sampling and estimation concepts

So, one might ask whether it really makes sense to pretend that an “infinite” sequence of coin flips is even a meaningful concept, or an objective one. That is all. Why? Firstly, in order to construct the rules I’m going to need a sample space $$X$$ that consists of a bunch of elementary events $$x$$, and two non-elementary events, which I’ll call $$A$$ and $$B$$. We’ve seen it already, but it’s worth looking at it one more time. The idea is quite simple. Statistical questions tend to look more like these: If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? Select Chapter 22 - Estimating … This kind of transformation just changes the scale of the numbers from between 0-1, and between 0-100. Each sample is taken from the normal distribution shown in red. So there’s a sense in which the 75th percentile should lie “in between” 3 and 4 skulls. Well, as Figure 4.4a shows, the main effect of this is to shift the whole distribution, as you’d expect. While the binomial distribution is conceptually the simplest distribution to understand, it’s not the most important one. This would show us a distribution of happiness scores from our sample. These values are now very close to the true population. As far as I can tell there’s nothing mathematically incorrect about the way frequentists think about sequences of events, and there’s nothing mathematically incorrect about the way that Bayesians define the beliefs of a rational agent. There’s more to the story, there always is. \end{array}\] Excellent. To help make sure you understand the importance of the sampling procedure, consider an alternative way in which the experiment could have been run. The notation that we sometimes use to say that a variable $$X$$ is normally distributed is as follows: $X \sim \mbox{Normal}(\mu,\sigma)$ Of course, that’s just notation. Everyone wins! Okay, what if we flipped a coin $$N=100$$ times? After all, the “population” is just too weird and abstract and useless and contentious. And how do we learn them? A sample statistic is a description of your data, whereas the estimate is a guess about the population. In this blog, we discuss the various probability and non-probability sampling methods that you can implement in any. The z-score for 50 is -2, because 50 is two 25s away from 100 in the opposite direction. When viewed from that perspective, I’d argue that we don’t need the sample to be randomly generated in every respect: we only need it to be random with respect to the psychologically-relevant phenomenon of interest. Why not? For this reason, statisticians really like it when a data set can be considered a simple random sample, because it makes the data analysis much easier. The more correct answer is that a 95% chance that a normally-distributed quantity will fall within 1.96 standard deviations of the true mean. Sampling & Sample Size Estimation Moazzam Ali MD, PhD, MPH Department of Reproductive Health and Research World Health Organization Geneva, Switzerland Presented at: GFMER September 16, 2014 . If all members of a population were identical, the population is considered to be homogenous. Finally, they are super useful when you are dealing with a normal distribution that has a known mean and standard deviation. $$z = \frac{\text{raw score} - \text{mean}}{\text{standard deviation}}$$, So, for example if we had these 10 scores from a normal distribution with mean = 100, and standard deviation =25. The section breakdown looks like this: Basic ideas about samples, sampling and populations. This makes it very simple for a survey creator to derive effective inference from the feedback. The mean of the sample is fairly close to the population mean 100 but not identical. This work helps maintain and develop the sampling and weighting methods used to derive the Office’s statistical outputs. That’s almost the right thing to do, but not quite. Probability sampling eliminates bias in the population and gives all members a fair chance to be included in the sample. In any case, now that we have all this terminology and notation, we can use it to state the problem a little more precisely. We will set our sample-size to 20. In other words, how people behave and answer questions when they are given a questionnaire. Let’s assume you’ve relied on a convenience sample, and as such you can assume it’s biased. Social networks are complex things, and just because you can use them to get data doesn’t always mean you should. For example, a researcher intends to collect a systematic sample of 500 people in a population of 5000. Sampling Distribution 5. So what is the true mean IQ for the entire population of Brooklyn? The relationship between the two depends on the procedure by which the sample was selected. In many scientific studies that level of precision is perfectly acceptable, but in other situations you need to be a lot more precise. It makes assumptions about the random variables, and sometimes parameters. The true population standard deviation is 15 (dashed line), but as you can see from the histogram, the vast majority of experiments will produce a much smaller sample standard deviation than this. However, that’s not answering the question that we’re actually interested in. Feel free to think of the “population” in different ways. The population can be defined in terms of geographical location, age, income, and many other characteristics. I don’t intend to subject you to a proof that the law of large numbers is true, but it’s one of the most important tools for statistical theory. Sampling Theory| Chapter 4 | Stratified Sampling | Shalabh, IIT Kanpur Page 1 Chapter 4 Stratified Sampling An important objective in any estimation problem is to obtain an estimator of a population parameter which can take care of the salient features of the population. Even a moment’s