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Nothing. One way to tell is to apply a better test. The null hypothesis is that each observation independently has a $1/2$ chance of being "right" and a $1/2$ chance of being "left." The total of th …

Draw a picture. This is a Venn diagram of the two events $\mathscr{i}$ (unit $i$ is included in a sample) and $\mathscr{j}$ (unit $j$ is included in a sample). On it I have posted the inclusion proba …

This question asks how to construct a rasterized two-dimensional kernel for a circular uniform distribution. The underlying mathematical problem is this: given a planar circle $R$ and a rectangle $X, …

Let's explore the possibilities by looking at a wide range of possible values of the spam proportion $p$. …

It's a truncated Normal distribution. This is a consequence of Boltzmann's Theorem. The following analysis provides the details needed to implement a practical solution. A Normal$(\mu,\sigma)$ di …

The proportion of silver balls in the urn is $p$ (but this is not the "proportion" we will be talking about). This urn provides a way to model a Bernoulli Trial. … A Simple Intuitive Analysis It is clear that we should expect the proportion of successes in the experiment to be close to $p$. …

In a simple random sample $X_1, \ldots, X_n$ where each $X_i$ independently has a Bernoulli$(p)$ distribution and weight $\omega_i$, the weighted sample proportion is $$\bar X = \sum_{i=1}^n \omega_i … X_i.$$ Since the $X_i$ are independent and each one has variance $\text{Var}(X_i) = p(1-p)$, the sampling variance of the proportion therefore is $$\text{Var}(\bar X) = \sum_{i=1}^n \text{Var}(\omega_i …

One (standard) interpretation of this question is that you seek a randomized sampling procedure that has at least a 95% chance of including at least one bad apple, should there happen to be a bad appl …

This question comes up in analyses of spatial relationships, especially in studying edge effects in spatial statistics. The answer depends intimately on the shape of the field, the shape of the water …

It would be erroneous to compute the ratios of the smoothed counts, because it's possible many of the ratios would not be true proportions--they could (easily) wind up outside the valid range from $0$ …

The unpooled variance tends to be too small. This is because under the null hypothesis there will still be chance variation in the two observed proportions, although the underlying probabilities are …

It can help to survey the scene: that is, to understand the consequences for your decision of the possible states of affairs. The missing information can be described by two integers: $m$, the number …

James' approach looks good: each observation, according to your description, might have a Binomial(n[i], p[i]) distribution where n[i] is known and--to be fully general--p[i] is a completely unknown f …