What Are the Contributions of Sampling Distribution Models - Assignment Example

Sampling Distribution Models: Research Analysis Student’s Name Institution Table of Contents Executive Summary Samples are used in statistics to prepare distribution models like histogram and then extrapolating or inferring the results into the entire set of population. For a distribution to be close to normal representation, then a larger sample size is deemed necessary. The sample must be selected from unbiased population. This means that the items of representation must be drawn randomly from a population whose all elements had an equal chance of being selected. Sampling distribution is defined as the distributional probability of a given statistics on the basis of a random sample. In statistics, these distributions are very critical because they offer a great simplification on the way to inferences or extrapolation from the distribution models. More critically, they provide for analytical considerations to be undertaken on the basis of sampling distributions of a given statistics instead of sample values of individual’s joint probability distribution. Introduction This report looks into the statistics topic of sampling distribution models. It provides for an extensive analysis on the topic and draws on the models that summarize the sampled data. This report has defined key terms in the topic and has evaluated critical areas of the topic and statistics like data collection process, inferences and has been able to evaluate the study with an approach that enhances mathematical reasoning. Sampling Distribution Models: Research Analysis Sampling distribution in statistics is also referred to as finite sample distribution. Sampling distribution is defined as the distributional probability of a given statistics on the basis of a random sample. Sampling distribution is defined as the distributional probability of a given statistics on the basis of a random sample. In statistics, these distributions are very critical because they offer a great simplification on the way to inferences or extrapolation from the distribution models. More critically, they provide for analytical considerations to be undertaken on the basis of sampling distributions of a given statistics instead of sample values of individual’s joint probability distribution (Quick, 2010). Data Collection Process Data collection is carried out early enough to ensure that the relevant information has been captured and is formalized more often through a collection plan of data that has the following activities. The pre collection activities, this will involve deciding on the goals to be achieved, the targeted data, methods of data collection and finally on the definitions of the terms used in the process (Quick, 2010). The second activity is the actual collection of data and lastly data presentation. Data presentation will involve elements of sorting the information and analyzing it. It should be noted that the activities prior to the actual collection of data are very critical. It normally comes into the knowledge of the individual collecting data, however too late that they have information whose value is discounted as a result of poor sampling from both the informants, the questions that have been used in the collection of such data and the techniques used (Kenkel, 1996). Then, the actual data collection is carried out in a more structured and systematic ways after the completion of the pre-collection activities. A process of data collection that is formal is very key in ensuring the kind of data that has been gathered is accurate and defined and that any decisions that will be arrived at basing on this data will be valid. The data collection process will involve a baseline from where a measurement can be done and in other cases considering a target on what can be improved (Kenkel, 1996). Other major collection of data methods will include sample survey, census and a by-product of administration, each having its unique advantages and disadvantages. Census means data collection concerning everything or any one on a statistical population. This is credited with advantages such as accuracy and in its detailed nature. It also has disadvantages like time consumption and cost impact. By-product administrative data are collected as an organization’s by product in the day to day operations and is credited with advantages like simplicity, time saving and accuracy and disadvantages like lack of control and less flexibility. Introduction to Sampling Distribution A statistic’s sampling distribution is that statistic’s distribution that is considered to as a random variable. That is when it is taken from a sample in random of size n. it can as well be considered as a statistical distribution for all the possible samples that have been taken from the same population all of the same size. The distribution of a sample will hugely depend on the populations underlying distribution, the considered statistics, and the procedure of sampling that is being employed and the size of the sample. This mostly is an interest that is considered in determining whether the sampled distribution is possible to be approximated by asymptotic distribution that corresponds to the case in limiting. That is n tending to infinity (n approaching ∞) (Kenkel, 1996). For instance, taking a normal population into consideration with a variance σ² and a mean μ. then assume that a given sample size of a population is taken repeatedly. Then an arithmetic mean is calculated for each sample. Note that this statistics is referred to as the sample mean, an average value is also there, for each value and it is the distribution of this averages that is referred to as the sampling distribution for the determined sample mean. Given that the underlying distribution is normal, the distribution is also normal. This normal distribution can be expressed as n (μ. σ² / n). It should however be noted that distributions of sampling may be close to normal when even the distribution for the population is not. The sample median is an alternative to the mean. The sample mean from a population that has a normal distribution represents a statistic that has been taken from the simplest statistical populations. For other populations and statistics, the formulas used are more complicated and in most of the time they do not exist in open form. In cases where they fail to exist in the closed form, then sampling distributions are approximated through bootstrap methods, Monte Carlo simulations and the asymptotic distributions (Kenkel, 1996). Preliminary Organization and Presentation of Data In