inspections makes clear that the larger sample is a much better approximation to the true population distribution than the smaller one. The name for this is a confidence interval for the mean. The x-axis corresponds to the value of some variable, and the y-axis tells us something about how likely we are to observe that value. All of these are good reasons to care about estimating population parameters. These samples are generally non-random in two respects: firstly, reliance on undergraduate psychology students automatically means that your data are restricted to a single sub-population. When applied to the sample mean, what the law of large numbers states is that as the sample gets larger, the sample mean tends to get closer to the true population mean. In fact, the vast majority of the content in this book relies on one of five distributions: the binomial distribution, the normal distribution, the $$t$$ distribution, the $$\chi^2$$ (“chi-square”) distribution and the $$F$$ distribution. is a method where the researchers divide the entire population into sections or clusters that represent a population. This type of sampling is entirely biased and hence the results are biased too, rendering the research speculative. The numbers that we measure come from somewhere, we have called this place “distributions”. Having decided to write down the definition of the $$E$$ this way, it’s pretty straightforward to state what the probability $$P(E)$$ is: we just add everything up. For example, startups and NGOs usually conduct convenience sampling at a mall to distribute leaflets of upcoming events or promotion of a cause – they do that by standing at the mall entrance and giving out pamphlets randomly. Here’s what I’d get (I did literally flip coins to produce this! Also, when N is large, it doesn’t matter too much. Employee survey software & tool to create, send and analyze employee surveys. For example, if you don’t think that what you are doing is estimating a population parameter, then why would you divide by N-1? Topics to be covered History of sampling Why sampling Sampling concepts and terminologies Types of sampling and factors affecting choice of sampling design Advantages of sampling . For example, if the United States government wishes to evaluate the number of immigrants living in the Mainland US, they can divide it into clusters based on states such as California, Texas, Florida, Massachusetts, Colorado, Hawaii, etc. This method is dependent on the ease of access to subjects such as surveying customers at a mall or passers-by on a busy street. No matter what distribution you’re talking about, there’s a d function, a p function, r a function and a q function. This way of conducting a survey will be more effective as the results will be organized into states and provide insightful immigration data. CONCEPT A sampling process that takes into consideration the chance of occurrence of each item being selected. This sampling method considers every member of the population and forms samples based on a fixed process. As you can see just by looking at the movie, as the sample size $$N$$ increases, the SEM decreases. Perhaps, but it’s not very concrete. Some people are very bi-modal, they are very happy and very unhappy, depending on time of day. It’s a branch of mathematics that tells you how often different kinds of events will happen. Intuitively, you already know part of the answer: if you only have a few observations, the sample mean is likely to be quite inaccurate (you’ve already seen it bounce around): if you replicate a small experiment and recalculate the mean you’ll get a very different answer. . For instance, “test taking” style might have taught the Australian participants how to direct their attention exclusively on fairly abstract test materials relative to people that haven’t grown up in a similar environment; leading to a misleading picture of what working memory capacity is. Parameter estimation is one of these tools. If I roll two six sided dice, how likely is it that I’ll roll two sixes? But if you’ve ever had that experience in real life, you might walk away from the conversation feeling like you didn’t quite get it right, and that (like many everyday concepts) it turns out that you don’t really know what it’s all about. Suppose that I believe that there’s a 60% probability of rain tomorrow. Picking up on that last point, there’s a sense in which this whole chapter is something of a digression. Probability sampling eliminates bias in the population and gives all members a fair chance to be included in the sample. 1. An elementary event $$x$$ has occurred. For the 2010 Federal election, the Australian Electoral Commission reported 4,610,795 enrolled voters in New South Whales; so the opinions of the remaining 4,609,795 voters (about 99.98% of voters) remain unknown to us. Well, in that case, we get Figure 4.4b. X &=& (x_1, x_2, x_3, x_4, x_5) \\ Now consider the evidentiary value of seeing 4 black chips and 0 white chips. Here are some steps expert researchers follow to decide the best sampling method. However, they start to make sense if you understand that there is this Bayesian/frequentist distinction. Maybe you noticed that I used $$p(X)$$ instead of $$P(X)$$ when giving the formula for the normal distribution. We’ll start with pbinom, and we’ll go back to the skull-dice example. This time around, my experiment involves flipping a fair coin repeatedly, and the outcome that I’m interested in is the number of heads that I observe. There’s only one city of Adelaide, and only 2 November 2048. This shared experience might easily translate into similar beliefs about how to “take a test”, a shared assumption about how psychological experimentation works, and so on. The samples are all very different from each other, but the red line doesn’t move around