Sample Survey Notes.ii.docx

PRINCIPLES OF SAMPLE SURVEY

Cont. ( II)

BASIC SYMBOLS: 1. SOME POPULATION VALUES i. N – The total number of elements in the population. ii. Yi –Value of the y variable for the ith population element. N

Y   y i - Population total for the y – variable.

iii.

i 1

Y 1 N   y i – Population mean for. y variable N N i1 N 2 1 S y2    y i  y - Population Variance for y variable N - 1 i1 N 2 1  y2    y i  y – Variance for the yi variables N i1 Sy or  y -standard deviation of the population elements as distinguish above. Y

iv.

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v.





vi. vii.

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2. SOME SAMPLE VALUES OR STATISTICS i.

yi – value of the yi variable for the ith sample element.

ii.

y   y i - Sample total.

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y 1 n y    y i - sample mean n n i 1

n

i 1

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In many but not all design of “epsem” probability sampling, y is used to estimate Y .

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Similarly N y is used to estimate N Y .

Ny  N NB.

 Since f 

N 1 y y n f

n is the sampling fraction or the finite population correction (f.p.c) N

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n 1 y y S   i n - 1 i 1 2 y

iv.

y n



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2 1 n 2 yi  n y    n  1  i 1 

1 n y2  y   i  n  i 1 n



-In simple random samples S y2 is an unbiased estimation of S Y2 1

THE PRINCIPLE STEPS IN A SAMPLE SURVEY -The principle steps in a sample survey are grouped under 11 headings i.e 1. Objective of The Survey The first step involves setting up a clear statement of the objective of the survey. Without it, it is easy to lose focus. 2. Identification of Population To Be Sampled - The word population is used to denote the aggregate from which the sample is to be chosen. - The population to be sampled must have the required information. 3. Identification of Data to Be Collected. - It is well to verify that all the data identified for collection are relevant to the purpose of the survey and that no essential data are omitted. - There is frequently a tendency especially with human population to ask too many questions, some of which are never subsequently analyzed. - An over- long questionnaire lowers the quality of the answers. 4. Decision On The Degree Of Precision Required - The results of sample surveys are always subject to some uncertainty because only part of the population has been measured and because of errors of measurements. - These uncertainties can be reduced by taking large samples and by using superior instruments of measurements. - Decision on the degree of precision required hence is an important step. 5. Choice of Methods of Measurements. - There is need, to make an informed choice on measuring instruments and method of approach to the population. - The approach may be by mail, telephone or personal visits or the combination of the three. - A major part of preliminary work at this stage is the construction of the record forms on which the questions and answers are to be entered. - With simple questionnaires the answers may be recorded (entered in a manner in which they can be routinely transferred to computer). - For instruction of good record forms, it is important to visualize the final summary tables that will be used for drawing conclusion. 6. Construction of the Sample Frame. - This involves the construction of list of all individual sampling units. The list of all sampling units in a population is called a sampling frame. - These units must cover the whole of the population and must not overlap i.e every element in the population belongs to one / appears only once. 7. Selection of a Sample. - There are varieties of plans by which a sample may be selected. For each plan that is considered, rough estimate of the size of sample can be made from knowledge on the degree of precision that is desired. 2

8. 9. -

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The relative cost and time required for each plan are always compared before making decision. The sample may be simple random, stratified, systematic, or cluster. The type of sample decided on, determines the method of selecting the sample. The Pretest It is necessary to try out the questionnaire and a field method on a chosen sample, just before the large scale sampling is carried out. Organization of the fieldwork. In extensive surveys many problems of business administration are made. The personnel must receive training on the purpose of survey and method of measurement to be employed and must be adequately supervised in their work. Plant must also be made for handling non-response. That is, in case there is a failure of the enumerator to obtain information from some of the units in the sample.

10. Collecting Data, Tabulation, Analysis and Presentation of Reports / Results - The first step here is to edit the completed questionnaires in the hope of amending recording errors, or at least deleting data that are obviously erroneous. - Decision about computing procedure are needed in cases where answers to certain questions were omitted by some respondents or were deleted in the editing process[missing data]. - There after computations which lead to estimates are performed. Different method s of estimation may be available for the same data. - In the presentation of results, it is good practice to report the amount of error to be expected in the evaluation of the most important estimates. 11. Data Archiving [Storage] - Information gained is then stored for future survey. Any complete survey is a potential guide to improved future sampling of the data and the presentation of results. It is a good practice to report the mean amount and the standard deviation to be expected on the cost. -It alerts the sampler on mistakes in execution of the survey and avoids making mistakes in future survey / repetition.

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SIMPLE RANDOM SAMPLING -

It is the technique of drawing a sample in such a way that each unit of the population has equal and independent chance of being included in the sample.

