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So, for instance, a t test on log-transformed data is really a test for location of the geometric mean. Further Reading Langley R.Practical Statistics Simply Explained1970, Dover Press. What Is Central Limit Theorem. One of the simplest versions of the theorem says that if is a random sample of size n say, n 30 from an infinite population finite standard deviationthen the standardized sample mean converges to a standard normal distribution or, equivalently, the sample mean approaches a normal distribution with mean equal to the population mean and standard deviation equal to standard deviation of the population divided by square root of sample size n.

In applications of the central limit theorem to practical problems in statistical inference, however, statisticians are more interested in how closely the approximate distribution of the sample mean follows a normal distribution for finite sample sizes, than the limiting distribution itself. Sufficiently close agreement with a normal distribution allows statisticians to use normal theory for making inferences about population parameters such as the mean using the sample mean, irrespective of the actual form of the parent population.

It is well known that whatever the parent population is, the standardized variable will have a distribution with a mean 0 and standard deviation 1 under random sampling. Moreover, if the parent population is normal, then is distributed exactly as a standard normal variable for any positive integer n. The central limit theorem states the remarkable result that, even when the parent population is non-normal, the standardized variable is approximately normal if the sample size is large enough say, 30.

It is generally not possible to state conditions under which the approximation given by the central limit theorem works and what sample sizes are needed before the approximation becomes good enough. As a general guideline, statisticians have used the prescription that if the parent distribution is symmetric and relatively short-tailed, then the sample mean reaches approximate normality for smaller samples than if the parent population is skewed or long-tailed.

On e must study the behavior of the mean of samples of different sizes drawn from a variety of parent populations. Examining sampling distributions of sample means computed from samples of different sizes drawn from a variety of distributions, allow us to gain some insight into the behavior of the sample mean under those specific conditions as well as examine the validity of the guidelines mentioned above for using the central limit theorem in practice.

The sample size needed for the approximation to be adequate depends strongly on the shape of the parent distribution. Symmetry or lack thereof is particularly important. Under certain conditions, in large samples, the sampling distribution of the sample mean can be approximated by a normal distribution. For a symmetric parent distribution, even if very different from the shape of a normal distribution, an adequate approximation can be obtained with small samples e.10 or 12 for the uniform distribution.

For symmetric short-tailed parent distributions, the sample mean reaches approximate normality for smaller samples than if the parent population is skewed and long-tailed. binomial with samples sizes far exceeding the typical guidelines say, 30 are needed for an adequate approximation. For some distributions without first and second moments e.Cauchythe central limit theorem does not hold.

What is a Sampling Distribution. We will study the behavior of the mean of sample values from a different specified populations. Because a sample examines only part of a population, the sample mean will not exactly equal the corresponding mean iq option é fraude the population. Thus, an important consideration for those planning and interpreting sampling results, is the degree to which sample estimates, such as the sample mean, will agree with the corresponding population characteristic.

In practice, only one sample is usually iq option é fraude in some cases a small pilot sample is used to test the data-gathering mechanisms and to get preliminary information for planning the main sampling scheme. However, for purposes of understanding the degree to which sample means will agree with the corresponding population mean, it is useful to consider what would happen if 10, or 50, or 100 separate sampling studies, of the same type, were conducted. How consistent would the results be across these different studies.

If we could see that the results from each iq option é fraude the samples would be nearly the same and nearly correct. In some extreme cases e.then we would have confidence in the single sample that will actually be used. On the other hand, seeing that answers from the repeated samples were too variable for the needed accuracy would suggest that a different sampling plan perhaps with a larger sample size should be used. A sampling distribution is used to describe the distribution of outcomes that one would observe from replication of a particular sampling plan.

Know that to estimate means to esteem to give value to. Know that estimates computed from one sample will be different from estimates that would be computed from another sample. Understand that estimates are expected to differ from the population characteristics parameters that we are trying to estimate, but that the properties of sampling distributions allow us to quantify, probabilistically, how they will differ. Understand that different statistics have different sampling distributions with distribution shape depending on a the specific statistic, b the sample size, and c the parent distribution.

