53
votes
Do we need hypothesis testing when we have all the population?
It all depends on your goal.
If you want to know how many people smoke and how many people die of lung cancer you can just count them, but if you want to know whether smoking increases the risk for ...
- 6,101
29
votes
What did statisticians use to quickly find p-values before applets were popularized in the late 2010s?
Simple tables of $p$-values were in lots of places, such as the back of every book. The industrial-grade reference for statisticians needing lots of distributions would be something like the ...
25
votes
Accepted
How can the central limit theorem hold for distributions which have limits on the random variable?
This is an excellent question, since it shows that you are thinking about the intuitive aspects of the theorems you are learning. That puts you ahead of most students who learn the CLT. Here I will ...
- 133k
24
votes
Accepted
Is the mean of samples still a valid sample?
No, $\bar x$ has its own sampling distribution. Take, for example, the variances of $\bar x$ and $x_i$, in which the former is always lower ($\leq$) than the latter, which means $\bar x$ is not ...
- 58.2k
22
votes
Is the mean of samples still a valid sample?
Good examples so far but consider $$X_i \sim Bernoulli(.5)$$
In that case the distribution of the data will only have support on 0 and 1. But the sample mean will have an ever decreasing probability ...
- 2,163
21
votes
Accepted
Do we need hypothesis testing when we have all the population?
To illustrate my points, I will assume that everybody has been asked whether they prefer Star Trek or Doctor Who and has to choose one of them (there is no neutral option).
To keep things simple, let’...
- 2,371
20
votes
Accepted
What is the difference between random variable and random sample?
A random variable, $X:\Omega \rightarrow \mathbb R$, is a function from the sample space to the real line. This is a deterministic formula that can be as simple as writing down the number a die lands ...
- 26.9k
18
votes
Accepted
How do sample weights work in classification models?
As Frans Rodenburg already correctly stated in his comment, in most cases instance or sample weights factor into the loss function that is being optimized by the method in question.
Consider the ...
- 2,864
18
votes
Using mean length and mean weight to calculate mean BMI?
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 ...
- 290k
16
votes
What did statisticians use to quickly find p-values before applets were popularized in the late 2010s?
I've never used an applet to get a p-value. I know they are there, but I've never needed them. The ones I've seen are too simplistic for nearly any of the work I do (I'm retired, but not completely ...
15
votes
Accepted
Are Kaggle competitions just won by chance?
Yes, your reasoning is correct. If a different test set was selected and the competition repeated, rankings would indeed change. Consider the following example. All entries to a Kaggle competition ...
- 815
15
votes
Detect if there is actually two populations in a sample
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 ...
- 5,130
15
votes
What did statisticians use to quickly find p-values before applets were popularized in the late 2010s?
Here's a short timeline based on personal experience.
Academics had access to SAS and SPSS on mainframes in the 1960's and 1970's. (My professional introduction to statistics was supporting PhD ...
14
votes
Is the mean of samples still a valid sample?
No. Suppose you have $X_1, X_2 \sim N(0,1)$. Then,
$$
\bar{X} = \dfrac{X_1 + X_2}{2} \sim N\left(0, \dfrac{1}{2} \right)\,.
$$
But $N(0,1) \ne N(0, 1/2)$.
- 16k
14
votes
Is the mean of samples still a valid sample?
No, it is only valid in cases as the Cauchy distribution, the means of samples of the Cauchy follow the same Cauchy dstribution.
- 633
14
votes
Is the mean of samples still a valid sample?
As an even more pathological example, consider a sample from the distribution which is uniform on the union of $[0,1]$ and $[3,4]$. As the sample size increases, the mean will tend to 2 which isn't ...
- 1,454
14
votes
Accepted
Why do we use term “population” instead of “Data-generating process”?
There are certainly already many contexts where statisticians do refer to a process rather than a population when discussing statistical analysis (e.g., when discussing a time-series process, ...
- 133k
14
votes
Accepted
Sample notation: When to use capital $N$ vs lowercase $n$?
I'd say the second notation is common, although evidently not universal, namely that $n$ is used for sample size and $N$ for population size.
By accident, perhaps, rather than design, it matches a ...
- 59.4k
13
votes
Accepted
Is $\overline{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n}$ an estimator of the mean in general (for random variables with any distribution)?
Estimators are random variables. They exhibit properties that we use to assess their quality, advantages, and disadvantages. So it depends what you mean by "is an estimate of." I can say $\hat{\mu}_0 =...
- 1,322
13
votes
How do sample weights work in classification models?
Rickyfox's answer is great in explaining how the weights influence the results of a classifier, but maybe could you be also interested in why / how we would need such weights in the first place (which ...
- 915
13
votes
Accepted
Confused when to use Population vs Sample standard deviation in engineering testing
The two forms of standard deviation are relevant to two different types of variability. One is the variability of values within a set of numbers and one is an estimate of the variability of a ...
- 17.2k
12
votes
Accepted
Random sampling and independence in a real world problem
This extract from the text suffers from ambiguity and incorrectness.
Let's deal with the latter first. Independence of two random variables $X$ and $Y$ is not about one variable "providing no ...
- 334k
12
votes
Is Kendall's tau uniquely determined by Pearson rho?
Can two different data sets have the same Pearson's ρ, but different Kendall's τ?
Anscombe's quartet gives you four two-dimensional datasets with (almost) identical Pearson correlations, but their ...
- 131k
11
votes
Accepted
About Sampling and Random Variables
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. ...
- 12.8k
11
votes
Accepted
Detect if there is actually two populations in a sample
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. ...
- 141k
10
votes
What's in a name: bootstrapped replications?
The bootstrap is done by taking your sample and resampling from it by sampling with replacement. Here is a small example.
Let's say you have a sample of 6 observations {$1,4,6,7,8,8.5$}.
A bootstrap ...
- 43.2k
10
votes
What is the distribution of the sample variance for a Poisson random variable?
The distribution of the sample variance is slightly tricky, particularly because of the way the sample mean comes into it.
Note that
it has a discrete distribution,
by taking deviations from the ...
- 290k
10
votes
Accepted
How is sample mean divided by sample STD distributed for normal distributions?
Let $X_1,...,X_n \sim \text{IID N}(\mu, \sigma)$ be your data points. It is well known from Cochran's theorem that the sample mean and sample variance are independent with distributions:
$$\bar{X}_n ...
- 133k
10
votes
Using mean length and mean weight to calculate mean BMI?
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, ...
- 131k
10
votes
Detect if there is actually two populations in a sample
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 ...
- 131k
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