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58

I will describe how a statistician interprets count data. With a tiny bit of practice you can do it, too. The basic analysis When cases arise randomly and independently, the times of their occurrences are reasonably accurately modeled with a Poisson process. This implies that the number of cases appearing in any predetermined interval has a Poisson ...

27

This is referred to as current status data. You get one cross sectional view of the data, and regarding the response, all you know is that at the observed age of each subject, the event (in your case: transitioning from A to B) has happened or not. This is a special case of interval censoring. To formally define it, let $T_i$ be the (unobserved) true event ...

18

Mathematically, it's not the case that these are necessarily close. It would work if it was the case that $E(Y/X^2) = E(Y)/E(X)^2$ but this is false in general and in some particular situations it might be quite far out. However, for a fairly realistic set of bivariate height and weight data it looks like the impact will be small. For example, consider ...

17

The concept of significance or hypothesis testing is not relevant for a whole population. Hypothesis testing is based on the assumption that you deal with a sample from a (usually) infinite population, and asks the question: what is the probability that we have drawn the sample by chance from a population that fulfills the assumptions of the null hypothesis? ...

15

This question goes to the heart of what statistics is and how to to conduct a good statistical analysis. It raises many issues, some of terminology and others of theory. To clarify them, let's begin by noting the implicit context of the question and go on from there to define the key terms "parameter," "property," and "estimator.&...

14

Why does a 95% CI not imply a 95% chance of containing the mean? There are many issues to be clarified in this question and in the majority of the given responses. I shall confine myself only to two of them. a. What is a population mean? Does exist a true population mean? The concept of population mean is model-dependent. As all models are wrong, but some ...

14

There is no way to do this by non-parametric paradigm, just think of it: the sampled distribution is a completely legit one, there is nothing preventing a single-population distribution from having two separate high density zones. But if you turn to parametric models, you may assume that your sub-populations are gaussian, and gaussian distribution has only ...

11

As with many questions on definitions, answers need to have an eye both on the underlying principles and on the ways terms are used in practice, which can often be at least a little loose or inconsistent, even by individuals who are well informed, and more importantly, variable from community to community. One common principle is that a statistic is a ...

10

I need to correct a number of mistaken or partly misplaced ideas in the question first (as well as some that aren't in the question but are commonly seen and may be indirectly influencing the way you ask your question), but I will return to the main issue. The answer you link to says: The t-test assumes that the means of the different samples are ...

10

It's not completely correct, but it will usually not make a huge difference. For example, suppose your population has weights 80, 90 and 100kg, and is 1.7, 1.8 and 1.9m tall. Then the BMIs are 27.68, 27.78 and 27.70. The mean of the BMIs is 27.72. If you calculate the BMI from the means of the weights and heights, you get 27.78, which is slightly different, ...

10

Let's start with terminology. Population in statistics is the "set of entities under study". When designing the study, we define the population of interest and then draw samples from this population. So sample cannot "consist" of multiple populations. More appropriate wording would be to talk about "groups", "clusters", or "subpopulations". To find clusters ...

9

Generally, when one has only a fraction of the population, i.e. a sample, you should divide by n-1. There is a good reason to do so, we know that the sample variance, which multiplies the mean squared deviation from the sample mean by (n−1)/n, is an unbiased estimator of the population variance. You can find a proof that the estimator of the sample variance ...

9

The first question is one that has no generally agreed upon answer. My own view is like yours, but others have argued that a population can be viewed as a sample from a "super-population" where the exact nature of a super-population varies depending on context: E.g. a census of all the people living in a building could be viewed as a sample from all the ...

9

The multivariate delta method has a heuristic justification here: https://en.wikipedia.org/wiki/Delta_method#Multivariate_delta_method. For the multivariate delta method you have a specific function $f$ that takes a vector argument which is $p$ dimensional and maps this to a $k$ dimensional space. In the case of a ratio estimator $p=2$ and $k=1$. The ...

9

The random variable $Y$ describes a relationship between events and the corresponding probabilities of those events. In more practical terms, a random variable describes a data-generating process. When you generate a random data point that is described by the random variable $Y$, the probability distribution of $Y$ describes the probability distribution of ...