Understand the relationship between sample size and the distribution of sample estimates. Understand that the variability in a sampling distribution can be reduced by increasing the sample size. Outlier Removal. Robust statistical techniques are needed to cope with any undetected outliers; otherwise the result will be misleading. Note that in large samples, many sampling distributions can be approximated with a normal distribution.

Because of the potentially large variance, outliers could be the outcome of sampling. It s perfectly correct to have such an observation that legitimately belongs to the study group by definition. For example, the usual stepwise regression is often used for the selection of an appropriate subset of explanatory variables to use in model; however, it could be invalidated even by the presence of a few outliers. Lognormally distributed data such as international exchange ratefor instance, will frequently exhibit such values.

Therefore, you must be very careful and cautious before declaring an observation an outlier, find out why and how such observation occurred. It could even be an error at the data entering stage. First, construct the BoxPlot of your data. Form the Q1, Q2, and Q3 points which divide the samples into four equally sized groups. Q2 median Let IQR Q3 - Q1. Outliers are defined as those points outside the values Q3 k IQR and Q1-k IQR.

For most case one sets k 1. Another alternative is the following algorithm. a Compute s of iq option é fraude sample. b Define a set of limits off the mean mean k smean - k s sigma Allow user to enter k. A typical value for k is 2. c Remove all sample values outside the limits. Now, iterate N times through the algorithm, each time replacing the sample set with the reduced samples after applying step c. Usually we need to iterate through this algorithm 4 times.

As mentioned earlier, a common standard is any observation falling beyond 1. 5 interquartile range i. 5 IQRs ranges above the third quartile or below the first quartile. The following SPSS program, helps you in determining the outliers. Outlier detection in the single population setting has been treated in detail in the literature. Quite often, however, one can argue that the detected outliers are not really outliers, but form a second population.

If this is the case, a cluster approach needs to be taken. Further Readings Hawkins D. It will be active areas of research to study the problem of how outliers can arise and be identified, when a cluster approach must be taken.Identification of OutliersChapman Hall, 1980. Rothamsted V. Barnett, and T. Lewis, Outliers in Statistical DataWiley, 1994. Least Squares Models. Realize that fitting the best line by eye is difficult, especially when there is a lot of residual variability in the data.

Know that there is a simple connection between the numerical coefficients in the regression equation and the slope and intercept of regression line. Know that a single summary statistic like a correlation coefficient or does not tell the whole story. A scatter plot is an essential complement to examining the relationship between the two variables. Know that the model checking is an essential part of the process of statistical modelling. After all, conclusions based on models that do not properly describe an observed set of data will be invalid.

Know the impact of violation of regression model assumptions i.conditions and possible solutions by analyzing the residuals. Least Median of Squares Models. What Is Sufficiency. A sufficient statistic t for a parameter q is a function of the sample data x1. xn, which contains all information in the sample about the parameter q. More formally, sufficiency is defined in terms of the likelihood function for q. For a sufficient statistic t, the Likelihood L x1. xn q can be written as.

Since the second term does not depend on qt is said to be a sufficient statistic for q. Another way of stating this for the usual problems is that one could construct a random process starting from the sufficient statistic, which will have exactly the same distribution as the full sample for all states of nature. To illustrate, let the observations be independent Bernoulli trials with the same probability of success.

Suppose that there are n trials, and that person A observes which observations are successes, and person B only finds out the number of successes. Then if B places these successes at random points without replication, the probability that B will now get any given set of successes is exactly the same as the probability that A will see that set, no matter what the true probability of success happens to be.

You Must Look at Your Scattergrams. All three sets have the same correlation and regression line. The important moral is look at your scattergrams. How to produce a numerical example where the two scatterplots show clearly different relationships strengths but yield the same covariance. Produce two sets of X, Y values that have different correlation s; 2. Perform the following steps.

