The Problem Of Planning - Essay Example

Order # 497111 Introduction Researches based on statistical analysis are reliable only when they are thoughtfully planned. The planning must be started from the very beginning of sample selection. The problem should be carefully defined and operationalized so that the researcher would be able to decide the methodology of the study- whether the study is based on sample or Census? If it is based on sample, which type of sampling., ie., random or purposive? Experimental observations must be selected from the appropriate population. That is, the study should be randomized correctly. The researcher may be able to obtain approximate results about the behaviour of the distribution of an estimator as the sample size becomes large. Random Sampling In statistical terms, random sample means the set of items that are drawn from a large set of items (population) and it is the subset of population. The method of sample satisfies the criterion of randomness, that is, each item has equal chance to be drawn or selected. The only factor an item to be selected is mere chance. But in order to ensure equal chance to all items, once an item is selected, it should be replaced from the population. Stratified Random Sample If the population consists of heterogeneous sub groups or different strata, it would be advisable to sample each subpopulation (stratum) separately. Stratification is the process of grouping members into relatively homogenous groups. The random sampling is applied within each stratum. This method of random sampling from different strata improves the representative nature of the sample by reducing the sampling error. Sample size Sample size is the number of observations that constitute the sample and it is normally represented by an integer (positive number). The sample size is determined by a number of factors like convenience, time, money and the purpose of the study. But in many cases, the decision of sample size becomes confusing. Relative to the objectives of the study, sample distribution must be of adequate size. It should be large enough that the effects of the magnitude will be statistically significant. The economic reason of the large sample size is that a small sized study may be a waste of resources for not having the capability to produce the useful results. Though the published literature on this issue is not very much rich in comparison to its importance, there are articles and books which are concerning the sample size determination. Some of them are Kraemer and Thiemann (1987), Cohen (1988), Lipsey (1990), Shuster (1990), and Odeh and Fox (1991). Theories advocating for large sample size Determining Sample Size is such an important issue that the reliability of the results mainly depends on this. The decision of how large would be a sample is so important that it enables the statistical judgments would be accurate and reliable. For this, the sample size should not be too small or too large. If the sample size is too small, the analysis will lack the precision to provide the reliable conclusions for the investigation. If sample size is too large, there would be wastage of time and resource. Law of Large Numbers (LLN) It is a theorem that describes the result of repeating an experiment a number of times. As per this law, large number of trials would result the average value very close to the expected value. For example, a single roll of six-sided die results one of the numbers 1,2,3,4,5, and 6 with equal probability. Therefore, the expected value of a single die roll is 3.5. According to the Law of Large Numbers (LLN), if a large number of dice are rolled, the sample mean of the values (the average of the values) is likely to be very close to 3.5, with the accuracy increasing as more dice are rolled. Another example of LLN is the flipping of coin. The expected value of the number of heads is equal to half when a coin is flipped once. According to the LLN, the proportion of heads in a large number of flips may be roughly one half. That is, the proportion of heads after n flips will converge to exact one half as n approaches infinity. LLN is important because it guarantees long-run stability in the random results. The theoretical foundations for the Law of Large Numbers were laid mainly by Geralamo Cardano, Jacob Bernoulli and S.D Poisson (Mlodinow, 2008). Limiting Theorem It is well known that the distribution of the mean of a sample tends to approximate normality as the sample size grows, regardless of the distribution of the individual observations. Knowledge about the limiting behaviour of the distribution of an estimator can be used to infer an approximate distribution for the estimator in a finite sample. In order to describe how this is done , it is necessary, first to present some results on convergence of random variables. This theory of Large Sample Distribution Theory (White 1984) thus explains three aspects of convergence 1) Convergence in probability : Convergence in the probability implies that the values that the variable may take that are not close to c (consatn) become increasingly unlikely. Limiting arguments will be with respect to the sample size, n. Let xn be a randome variable indexed by the size of a sample. 2) Convergence in Distribution- Limiting Distributions A second form of convergence is convergence in distribution. Xn are the sequence of random variables indexed by the sample sample size. The probability distribution is associated with Xn; it does not imply that Xn converges to all. 3) Asymptotic distributions: an asymptotic distribution is a distribution that is used to approximate the true finite sample distribution of a random sample. Central Limit Theorem According to this theorem, sampling distribution of the mean approaches to the normal distribution when the sample size increases. The amazing thing about the Central Limit Theorem (CLT) is that no matter what the shape of original distribution, sampling distribution approaches to the normal distribution with large sample size. That is, the distribution of an average tends to be normal even though the distribution from which the average is taken is non-normal. Under CLT the conditions are stated for the mean of large number of independent random variables will be normally distributed (Rice 1995). The Central Limit Theorem requires that the random numbers should be identically distributed. It provides the partial explanation for the prevalence of the normal probability distribution and justifies the large sample statistics. Power Analysis and Sample Size Calculation Though the above discussed theorems point out the need for large sample size, decision should be made on how large a sample is needed to enable statistical judgements that are accurate and reliable. Hence there is a growing importance to the Power Analysis and sample size calculation. Software programmes are available for power analysis where a number of graphical and analytical tools enables precise evaluation of the factors affecting power and sample size in many of the most commonly statistical analyses. There is a growing amount of software for sample-size determination, including nQuery Advisor (Elashoff, 2000), PASS (Hintze, 2000), UnifyPow (O’Brien, 1998), and Powerand Precision (Borenstein et al., 1997) and Cohen (1998). References 1. Mlodinow, L. (2008). The Drunkard ’s Walk, New York: Random House 2. White, H (1984). The Assimtotic Theory for Econometricians, New York: Academic Press. 3. Stark, P B. The Law of Large Numbers, http://stat-www.berkeley.edu/~stark/Java/lln.htm 4. Cochran, W.G. (1977). Sampling Techniques (3rd Ed.), New York: John Wiley & Sons. 5. Donald, M.N (1967): Implications of Non Response for the Interpretation of Mail Questionnaire data, Public Opinion Quarterly, 24(1), 99-114. 6. Fink, A (1995). The Survey Handbook, Thousand Oaks, CA: Sage Publications 7. Hintze, J. (2000). Pass 2000, Number Cruncher Statistical Systems, Kaysville, UT, Software for MS-DOS systems 8. Elashoff, J (2000). nQuery Advisor Release 4.0, Statistical Solutions, Cork, Ireland, Software for MS-DOS systems 9. Borenstein, M; Rothstein, H and Cohen J (1997). Power and Precision, Biostat, Teaneck, NJ, Software for MS-DOS systems. 10. Cohen, J (1988). Statistical Power Analysis for the Behavioural Sciences, Academic Press, NewYork 11. Le, Cam Lucien (1986). The Central Limit Theorems Around 1935, Statistical Science, 1:1, 78-91. 12. Behrens, J T (1997). 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