statistics, data can take the form or grouped or ungrouped, it can represent as: Pie Charts, frequency polygons, frequency histograms, box plots and ogives. The following recommendations should be followed when presenting statistical data: The method of presentation should be made simple; this is achieved by avoiding lot information. This requires that only a summary of those features considered essential are included. The presentation need to be self explanatory. This means that the user of the presented need to understand it without having to refer to any other source. The title of the presentation should concise and clear and should indicate when, what and where data was obtained. The labels, codes and legends need to be clear and the standard format needs to be paid attention to And finally the foot notes are required in order to create room for correct interpretations of the data. Standard Error Standard error is the statistic’s sampling distribution standard deviation. In a situation where the samples mean is the statistics, and the samples collected have no correlation, then the standard error is given as Б x = б / n 0.5 Whereby, б is the population distribution’s standard deviation of the given quantity. N is the number of times or the sample size. This formula has a very important implication in that the size of the sample requires to be quadrupled. That is multiplied by 4 in order to get the half of the error of measurement. In the process of designing statistical studies that has cost as a factor this will have a major role to enable an understanding of the tradeoffs between costs and benefits Examples; Population Statistic Sampling Distribution Normal population N {μ, σ ²) From the sample of size n, the sample mean is x 1 X1 ={ μ ( σ ² /n 0.5)} In the case of two normal independent populations N {μ 1 σ 1²} and N { μ 2 σ 2²} Sample mean difference between the two X 1 – x 2 X 1 – x 2 = n { μ 1 - μ 2, σ1² / n 1 + σ2² / n 2 } Bernoulli (p ) Successful trials in the sample proportion. N X 1= binomial (n, p) Distribution f that has f density found in any absolute continuous distribution From a sample with size of n = -1 + 2 g, with an ordered sample X 1 to x n, the median is X g F x (g) (x) = [(2 g – 1)! / ( g – 1 )!2] f ( x ){ f ( x ) (1 – f ( x ) ) } g - 1 Any distribution that has an F distribution function Maximum That is, L = max X g from a size n random sample. F l ( x ) = p ( l < x ) = π p ( x g < x ) = ( f ( x )) n Statistical Inference In the statistical inference theory, the sufficient statistics idea offers a basis of selecting a statistics as a sample data points in that there is no information that gets lost in the process of replacing the sample’s full probabilistic description with the selected statistic’s sampling distribution. In the frequents inference, for instance in the confidence interval or statistical hypothesis test development, the availability of a statistic’s sampling distribution or by using an asymptotic distribution to give a very close estimation, such procedures can easily be formulated however, the development of those procedures that start from a joint distribution of a given sample will be less straight forward. When a statistic’s sampling distribution is available in the case of a Bayesian distribution, the final outcome of the procedure carried out can easily be replaced. This is especially more possible in the conditional distributions of an unknown quantity given a data sample by the conditional distribution from any quantity that is not known given certain selected sample of statistics. In this case, a statistic’s sampling distribution would be involved in such procedures. If the chosen statistics are sufficient jointly, then the outcome of the results would be identical. Kinds of Sample It should be noted that a typical population is very large,. This makes it very difficult to enumerate the entire population. Then a sample that represents a small size that is manageable is chosen. Then from the collected samples, the calculations of the statistics are done and then this allows for inferences and extrapolations to be done by generalizing the sample results into the entire population. A sample that is complete contains all the population objects found in the parent population. For example for a complete sample of all men of country x less short than 2m will have to contain a full list of all the men in that country who are short than the 2m set. This means that for this data to be compiled, then a full list that indicates all the male members of country x must be prepared clearly indicating the gender and the height of each individual. It should be noted that such a complete set of data for human beings would be impossible. However for other disciplines, the list will be available. A representative or unbiased sample is defined as a set of objects that are drawn from a sample that is complete by use of a process of selection which is not dependent on the characteristics of the objects. For instance, an example of unbiased sample of men shorter than 2m in country x can be taken by only studying a subset randomly sampled of 1.5% of the country x male who are shorter than 2m. It should be noted that if an electrical register was used, it would have not given unbiased data as those ale in country x who have not attained the age of majority to be registered would have not been represented. This means that the source of the data should be chosen carefully to avoid giving data that is biased. By selecting a random sample, the possibility of selecting unrepresentative or biased sample is eliminated. The random sample is also called the probability sample. In a random sample, all the items of a population are said to have an equal chance of being elected to represent the entire population. Common types of random samples include the simple random samples, stratified random samples and the cluster random samples. A sample that is not selected randomly is called non probability sample or non random sample. Some of the examples of non probability sampling are the purposive samples, quota samples, judgment samples and snow ball samples. Mathematical Reasoning and Description of Random Sample In the mathematical terms, if there is a given random x variable that has an f distribution and an n random length of a sample. Then there is a concrete representation of the experiments n in which the quantity measured is the same, to illustrate this, assume that the an individual’s height is represented by x and the number of individuals measured is n, x 1 is the last individual’s height. It should be noted that there should be no