Calculate the two covariances, say C1 and C2; 3. Suppose you want to make C2 equal to C1. Then you want to multiply C2 by C1 C2 ; 4. S yyou want two numbers one of them might be 1a and b such that a. Multiply all values of X in set 2 by a, and all values of Y by b for the new variables, C r. An interesting numerical example showing two identical scatterplots but with differing covariance is the following Consider a data set of X, Y values, with covariance C1.

Now let V 2X, and W 3Y. The covariance of V and W will be 2 3 6 times C1, but the correlation between V and W is the same as the correlation between X and Y. Power of a Test. Power of a test is the probability of correctly rejecting a false null hypothesis. This probability is one minus the probability of making a Type II error b.

Recall also that we choose the probability of making a Type I error when we set a and that if we decrease the probability of making a Type I error we increase the probability of making a Type II error. Power and Alpha. Power and the True Difference between Population Means Anytime we test whether a sample differs from a population or whether two sample come from 2 separate populations, there is the assumption that each of the populations we are comparing has it s own mean and standard deviation even if we do not know it.

The distance between the two population means will affect the power of our test. Power as a Function of Sample Size and Variance You should notice that what really made the difference in the size of b is how much overlap there is in the two distributions. When the means are close together the two distributions overlap a great deal compared to when the means are farther apart.

Thus, anything that effects the extent the two distributions share common values will increase b the likelihood of making a Type II error. Sample size has an indirect effect on power because it affects the measure of variance we use to calculate the t-test statistic. Thus, sample size is of interest because it modifies our estimate of the standard deviation. Since we are calculating the power of a test that involves the comparison of sample means, we will be more interested in the standard error the average difference in sample values than standard deviation or variance by itself.

When n is large we will have a lower standard error than when n is small. In turn, when N is large well have a smaller b region than when n is small. Pilot Studies When the needed estimates for sample size calculation is not available from existing database, a pilot study is needed for adequate estimation with a given precision. Further Readings Cohen J.Statistical Power Analysis for the Behavioral SciencesL. Erlbaum Associates, 1988. Thiemann, How Many Subjects. Provides basic sample size tablesexplanations, and power analysis.

Myors, Statistical Power AnalysisL. Erlbaum Associates, 1998. Provides a simple and general sample size determination for hypothesis tests. ANOVA Analysis of Variance. Thus, when the variability that we predict between the two groups is much greater than the variability we don t predict within each group then we will conclude that our treatments produce different results.

Levene s Test Suppose that the sample data does not support the homogeneity of variance assumption, however, there is a good reason that the variations in the population are almost the same, then in such a situation you may like to use the Levene s modified test In each group first compute the absolute deviation of the individual values from the median in that group.

Apply the usual one way ANOVA on the set of deviation values and then interpret the results. The Procedure for Two Populations Independent Means Test Click on the image to enlarge it and THEN print it. You may use the following JavaScript to Test of Hypothesis for Two Populations. The Procedure for Two Dependent Means Test Click on the image to enlarge it and THEN print it.

You may use the following JavaScript to Two Dependent Populations Testing. The Procedure for More Than Two Independent Means Test Click on the image to enlarge it and THEN print it. The Procedure for More Than Two Dependent Populations Test Click on the image to enlarge it and THEN print it. You may use the following JavaScript to Three Dependent Means Comparison.

Orthogonal Contrasts of Means in ANOVA. Further Readings Kachigan S.Multivariate Statistical Analysis A Conceptual IntroductionRadius Press, 1991. The Six-Sigma Quality. Sigma is a Greek symbol, which is used in statistics to represent standard deviation of a population.Statistical Analysis An Interdisciplinary Introduction to Univariate Multivariate MethodsRadius Press, 1986. When a large enough random sample data are close to their mean i.the averagethen the population has a small deviation.

If the data varies significantly from the mean, the data has a large deviation. In quality control measurement terms, you want to see that the sample is as close as possible to the mean and that the mean meets or exceed specifications. A large sigma means that there is a large amount of variation within the data.

