Binomial statistics

In this problem you will collect and analyze data in a simulation of the sampling process. Obtain a pack of M M s or other similar candy. Obtain a sample of five candies, and count the number that are red. Report the result of your analysis as % red. Return the candies to the bag, mix thoroughly, and repeat the analysis for a total of 20 determinations. Calculate the mean and standard deviation for your data. Remove all candies, and determine the true % red for the population. Sampling in this exercise should follow binomial statistics. Calculate the expected mean value and expected standard deviation, and compare to your experimental results. 

EXAMPLE 3.4 Why does energy exchange Consider the two systems, A and B, shown in Figure 3.9. Each system has ten particles and only two energy levels, f = 0 or f = 1 for each particle. The binomial statistics of coin flips applies to this simple model. 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

The binomial distribution function is one of the most fundamental equations in statistics and finds several applications in this volume. To be sure that we appreciate its significance, we make the following observations about the plausibility of Eq. (1.21) ... 

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

It would be of obvious interest to have a theoretically underpinned function that describes the observed frequency distribution shown in Fig. 1.9. A number of such distributions (symmetrical or skewed) are described in the statistical literature in full mathematical detail apart from the normal- and the f-distributions, none is used in analytical chemistry except under very special circumstances, e.g. the Poisson and the binomial distributions. Instrumental methods of analysis that have Powjon-distributed noise are optical and mass spectroscopy, for instance. For an introduction to parameter estimation under conditions of linked mean and variance, see Ref. 41. 

A very common analysis in clinical trials involves the analysis of two binomial variables to see if there is a statistically significant association between them. A binomial variable is one that can have only one of two values. For example, let s assume that we have a variable called treatment whose value is either a 1 to indicate active drug therapy or a 0 to indicate placebo. We also have a variable called headache whose value is a 1 if the patient experiences headache after therapy and a 0 if not. What we want to know is whether a change in the level of therapy is significantly associated with a change in the level of headache. The 2x2 table looks like this ... 

The statistics of this process is identical to those pertaining to the tossing of a coin. The mathematics was first worked out with respect to games of chance by de Moivre, in 1733. It is formally described by the binomial distribution. 

The random-walk model of diffusion can also be applied to derive the shape of the bell-shaped concentration profile characteristic of bulk diffusion. As in the previous section, a planar layer of N tracer atoms is the starting point. Each atom diffuses from the interface by a random walk of n steps in a direction perpendicular to the interface. As mentioned (see footnote 5) the statistics are well known and described by the binomial distribution (Fig. S5.5a-S5.5c). At large values of N, this discrete distribution can be approximated by a continuous function, the Gaussian distribution curve7 with a form ... 

Figure S5.5 Random-walk statistics Each plot shows the number of atoms, N, reaching a distance d in a random walk, for walks (a) 100 atoms and 200 steps, (b) 500 atoms and 200 steps, and (c) 10,000 atoms and 400 steps. The curve approximates to the binomial distribution as the number of atoms and steps increases.

The above treatment has implicitly assumed that the experimental design was such that the number of trials was fixed at 12 and the observation was the number of heads. However, an alternative design could have been to continue tossing the coin until 3 tails were obtained, and the observation would be n, the number of tosses required to produce the 3 tails. In this case, the statistic for judging the data is just n. But the distribution of n, the number of tosses to produce 3 tails, is given by the negative binomial ... 

The split-sample method is often used with so few samples in the test set, however, that the validation is almost meaningless. One can evaluate the adequacy of the size of the test set by computing the statistical significance of the classification error rate on the test set or by computing a confidence interval for the test set error rate. Because the test set is separate from the training set, the number of errors on the test set has a binomial distribution. 

There are many other distributions used in statistics besides the normal distribution. Common ones are the yl and the F-distributions (see later) and the binomial distribution. The binomial distribution involves binomial events, i.e. events for which there are only two possible outcomes (yes/no, success/failure). The binomial distribution is skewed to the right, and is characterised by two parameters n, the number of individuals in the sample (or repetitions of a trial), and n, the true probability of success for each individual or trial. The mean is n n and the variance is nn(l-n). The binomial test, based on the binomial distribution, can be used to make inferences about probabilities. If we toss a true coin a iarge number of times we expect the coin to faii heads up on 50% of the tosses. Suppose we toss the coin 10 times and get 7 heads, does this mean that the coin is biased. From a binomiai tabie we can find that P(x=7)=0.117 for n=10 and n=0.5. Since 0.117>0.05 (P=0.05 is the commoniy... 

For the statistical purest, chi square tests may be used to determine significance levels, binomial distribution for confidence limits on the probabilities. 

An analysis of the [Co-( )-pn3]3+ system may be carried out if the statistical term is considered solely an entropy effect and the conformational term an enthalpy contribution. Also since the four tris and four mixed species are not differentiated statistically, only the equilibrium constant k = tris/mixed is considered. For (4-)-pn/(—)-pn= 1 and assuming the ligands are distributed binomially around the metal ion, the statistical factor gives fc = 0.33 (J7/=0 assumed) which leads to TAS= -0.66 kcal/mole at 25°. 

Construct the Lagrange multiplier statistic for testing the hypothesis that all of the slopes (but not the constant term) equal zero in the binomial logit model. Prove that the Lagrange multiplier statistic is iiR2 in the regression of (y - P) on the xs, where P is the sample proportion of ones. 

The following examples illustrate the procedure for a statistical test [7]. In the first, we consider a very simple test on a single observation. The second applies the seven-step procedure to a test on the mean of a binomial population using a normal approximation. Here, and in the third example, we introduce the idea of one-sided and two-sided tests, while in the fourth example we illustrate the calculation of Type II error, and the power function of a test. 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

For a binomial distribution, the normal approximation can be used with good accuracy for sample sizes as low as 8, providing the binomial k is arbitrarily increased by 0.5 in calculating the approximate normal statistic. For values of the parameter p near 0 or 1, a larger sample must be used to obtain an accurate approximation. 

In practice, simpler though less reliable tests are used for evaluating the state of mixing. One of these involves calculating certain mixing indices that relate representative statistical parameters of the samples, such as the variance and mean, to the corresponding parameters of the binomial distribution. One such index is defined as follows ... 

This probability distribution is called the (equal probability) binomial distribution, and is the same distribution that is obtained for tossing an unbiased coin. In books on statistics,19 it is shown that for large values of N, the binomial distribution approaches the continuous normal distribution ... 

---