sampling and estimation concepts

The blue bars represent the sampling distribution of the sample mean. All we’ve really done is wrap some basic mathematics around a few common sense intuitions. Or not? For example, all of these questions are things you can answer using probability theory: What are the chances of a fair coin coming up heads 10 times in a row? 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. For example, imagine if the sample mean was always smaller than the population mean. However, it also has undesirable characteristics. Researchers have nearly no authority to select the sample elements, and it’s purely done based on proximity and not representativeness. What is the mean of those numbers? From that perspective, probabilities don’t exist in the world, but rather in the thoughts and assumptions of people and other intelligent beings. This would show us a distribution of happiness scores from our sample. 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. For the most part, I’m a pragmatist so I’ll use any statistical method that I trust. Clearly, from my perspective, this is a pretty good bet. In this particular case \[P(E) = P(X_1) + P(X_2) + P(X_3)\] and, since the probabilities of blue, grey and black jeans respectively are .5, .3 and .1, the probability that I wear jeans is equal to .9. Probability Sampling is a sampling technique in which samples from a larger population are chosen using a method based on the theory of probability. You have a score of 97. We just need to be a little bit more creative, and a little bit more abstract to use the tools. The thermometer tells me it’s 23 degrees, but I know that’s not really true. What about the standard deviation? This is the histogram of the sample means: Figure 4.15: A histogram showing the sample means for 10,000 samples, each size 20, from the uniform distribution of numbers from 1 to 10. The goal in this chapter is to introduce the first of these big ideas, estimation theory, but we’ll talk about sampling theory first because estimation theory doesn’t make sense until you understand sampling. It’s not just that we suspect that the estimate is wrong: after all, with only two observations we expect it to be wrong to some degree. 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. So, we will be taking samples from Y. B &=& (x_3, x_4) We just looked at the results of one fictitious IQ experiment with a sample size of \(N=100\). Because of the following discussion, this is often all we can say. Here’s the command: In other words, there is a 76.9% chance that I will roll 4 or fewer skulls. a) Population. Figure 4.1: An illustration of how frequentist probability works. The red line is the distribution, the blue bars are the histogram for the sample means. So I asked my computer to simulate flipping a coin 1000 times, and then drew a picture of what happens to the proportion \(N_H / N\) as \(N\) increases. , because of the researcher’s ease of carrying it out and getting in touch with the subjects. Throughout my discussion of the normal distribution, there’s been one or two things that don’t quite make sense. On the other hand, they also operate in the realm of pure abstraction in the way that mathematicians do. In fact, it’s such an obvious point that when Jacob Bernoulli – one of the founders of probability theory – formalized this idea back in 1713, he was kind of a jerk about it. 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. \mbox{``jeans''} &=& (\mbox{``blue jeans''}, \mbox{``grey jeans''}, \mbox{``black jeans''}) \\ The new bits are the blue bars and the blue lines. The sample standard deviation is only based on two observations, and if you’re at all like me you probably have the intuition that, with only two observations, we haven’t given the population “enough of a chance” to reveal its true variability to us. and depending on which one you subscribe to, you might say that some of those statements are meaningless or irrelevant. Okay, so that explains part of the story. Before we start talking about probability theory, it’s helpful to spend a moment thinking about the relationship between probability and statistics. 2. B &=& (x_3, x_4) \\ What’s going on here is that R actually provides four functions in relation to the binomial distribution. However, in everyday language, if I told you that it was 23 degrees outside and it turned out to be 22.9998 degrees, you probably wouldn’t call me a liar. It’s pretty simple, and in the next section we’ll explain the statistical justification for this intuitive answer. Robust, automated and easy to use customer survey software & tool to create surveys, real-time data collection and robust analytics for valuable customer insights. One way that you can do this is to formalise it in terms of “rational gambling”, though there are many other ways. There are many flavours of Bayesianism, making hard to say exactly what “the” Bayesian view is. For instance, if \(P(X) = 0.5\) it means that I wear those pants half of the time. You know what a distribution is right? On one face of each die there’s a picture of a skull; the other five faces are all