A lower sigma value corresponds to a small variation, and therefore a controlled process with a good quality. The Six-Sigma means a measure of quality that strives for near perfection. Six-Sigma is a data-driven approach and methodology for eliminating defects to achieve six sigmas between lower and upper specification limits. 4 defects per million opportunities. Accordingly, to achieve Six-Sigma, e. Therefore, a Six-Sigma defect is defined for not meeting the customer s specifications.

A Six-Sigma opportunity is then the total quantity of chances for a defect. One sigma means only 68 of products are acceptable; three sigma means 99. 7 are acceptable. Six-Sigma is 99. Six-Sigma is a statistical measure expressing how close a product comes to its quality goal. 4 defects per million parts or opportunities. 9997 perfect or 3. The natural spread is 6 times the sample standard deviation.

The natural spread is centered on the sample mean, and all weights in the sample fall within the natural spread, meaning the process will produce relatively few out-of-specification products. Six-Sigma does not necessarily imply 3 defective units per million made; it also signifies 3 defects per million opportunities when used to describe a process. Some products may have tens of thousands of opportunities for defects per finished item, so the proportion of defective opportunities may actually be quite large.

Six-Sigma Quality is a fundamental approach to delivering very high levels of customer satisfaction through disciplined use of data and statistical analysis for maximizing and sustaining business success. What that means is that all business decisions are made based on statistical analysis, not instinct or past history. Using the Six-Sigma approach will result in a significant, quantifiable improvement.in a manufacturing process it must not produce more than 3.

6 sigma defect-free good enough. Here are some examples of what life would be like if 99. 9 were good enough 1 hour of unsafe drinking water every month 2 long or short landings at every American cities airport each day 400 letters per hour which never arrive at their destination 3,000 newborns accidentally falling from the hands of nurses or doctors each year 4,000 incorrect drug prescriptions per year 22,000 checks deducted from the wrong bank account each hour As you can see, sometimes 99.

9 good just isn t good enough. Is it truly necessary to go for zero defects. Here are some examples of what life would be still like at Six-Sigma, 99. 9997 defect-free 13 wrong drug prescriptions per year 10 newborns accidentally falling from the hands of nurses or doctors each year 1 lost article of mail per hour. Now we see why the quest for Six-Sigma quality is necessary. Six-Sigma is the application of statistical methods to business processes to improve operating efficiencies.

It provides companies with a series of interventions and statistical tools that can lead to breakthrough profitability and quantum gains in quality. Six-Sigma allows us to take a real world problem with many potential answers, and translate it to a math problem, which will have only one answer. We then convert that one mathematical solution back to a real world solution. Six-Sigma goes beyond defect reduction to emphasize business process improvement in general, which includes total cost reduction, cycle-time improvement, increased customer satisfaction, and any other metric important to the customer and the company.

An objective of Six-Sigma is to eliminate any waste in the organization s processes by creating a road map for changing data into knowledge, reducing the amount of stress companies experience when they are overwhelmed with day-to-day activities and proactively uncovering opportunities that impact the customer and the company itself. The key to the Six-Sigma process is in eliminating defects.

Organizations often waste time creating metrics that are not appropriate for the outputs being measured. Executives can get deceptive results if they force all projects to determine a one size fits all metric in order to compare the quality of products and services from various departments. From a managerial standpoint, having one universal tool seems beneficial; however, it is not always feasible.

In the airline industry, the US Air Traffic Control System Command Center measures companies on their rate of on time departure. Below is an example of the deceptiveness of metrics. This would obviously be a critical measurement to customers the flying public. Whenever an airplane departs 15 minutes or more later than scheduled, that event is considered as a defect. Unfortunately, the government measures the airlines on whether the plane pulls away from the airport gate within 15 minutes of scheduled departure, not when it actually takes off.

Airlines know this, so they pull away from the gate on time but let the plane sit on the runway as long as necessary before take off. The result to the customer is still a late departure. This defect metric is therefore not an accurate representation of the desires of the customers who are impacted by the process. If this were a good descriptive metric, airlines would be measured by the actual delay experienced by passengers. This example shows the importance of having the right metrics for each process.