Random Sample - A random sample is one chosen by a method involving an unpredictable components - A sample is a subject chosen from a population for investigation. - To select an item at random from a population, one must ensure that each item in the population has an equal chance of being selected. - To obtain a random sample of size n items from a population, each selection must be obtained using any of the following methods. 1. Physical randomization devices such as coin, playing cards or sophisticated devices [Eletronic Random number indicated Equipment] used in lotteries] 2. Mathematical algorithms for Pseudoration number generator [PRNG] 3. Random number tables Simple Random Sampling Procedure: -

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Random sampling from a finite population, refers to that method of sample selection which gives each possible sample combination an equal probability of being picked up and each item in entire population to have an equal chance of being included in the sample. It refers to sampling without replacement. N This is a method of selecting n units out of the N such that every one of the NCn or   n  distinct samples has an equal chances of being drawn. Thus in simple random sampling from a population of N units. The probability of drawing any unit as the first drawn is 1/N. The probability of drawing any unit in the second draw from the available [N – 1] unit is 1/(N – 1) and so on. Suppose the sequential method yields n distinct population members whose x values are x1, x2, ---- xn where xi refers to the ith chosen member [i= 1, 2, ----n]. The probability of obtaining this ordered sequence is: Pr [x1,x2 ---nx] = N – n N!

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Proof: Probability of 1st draw = 1/N Probability of 2nd draw = 1/N-1 Probability of 3rd draw= 1/N-2 Generally Probability of the nth draw =1/N-n+1 Therefore, Pr(1st ,2nd , . . . .nth) = 1 1 1 1  . . . .. . . N N 1 n  2 N  n 1 1.(N  N)  N! (N  n)  N! Verify that all the NCn distinct possible samples [S1,S2,--------- Sn] have an equal chance of being n !(N  n)! selected as the study sample = N! Proof : Consider one distinct sample i.e one set of n specified units. n - At the 1st draw, the probability that some one of the n specified units is selected is N - The 2nd draw the probability that some one of the remaining n – 1 specified units if drawn is n -1 N -1 - The 3rd draw the probability that some one of the remaining n – 2 specified units is draw is n-2 and so on. N-2 Hence the probability that all n specified units are selected in n draws to make any Si is n !(N  n)! n (n  1) (n  2) 1 . . ............  N (N  1) (N  2) N  n 1 N! 1  N Cn  N    n 

=

1

n !(N  n)! N!

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Example Consider a case when N=6 and a sample of size n=4 is desired. The probability of any possible unit being in sample is. Probability of 1st draw = 1/6 Probability of 2nd draw = 1/5 Probability of 3rd draw= 1/4 Probability of 3rd draw= 1/3 Therefore Pr (4draws) = 1/6 x1/5 x1/4 x1/3 =(1/360) 1!.(6  4)1  6! (N  n) i.e. N! - And the probability of any of the distinct sample being chosen is. 1 n !(N  n)! (4) !(2)!   N Cn N! 6! 

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(4x3x2x1)x (2x1) 6x5x4x3x2x 1

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1 15

There are 15 possible samples.

Example 2. Suppose N=6 and a sample of size n=3 is desired. How many possible samples would a sampler get and what are the chances that any one of the units in the population is selected into sample? Solutions a. Distinct samples = N C n 

N! 6!  n !N  n ! 3!(6  3)!  20

20 possible samples b. P[xi into the sample in 3 draws

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( N  n)! (6  3)! 1   N! 6! 120

Definition of Notations - In a sample survey we decide on certain properties that we attempt to measure and record for every unit that comes into the sample. - The properties of the units are referred to as characteristics or items - The values attained for any specific items in the N units that comprise the population are denoted by Y1,Y2 ------------ Yn [ characteristics of population] - The corresponding values for the units in the sample are denoted by y1, y2 --------- yn or yi for [i= 1, 2, ------n] 6

NB: The capital letters refers to characteristics of the population and lower case letters refers to characteristics of sample. Statistic and parameters - A statistic is a characteristic of sample where as a parameter is a characteristic of population. - Common characteristics; the mean, median and totals of a sample are the statistics of the samples. When such measurements describe the characteristic of the population they are known as parameter. e.g Population mean X is a parameter whereas the mean x is a statistic. - A sample statistic is usually used to estimate a corresponding population parameter and this is the prime objective of sample analysis. Properties of The Estimators - The estimates of a population parameters may be one single value or it could be a range of values. - In the former case, it is referred to as a point estimate and the latter is an interval estimate. - A researcher usually makes two types of estimates through sampling analysis. - It is important to give only the best point estimates or else speak in terms of intervals and probabilities for it is never possible to estimate with certainty the exact values of population parameters. - A researcher must hence know the properties of good estimators for his study. - A good estimator possess the following properties 1. Property of unbiasedness - An estimator should on average be equal to the value of the parameter being estimated. This is the property of unbiasedness. - An estimation is said to be unbiased if the expected value of the estimate is equal to the parameter being estimated. e.g . The sample mean x is the most widely used estimation for the population mean (u) because of the fact that it provides an unbiased estimate of the population mean. 2. Property of efficiency - An estimation should have a relatively small variance i.e most efficient estimators among a group of unbiased estimator is one which has the smallest variance. 3. Property of sufficiency -

An estimator should use as much as possible the information available from the sample

4. Property of consistency. -

An estimator should approach the value of the population parameter as the sample size becomes larger and larger.

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