confusion between the random variable samples and the realizations resulting from these variables. This means that the function x 1 is the representative measure of the last experiment and x 1 = x 1(w) is the obtained value during the measurement making process (Selvanathan, Keller & Warrack, 2007). The sampling distribution therefore contains the data collection process. This is very critical in order for the mathematical statements to be made about the statistics and the sample used to compute it. Examples include the covariance and the simple mean. Models In the past poll a random sample of individuals from country x gave a report that 90% believe that the country’s system of health care is better than that of other countries. The question asked is on how much that this statistics can be trusted. In fact, the poll was only conducted on only 100 individuals, this left out millions of other citizens of that country. There is possibility that 96 % of the citizens of the country have the same feeling or 90% or even as little as 65 %. All the above percentages are possibilities. What degree of reliability that can be placed on these random samples? This will call for an extrapolation to the entire population (Selvanathan, Keller & Warrack, 2007). Example of a Military Mission of County X in Country Y After a high rate of fatalities of the military soldiers in their mission, a random poll of 1009 citizens of country X issued a report indicating that 53% of the citizens favored an early withdrawal of soldiers from country Y. around the same time, another poll conducted by a strategic counsel poll that examined 1000 citizens in country x indicated that 65 % of the citizens supported an early withdrawal. The questions that lingers in the minds of many individuals is on why the variability. It is possible to observe variations in various reports but to what variability should the individuals expect to observe. Sample Proportion’s Central Limit Theorem In this case, from the troops in country y, it is assumed that 60 % of all the country’s adult population supported an early withdrawal. Then for this example, let p = 0.60. For the shape of the histogram, it cannot just be imagined but instead it must be simulated. In this case, a bunch of samples taken randomly is simulated that have been actually drawn. For this to be effected, then random samples of size 1000 that are many and independent are simulated keeping in mind the same success probability. P has been assigned the value of 0.60. Below is a histogram of proportions sample that supports an early withdrawal of 1000 adults for 2000 samples that are independent when the proportion that is true is actually p = 0.60. Number Of Samples 200 150 100 50 0.05 0.56 0.54 0.61 0.62 0.63 0.66 Simulated P’s It should be noted that the results for each sample does not give the same results even if the true values underlying is for the population is the same. Each value of the p comes from a different sample after being simulated (Selvanathan, Selvanathan, Keller & Warrack, 2007). It should be noted that the above histogram is a simulation of what would have been found if all the proportions would have been seen from all the samples. There is a special name that is given to that kind distribution; it is called the proportion’s sampling distribution. It should be noted just as was discussed that the histogram is symmetric and unimodal. It has its centre at p. this is a normal model (Quick, 2010) A model in sampling distribution on how a proportion of a sample varies from one sample to another allows individual users to be able to quantify the extent of variation and discuss on how likely of the possibility for them having observed a proportions sample in any particular interval. In order to make use of a normal model, then two parameters need to be specified, its standard deviation and the mean, p is the natural centre of the histogram, thus the mean μ of the normal will be put at p. note that it is the mean that gives information on the standard deviation. Given the proportions, then the standard deviation can be determined. σ P = s d (p) = [ p (1-p) /n ] 0.5 = p s / n 0.5 (Quick, 2010) Benefits of the Model Models are supposed to be a close approximation to the true representation. It should be noted that distributions continuous to be true with an increase in the size of the sample. One or two sample size might not work very well but proportion’s distributions of a number of samples that are large have close to normal histograms. Assumptions and Conditions; In order to make use of any model, then assumptions must be made, in order to have sampling distribution model useful, then two key assumptions must be made. The first one is independence assumption; this assumes that the values sampled must be independent draws that are random from the population under study (Quick, 2010) The second is the sample size assumption. The size of the sample given as n must be large enough to be a good population representative. Recommendations and Conclusion Sampling is key and necessary tool in data collection, analysis and extrapolation. From the target population, an individual is able to determine how many respondents to be examined. Sampling makes it possible for research studies to be carried out and the results generalized to the entire population. Sampling is practical and economical faster and cheaper it yields more information. It is also has the advantage of accuracy. It is faster hence saves on time and costs associated with doing the research for many days. The sampling distributions represent and summarize data such that at a glance an individual is at a position seeing the trends, it is easier and simple to read and interpret like graphs. Because of the above numerous positive factors, sample distributions have been used and continues to be an important tool in businesses and other institutions to summarize complex data into simpler, easy to understand and interpret and extrapolate. References Black, Asafu-Adjaye, Khan, P, Edwards, H, (2007), Australasian Business Statistics, John Wiley Kenkel, J. L. (1996), Introductory Statistics for Management and Economics, 4th Edition, Duxbury Keller, G. (2009), Statistics for Management and Economics, 8th Edition, South- Western Engage Learning. Quick, J (2010), Statistical Analysis with R, Packt Publishing Selvanathan, E.A., Selvanathan, S., Keller, A., & Warrack, B. (2007). Australian Business Statistics, Abridged (4th Ed.).Australia: Nelson Thomson Learning Selvanathan, S., & Selvanathan, E.A. (2007), Learning Statistics and Excel in Tandem (2nd Ed.). Melbourne: Nelson ITP Read More