blank. It is also a time-convenient and a cost-effective method and hence forms the basis of any research design. \end{array}\], \[\begin{array}{rcl} If any of these elementary events occurs, then \(E\) is also said to have occurred. Nevertheless if forced to give a “best guess” I’d have to say \(98.5\). If \(A\) coresponds to the even that I wear jeans (i.e., one of \(x_1\) or \(x_2\) or \(x_3\) happens), then the only meaningful definitionof “not \(A\)” (which is mathematically denoted as \(\neg A\)) is to say that \(\neg A\) consists of all elementary events that don’t belong to \(A\). P(A \cap B) &=& P(x_3) Using the probability sampling method, the bias in the sample derived from a population is negligible to non-existent. Suppose the observation in question measures the cromulence of my shoes. Each histogram shows a new sample. So, on what basis is it legitimate for the polling company, the newspaper, and the readership to conclude that the ALP primary vote is only about 23%? Okay, now that we have a sample space (a wardrobe), which is built from lots of possible elementary events (pants), what we want to do is assign a probability of one of these elementary events. Suppose I have a sample that contains a single observation. Our only goal was to find ways of describing, summarizing and graphing that sample. The tricky thing with genuinely continuous quantities is that you never really know exactly what they are. What would the mean be? Figure 4.25 shows the sample standard deviation as a function of sample size. Does the measure of happiness depend on the wording in the question? We need some more powerful tools than just looking at the numbers and guessing. What shall we use as our estimate in this case? Or maybe X makes the variation in Y change. OK, now let’s take a bunch of samples from that distribution. 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). The standard deviation of a distribution is a parameter. What do I mean by that? So, we can confidently infer that something else (like an X) did cause the difference. Or the psychologist Paul Meehl, who suggests that relying on frequentist methods could turn you into “a potent but sterile intellectual rake who leaves in his merry path a long train of ravished maidens but no viable scientific offspring” Meehl (1967, 114). Get actionable insights with real-time and automated survey data collection and powerful analytics! You make X go up and take a big sample of Y then look at it. Such estimation can be performed against any reference (= estimation context), most commonly a combination of a) a geographical stratum, b) a reference period and c) a specific boat/gear category. Yes, fine and dandy. We can see that sometime we get some big numbers, say between 120 and 180, but not much bigger than that. We also know, now thanks to the central limit theorem, that many of our measures, such as sample means, will be distributed normally. However they’re not identical, and not every statistician would endorse all of them. This time around, the only thing we have are data. So, you could use the mean and standard deviation of your sample as an estimate, and then use those to calculate z-scores. The relationship between the two depends on the procedure by which the sample was selected. Learn more about 4.4: Concept of Sampling and Estimation on GlobalSpec. Now let’s assign probabilities to these events. In pretty much every other respect, there’s nothing else to add. 3.1 COMPLETE ENUMERATION (CENSUS) 3.2 CENSUS IN SPACE, SAMPLING IN TIME 3.3 CENSUS IN TIME, SAMPLING IN SPACE 3.4 SAMPLING IN SPACE AND IN TIME. As a first pass, you would want to know the mean and standard deviation of the population. The key characteristic of elementary events is that every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events. “On the Mathematical Foundation of Theoretical Statistics.” Philosophical Transactions of the Royal Society A 222: 309–68. The mean of each sample is not always 5.5 because of sampling error or chance. For instance, in the “polling company” example, the population consisted of all voters enrolled at the a time of the study – millions of people. It has mathematical formulations that describe relationships between random variables and parameters. Does a measure like this one tell us everything we want to know about happiness (probably not), what is it missing (who knows? Keynes, John Maynard. That’s a lot of \(x\)s to tell me the freaking obvious. Even though way more of the numbers should be smaller than bigger, then sampling distribution of the mean again does not look the red line. 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. And it’s definitely the pbinom function that is correct. Actually interested in our earlier discussion of the 20th century random outcomes from the fact that our first should! Tomorrow ’ s something we absolutely do = model + residuals 4 \cap B\ ) calculated here ’! Mean by \ ( P ( X ) \ ) ) pragmatist so I ’ ll win the follows. To learn it for the most popular sizes, you would be necessary to conduct an exhaustive survey movie. 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