very much, it always stays near the middle. The sample would consist of a collection of numbers like this: Each of these IQ scores is sampled from a normal distribution with mean 100 and standard deviation 15. Sampling - Concepts and Definitions. Select the method that works best for the research. For example, the sample mean goes from about 90 to 110, whereas the standard deviation goes from 15 to 25. In contrast, the purpose of inferential statistics is to “learn what we do not know from what we do”. If I proceed to roll all 20 dice, what’s the probability that I’ll get exactly 4 skulls? But it’s not clear how to define this in frequentist terms. B &=& (x_3, x_4) \\ What is X? Let’s give a go at being abstract. Or, to say it a little bit more precisely, as the sample size “approaches” infinity (written as $$N \rightarrow \infty$$) the sample mean approaches the population mean ($$\bar{X} \rightarrow \mu$$). First question. This may well be the last time we talk about z-scores in this book. . Using the probability sampling method, the bias in the sample derived from a population is negligible to non-existent. What might we observe? Let’s look at this thing. We can simulate the results of this experiment using R, using the rnorm() function, which generates random numbers sampled from a normal distribution. As you might imagine, probability distributions vary enormously, and there’s an enormous range of distributions out there. Even sadder, I’ve given them names: I call them $$X_1$$, $$X_2$$, $$X_3$$, $$X_4$$ and $$X_5$$. We have already done lots of sampling, so you are already familiar with some of the big ideas. Unfortunately, I don’t have an infinite number of coins, or the infinite patience required to flip a coin an infinite number of times. There are real populations out there, and sometimes you want to know the parameters of them. For most applied researchers you won’t need much more theory than this. The logic of sampling gives you a way to test conclusions about such groups using only a small portion of its members. What do I mean by that? If you want to turn percentages back into proportions, you divide by a constant of 100. A &=& (x_1, x_2, x_3) \\ However, what we can talk about is the probability that the value lies within a particular range of values. When we find that two samples are different, we need to find out if the size of the difference is consistent with what sampling error can produce, or if the difference is bigger than that. Since the sampling method is arbitrary, the population demographics representation is almost always skewed. "It helped me a lot in clearing my concepts on estimation of a sample size, thanks a lot!" Sometimes the population is obvious. How happy are you in the afternoons on a scale from 1 to 7? The unit of analysis may be a person, group, organization, country, object, or any other entity that you wish to draw scientific inferences about. If we find any big changes that can’t be explained by sampling error, then we can conclude that something about X caused a change in Y! Let’s now define two new events that correspond to important everyday concepts: $$A$$ and $$B$$, and $$A$$ or $$B$$. This non-probability sampling method is used when there are time and cost limitations in collecting feedback. It is not enough to know that we will eventually arrive at the right answer when calculating the sample mean. For example, we see the proportion is .341 for scores that fall between the range 0 and 1. There are four types of probability sampling techniques: There are multiple uses of probability sampling: The non-probability method is a sampling method that involves a collection of feedback based on a researcher or statistician’s sample selection capabilities and not on a fixed selection process. The first problem is figuring out how to measure happiness. Consequently, every single elementary event belongs to either $$A$$ or $$\neg A$$, but not both. To encapsulate the whole discussion, though, the significant differences between probability sampling methods and non-probability sampling methods are as below: Creating a survey with QuestionPro is optimized for use on larger screens -. Does that matter? Here’s how he described the fact that we all share this intuition: For even the most stupid of men, by some instinct of nature, by himself and without any instruction (which is a remarkable thing), is convinced that the more observations have been made, the less danger there is of wandering from one’s goal (see Stigler, 1986, p65). For example, take a look at this normal distribution, it has a mean =100, and standard deviation =25. Figure 4.8: A normal distribution with a shifting sd. This specific kind of of stratified sampling is referred to as oversampling because it makes a deliberate attempt to over-represent rare groups. Keynes (1923, 80). Recall, a statistical inference aims at learning characteristics of the population from a sample; the population characteristics are parameters and sample characteristics are statistics. So, the probability of rolling 4 skulls out of 20 times is about 0.20 (the actual answer is 0.2022036, as we’ll see in a moment). Hold up now. &=& P(A \cup B) “Theory Testing in Psychology and Physics: A Methodological Paradox.” Philosophy of Science 34: 103–15. We refer to this range as a 95% confidence interval, denoted $$\mbox{CI}_{95}$$. To finish this section off, here’s another couple of tables to help keep things clear: Statistics means never having to say you’re certain – Unknown origin. My goal, as a cognitive scientist, is to try to learn something about how the mind works. These distributions are telling you what to expect from your sample. \end{array}\], \[\begin{array}{rcl} The labels show the proportions of scores that fall between each bar. 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