The short version is that the Beta distribution can be understood as representing a distribution of probabilities, that is, it represents all the possible values of a probability when we don't know ...

What a great question- it's a chance to show how one would inspect the drawbacks and assumptions of any statistical method. Namely: make up some data and try the algorithm on it! We'll consider two ...

The difference is not the mathematical expression, but rather what you are measuring. Mean squared error measures the expected squared distance between an estimator and the true underlying parameter: ...

Notice that: \begin{equation} \frac{\alpha\cdot\beta}{(\alpha+\beta)^2}=(\frac{\alpha}{\alpha+\beta})\cdot(1-\frac{\alpha}{\alpha+\beta}) \end{equation} This means the variance can therefore be ...

A Poisson distribution is discrete while a normal distribution is continuous, and a Poisson random variable is always >= 0. Thus, a Kolgomorov-Smirnov test will often be able to tell the difference. ...

rnorm generates a random value from the normal distribution. runif generates a random value from the uniform.

Your question is based on a false premise: isn't the null hypothesis still more likely than not to be wrong when p < 0.50 A p-value is not a probability that the null hypothesis is true. For ...

You can estimate them. The best estimate of the mean of the Gaussian distribution is the mean of your sample- that is, the sum of your sample divided by the number of elements in it. \bar{x} = \...

Yes, this is possible, if the proportion of null hypotheses (which is estimated by the qvalue package based on your p-value distribution) is small and your test is powerful. Here's an example. Let's ...

First, note that your understanding of PCA is slightly off. PCA doesn't "group" variables into principal components. Each principal component is, rather, a new variable (a "new metabolite") of the ...

I'll start by answering your question about updating events with the "fourth and fifth extensions." As you suspected, the arithmetic is indeed quite simple. First, recall how Bayes theorem is derived ...

The reason for the difference is that you have + 0 in your formula, setting an intercept of 0. This means it is comparing the model not to one with y equal to mean(Elapsed), but one with y equal to 0. ...

First, let's rephrase your alternative hypothesis. You phrased it as "less than 50% of the individuals tested..." But by talking specifically about the individuals tested, we know whether it's true or ...

You can convert from a q-value distribution to a p-value distribution rather simply (indeed, it's easier than the other way around!). The way to do this in R is (explanation is in the comments): ...

The only way that the product $X_iX_j=1$ is if both coupons i and j are selected. (Otherwise, either $X_i$ or $X_j$ will be 0, and the product will as well). Thus, $E[X_iX_j]$ is the probability ...

There are no solutions without dropping one of the numbers below 0. Besides your arithmetic solution to show this, you can run a simulation in R. The following simulation tries adding various numbers ...

It sounds like your problem is analogous to "Predict who will win the lottery based on whether they buy a lottery ticket." Now, whether someone buys lottery tickets is a very significant predictor of ...

You can do this with an ANOVA analysis: my.locs$cluster = factor(rep(c(1, 2), each=5)) anova(lm(attribute ~ cluster, my.locs)) # Analysis of Variance Table # # Response: attribute # Df ... View answer Accepted answer 4 votes Here's how the binary values for Apple, Orange and Pear are calculated. You know it has two binary digits (since k=3 and it has k-1 digits), so you compute each of those two digits: Apple is one of ... View answer Accepted answer 4 votes Imagine that this person is planning out his choices in advance. He has 10 slots and has to assign each a letter- A, B, C, or D. First, he must put an A somewhere. He has 10 choices. Second, he must ... View answer Accepted answer 3 votes Why does all of this mean that the bigger n is, the more easier it will be to reject the null hypothesis? That's true only when the null hypothesis is false (where$\mu_0\$ is not the true mean). ...

You didn't get the same information: you got a confidence interval of a certain width from measuring 40 people, and then after measuring 100 people you got a confidence interval of a (probably) much ...

Based on Very small p-values (less than 2e-16) Large effect size estimations (log-odds ratios ranging from .68 to 1.06) Small standard errors (around .015) All evidence indicates there is a ...

You're interested in the power of the t-test, given a certain effect size and standard deviation (that you've estimated with a smaller experiment). In R, this can be calculated with the power.t.test ...

I would use the maximum of the X vector as the total possible number of successes. (This is a biased estimate of the true maximum number of successes, but it should work fairly well if you have enough ...