. When the probabilities are revise with the use of Bayes; rule, they are called posterior probabilities. Suppose, a random experiment having several mutually exclusive events E1, E and the probabilities of each event p(e1), p(e2) have been obtained. These probabilities are referred to as prior probabilities, because they represent the chances that events before results from empirical investigation are obtained. The investigation itself may have several possible outcomes, each spastically dependent upon Es. For any particularly result, which we may designate by the letter R, the conditional probabilities P(R/E), P(R/e2) are often available. The result itself serves to revise the event probabilities upward or downward. The resulting values are called posterior since they apply after the information result has been learned. The posterior probability values are actually conditional probabilities of the form p(e1/r), P(E2/R) that may be found according to Bayes theorem Probability Distribution:- there are 3 types main frequency in probability distribution:- I. Binomial Distribution:- B.D. is associated with the name of James Bernoulli ( ). It is also known as Berne wall Distribution. Binomial means two names. Hence the frequency distribution falls into 2 categories a dichotomous process. A binomial distribution is a probability expressing the probability of one set of achromous alternatives e.g. success or failure (CP-205) 16/17

17 II. III. AIM COLLEGE-HISAR Contact: , Poisson Distribution:- PD developed by Simeon Danispoisson ( ). The Poisson distribution is based on the same assumptions as the binomial distribution. This means that in a Poisson experiment we deal with either success or failure that the success are independent of each other & thus probability of success throughout the entire process remain constant. Normal Distribution:- ND was first discovered by the english mathematician De Mayer ( ) Normal distribution of Sample means and many other statistics for large sample sizes is approximately normal, even though the original population may not be normal. Advantages of Distribution- 1. Easy to sort. 2. Mutually cases. 3. Accurate value. 4. Logical Conditions. 5. Assumptions.(Some time). 6. Trial are independence. 7. Stability. 8. Parameters. Disadvantages of Distribution- 1, Time consuming. 2. Costly. 3. Based on assumption. 4. Non Reliability. 5. Difficult to analyze. 6. Biasness. (CP-205) 17/17