9

In statistical terms, you are wondering whether your data comes from a mixture of two (or more) populations, as against coming from a single population. Looking at the mixture or more specifically the gaussian-mixture tags will be helpful. Number of components for Gaussian mixture model? includes a very good approach for deciding between one or two ...

8

I don't know whether this should be asked as a new question but it is addressing the very same question asked above by proposing a thought experiment. Firstly, I'm going to assume that if I select a playing card at random from a standard deck, the probability that I've selected a club (without looking at it) is 13 / 52 = 25%. And secondly, it's been stated ...

8

I think you are misreading the paper, they do not claim what you say. Their claims are not based on number of top players, but on their ratings. If the statistical distribution of strength is the same among men and women, then the expected number of women among the top 100 is 6, if their proportion of the total population is 6%. Some citations from the ...

8

The word "population" does not refer to all living people in the world. That is the generic understanding of the word, and not the statistical understanding. Statistically, population refers to the class/group of units (or individuals in this case) about whom you want to make some inference. If the group you want to make the inference on is "women in Math ...

7

Suppose only 2 out of 12 committee members are women. The proportion $\frac{1}{6}$ can be taken as a statistic descriptive of the whole population (the committee). Perhaps something ought to be done to correct the imbalance, regardless of how it arose. Or it can be taken as an estimate of the probability of a woman's being selected for the committee—...

7

You are very close... If $X_1, \dots, X_n$ is a sample of i.i.d normal observations with mean $\mu$ and variance $\sigma^2$, then the standardized mean $$\frac{\bar X_n-\mu}{\sigma/\sqrt{n}}$$ is standard normal. Now, as you pointed out, in reality we never know $\sigma$. So we replace $\sigma$ by its sample estimate $S$ and consider the "studentized" ...

6

I tend to think of parameters by analogy by thinking about the normal distribution: $$\text{pdf}=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{1}{2}\frac{(x_i-\mu)^2}{\sigma^2}}$$ What's important to recognize about this function is that, as ugly as it is, I pretty much know what most of the parts are. For example, I know what the numbers $1$ and $2$ are, what $\... 6 Base SPSS isn't very deft at handling sampling weights. See this website for some details on weighting (including caveats on SPSS): http://www.ats.ucla.edu/stat/spss/faq/weights.htm Weight cases in SPSS treats each line as representative of a certain number of observed samples. (e.g. if you assign a weight of 100 to a particular line (case), then that line ... 6 By selecting data within a certain range, it no longer has the original distribution. You have changed it so it is no longer exponential (It actually becomes a doubly-truncated exponential; equivalently an exponential that's shifted and truncated. No matter how you describe it, it's not exponential.)$\hspace{2cm}^\text{Histogram of large random sample, ...

6

The confidence interval with $\frac{1}{\sqrt{n}}$ is based on the same idea as the confidence interval with $1.96\sqrt{\frac{p(1-p)}{n}}$ but is more "conservative", in the sense that it is larger. The reason for that is that the function $$f(x) = x \left(1-x\right), x \in [0,1]$$ can be shown (with elementary calculus) to be maximal for $x= \frac{1}{2}$....

6

There is no way - the "population" of interest is part of the specification of the problem. Statistical problems involving inference to a "population" require specification of the group of interest, about which we are making an inference. Only a proper specification of the problem ---in this case, from a briefing from the client--- can give you this. Of ...

6

When writing the lower part, I missed that the population size is 21. I somehow thought the question was for general population size. For known population size 21 there should be a mathematical maximum range, not only a minimum one. First now considerations regarding the maximum possible range: Note that it is possible to have a very small observation if all ...

5

I suspect that the second example's description of the standard error is incorrect -- I've never seen the normal approximation approach to getting a standard error for a proportion reported without using the square root. [see e.g. http://www.stats.org.uk/statistical-inference/Newcombe1998.pdf for a description of different approaches to confidence interval ...

5

I believe in this kind of situations it's more important to ask "what do I need to show?" rather than "what should the picture look like?" There are times when map with a sea of white and pink and a couple of big red dots being very useful; there are also times that the same design can lead to biased decision. It all depends on what do you mean by meaningful....

5

There really are 2 concepts here. The exact or theoretical sampling distribution consists of every possible sample from the population. The approximate sampling distribution is the result of a large number of samples. When we make statements like the mean of the sampling distribution is equal to the mean of the population then we are talking about the ...

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