The method above creates no incentive to reduce actual delays, so the customer and ultimately the industry still suffers. With a Six-Sigma business strategy, we want to see a picture that describes the true output of a process over time, along with additional metrics, to give an insight as to where the management has to focus its improvement efforts for the customer. The Six Steps of Six-Sigma Loop Process The process is identified by the following five major activities for each project Identify the product or service you provide What do you do.

Identify your customer base, and determine what they care about Who uses your products and services. What is really important to them. Identify your needs What do you need to do your work. Define the process for doing your work How do you do your work. Ensure continuous improvement by measuring, analyzing, and controlling the improved process How perfectly are you doing your customer-focused work.

Often each step can create dozens of individual improvement projects and can last for several months. It is important to go back to each step from time to time in order to determine actual data maybe with improved measurement systems. Eliminate wasted efforts How can you do your work better. Once we know the answers to the above questions, we can begin to improve the process.

The following case study will further explain the steps applied in Six-Sigma to Measure, Analyze, Improve, and Control a process to ensure customer satisfaction. The Six Sigma General Process and Its Implementation The Six-Sigma means a measure of quality that strives for near perfection. Six-Sigma is a data-driven approach and methodology for eliminating defects to achieve six-sigma s between lower and upper specification limits.

The implementation of the Six Sigma system starts normally with a few days workshop of the top level management of the organization. Only if the advantages of Six Sigma can be clearly stated and supported of the entire Management, then it makes sense to determine together the first project surrounding field and the pilot project team. The pilot project team members participate is a few days Six Sigma workshop to learn the system principals, the process, the tools and the methodology.

The project team meets to compiles main decisions and identifying key stakeholders in the pilot surrounding field. Within the next days the requirements of the stakeholders are collected for the main decision processes by face-to-face interviews. By now, the workshop of the top management must be ready for the next iq option é fraude. The next step for the project team is to decide which and how the achievements should be measured and then begin with the data collection and analysis.

Whenever the results are understood well then suggestions for improvement will be collected, analyzed, and prioritized based on the urgency and inter-dependencies. As the main outcome, the project team members will determine which improvements should be realized first. The activities must be carried out in parallel whenever possible by a network activity chart.

The activity chart will become more and more realistic by a loop-process while spread the improvement throughout the organization. In this phase it is important that rapid successes are obtained, in order to even the soil for other Six Sigma projects in the organization. The main objective of the Six-Sigma approach is the implementation of a measurement-based strategy that focuses on process improvement.

The aim is variation reduction, which can be accomplished by Six-Sigma methodology. More and more processes will be included and employees are trained including Black Belts who are the six sigma masters, and the dependency of external advisors will be reduced. The Six-Sigma is a business strategy aimed at the near-elimination of defects from every manufacturing, service and transactional process. The concept of Six-Sigma was introduced and popularized for reducing defect rate of manufactured electronic boards.

Although the original goal of Six-Sigma was to focus on manufacturing process, today the marketing, purchasing, customer order, financial and health care processing functions also embarked on Six Sigma programs. Motorola Inc. Case Motorola is a role model for modern manufacturers. There is a reason for this reputation. The maker of wireless communications products, semiconductors, and electronic equipment enjoys a stellar reputation for high-tech, high-quality products.

A participative-management process emphasizing employee involvement is a key factor in Motorola s quality push. In 1987, Motorola invested 44 million in employee training and education in a new quality program called Six-Sigma. Motorola measures its internal quality based on the number of defects in its products and processes. Motorola conceptualized Six-Sigma as a quality goal in the mid-1980. Their target was Six-Sigma quality, or 99.

9997 defect free products which is equivalent to 3. 4 defects or less per 1 million parts. Quality is a competitive advantage because Motorola s reputation opens markets. When Motorola Inc. won the Malcolm Baldridge National Quality Award in 1988; it was in the early stages of a plan that, by 1992, would achieve Six-Sigma Quality. It is estimated that of 9.

