2search]searchw 50i .a 25p Szh v Over r Searchforsinglesmeetsingles ne Szh S2a Searchforsinglesmeetsingles c Search Szh Szh z 50i ea Search c Www fo 50i sn Searchforsinglesmeetsingles lsearchssearchesearcht Over in 50i lsearchs Szh Searchforsinglesmeetsingles 5 25p rhijap. 25p osearch.n/ Over [ Search /s12 + 1/s22]

The combined variance is:

2 / [1/s12 + 1/s22]

For our application, using the above tabular information, the combined estimate of expected sales is 83.15 units with combined variance of 65.77.

You might like to use Revising the Mean and Variance JavaScript in performing some numerical experimentation. You may apply it for validating the above example and for a deeper understanding of the concept where more than two sources of information are to be combined.

Subjective Assessment of Several Estimates Based on Relative Precision

In many cases, we may wish to compare several estimates of the same parameter. The simplest approach is to measure the closeness among the estimates in an attempt to determine that at least one of the estimates is more than r times the parameter away from the parameter, where r is a subjective, non-negative number less than one.

You might like to use Subjective Assessment of Estimates JavaScript to isolate any inaccurate estimate. By repeating the same process you might be able to remove all inaccurate estimates.

Further Reading:
Tsao H. and T. Wright, On the maximum ratio: A tool for assisting inaccuracy assessment, The American Statistician, 37(4), 1983.

Bayesian Statistical Inference: An Introduction

Statistical inference describes the procedures by which we use the observed data to draw conclusions about the population from which the data came or about the process by which the data were generated. Our assumption is that there is an unknown process that generates the data we have and that this process can be described by a probability distribution, which, in turn, can be characterized by some unknown parameters. For instance, for a normal distribution the unknown parameters are m and s2.

Broadly speaking, statistical inference can be classified under two headings: classical inference and Bayesian inference. Classical statistical inference is based on two premises:

  1. The sample data constitute the only relevant information.
  2. The construction and assessment of the different procedures for inference are based on long-run behavior under essentially similar circumstances.

In Bayesian inference we combine sample information with prior information. Suppose that we draw a random sample x1, x2,....xn of size n from a normal population.

In statistical inference we take the sample mean as our estimate of m. Its variance is s2 / n. The inverse of this variance is known as the sample precision. Thus the sample precision is n / s2.

In the Bayesian inference we have prior information on m. This is expressed in terms of a probability distribution known as the prior distribution. Suppose that the prior distribution is normal with mean m0 and variance s02, that is, precision 1/ s02. We now combine this with the sample information to obtain what is known as the posterior distribution of µ. This distribution can be shown to be normal. Its mean is a weighted average of the sample mean and the prior mean, weighted by the sample precision and prior precision, respectively. Thus,

Posterior mean = (W1 + W2 m0) / (W1 + W2)

Posterior variance = 1 / (W1 + W2)

where

W1 = Sample precision = n/S2, and W1 = Prior precision = n/s02

Also, the precision (or inverse of the variance) of the posterior distribution of m is W1 + W2, that is, the sum of the sample precision and prior precision.

The posterior mean will lie between the sample mean and the prior mean. The posterior variance will be less than both the sample and prior variances.

In this Web site do not discuss Bayesian inference because this would take us into a lot more detail than we intend to cover. However, the basic notion of combining the sample mean and prior mean in inverse proportion to their variances will be of interest while being useful.

You may like using the Bayesian Statistical Inference JavaScript for checking your computation and performing some experiment.

Further Reading:
Ghosh M., and G. Meeden, Bayesian Methods for Finite Population Sampling, Chapman & Hall/CRC, 1997.

Managing the Producer's or the Consumer's Risk

The logic behind a statistical test of hypothesis is similar to the following logic. Draw two lines on a paper and determine whether they are of different lengths. You compare them and say,"Well, certainly they are not equal. Therefore they must be of different lengths. By rejecting equality, that is, the null hypothesis, you assert that there is a difference.

The power of a statistical test is best explained by the overview of the Type I and Type II errors. The following matrix shows the basic representation of these errors.


Click on the image to enlarge it and THEN print it.
The Type-I and Type-II Errors

As indicated in the above matrix a Type-I error occurs when, based on your data, you reject the null hypothesis when in fact it is true. The probability of a type-I error is the level of significance of the test of hypothesis and is denoted by a .

Type-I error is often called the producer's risk that consumers reject a good product or service indicated by the null hypothesis. That is, a producer introduces a good product, in doing so, he or she take a risk that consumer will reject it.

A type II error occurs when you do not reject the null hypothesis when it is in fact false. The probability of a type-II error is denoted by b . The quantity 1 - b is known as the Power of a Test. A Type-II error can be evaluated for any specific alternative hypotheses stated in the form"Not Equal to" as a competing hypothesis.

Type-II error is often called the consumer's risk for not rejecting possibly a worthless product or service indicated by the null hypothesis.

Students often raise questions, such as what are the 'right' confidence intervals, and why do most people use the 95% level? The answer is that the decision-maker must consider both the Type I and II errors and work out the best tradeoff. Ideally one wishes to reduce the probability of making these types of error; however, for a fixed sample size, we cannot reduce one type of error without at the same time increasing the probability of another type of error. Nevertheless, to reduce the probabilities of both types of error simultaneously is to increase the sample size. That is, by having more information one makes a better decision.

The following example highlights this concept. A electronics firm, Big Z, manufactures and sells a component part to a radio manufacturer, Big Y. Big Z consistently maintain a component part failure rate of 10% per 1000 parts produced. Here Big Z is the producer and Big Y is the consumer. Big Y, for reasons of practicality, will test sample of 10 parts out of lots of 1000. Big Y will adopt one of two rules regarding lot acceptance:

On the basis of the binomial distribution, the P(0 or 1) is 0.7367. This means that, with a defective rate of 0.10, the Big Y will accept 74% of tested lots and will reject 26% of the lots even though they are good lots. The 26% is the producer's risk or the a level. This a