Shortly thereafter, many US firms were following Motorola s lead. Control Charts, and the CUSUM. 2 billion in 1989 sales, 480 million was saved as a result of Motorola s Six-Sigma program. Developing quality control charts for variables X-Chart The following steps are required for developing quality control charts for variables. Decide what should be measured. Determine the sample size. Collect random sample and record the measurements counts. Calculate the average for each sample.

Calculate the overall average. This is the average of all the sample averages X-double bar. Calculate the average range R-bar. Determine the upper control limit UCL and lower control limit LCL for the average and for the range. Determine the range for each sample. Plot the chart. Determine if the average and range values are in statistical control.

Take necessary action based on your interpretation of the charts. Developing control charts for attributes P-Chart Control charts for attributes are called P-charts. The following steps are required to set up P-charts. Determine what should be measured. Collect sample data and record the data. Determine the required sample size. Calculate the average percent defective for the process p. Determine the control limits by determining the upper control limit UCL and the lower control limit LCL values for the chart.

Plot the data. Determine if the percent defectives are within control. Control charts are also used in industry to monitor processes that are far from Zero-Defect. However, among the powerful techniques is the counting of the cumulative conforming items between two nonconforming and its combined techniques based on cumulative sum and exponentially weighted moving average smoothing methods. The general CUSUM is a statistical process control when measurements are multivariate. It is an effective tool in detecting a shift in the mean vector of the measurements, which is based on the cross-sectional antiranks of the measurements At each time point, the measurements, after being appropriately transformed, are ordered and their antiranks are recorded.

When the process is in-control under some mild regularity conditions the antirank vector at each time point has a given distribution, which changes to some other distribution when the process is out-of-control and the components of the mean vector of the process are not all the same. This latter shift, however, can be easily detected by a univariate CUSUM.

Further Readings Breyfogle F. Therefore it detects shifts in all directions except the one that the components of the mean vector are all the same but not zero.Implementing Six Sigma Smarter Solutions Using Statistical Methods, Wiley, 1999. del Castillo E.Statistical Process and Adjustment Methods for Quality ControlWiley, 2002. Juran J, and A. Godfreym, Juran s Quality Handbook, McGraw-Hill, 1999. Kuralmani, Statistical Models and Control Charts for High Quality ProcessesKluwer, 2002.

Repeatability and Reproducibility.Concepts for R R StudiesASQ Quality Press, 1991. Lyday, Evaluating the Measurement ProcessStatistical Process Control Press, 1990. Further Readings Barrentine L. Statistical Instrument, Grab Sampling, and Passive Sampling Techniques. What is a statistical instrument. A statistical instrument is any process that aim at describing a phenomena by using any instrument or device, however the results may be used as a control tool.

Examples of statistical instruments are questionnaire and surveys sampling. What is grab sampling technique. The grab sampling technique is to take a relatively small sample over a very short period of time, the result obtained are usually instantaneous. However, the Passive Sampling is a technique where a sampling device is used for an extended time under similar conditions. Depending on the desirable statistical investigation, the Passive Sampling may be a useful alternative or even more appropriate than grab sampling.

However, a passive sampling technique needs to be developed and tested in the field. line transect sampling, in which the distances sampled are distances of detected objects usually animals from the line along which the observer travels. Distance Sampling. point transect sampling, in which the distances sampled are distances of detected objects usually birds from the point at which the observer stands.

cue counting, in which iq option é fraude distances sampled are distances from a moving observer to each detected cue given by the objects of interest usually whales. trapping webs, in which the distances sampled are from the web center to trapped objects usually invertebrates or small terrestrial vertebrates. migration counts, in which the distances sampled are actually times of detection during the migration of objects usually whales past a watch point.

Many mark-recapture models have been developed over the past 40 years. Monitoring of biological populations is receiving increasing emphasis in many countries. Data from marked populations can be used for the estimation of survival probabilities, how these vary by age, sex and time, and how they correlate with external variables. Estimation of the finite rate of population change and fitness are still more difficult to address in a rigorous manner. Estimation of immigration and emigration rates, population size and the proportion of age classes that enter the breeding population are often important and difficult to estimate with precision for free-ranging populations.

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