For Example 15-1 suppose that a third sample is obtained. Three out of 10 people in the sample are product users. Update the probabilities of market share after the third sampling, and produce the new posterior distribution.

Problem 15-1

A market research analyst is interested in estimating the proportion of people in a certain area who use a product made by her client. That is, the analyst is interested in estimating her client's market share. The analyst denotes the parameter in question- the true (population) market share of her client-by S. From previous studies of a similar nature, and from other sources of information about the industry, the analyst constructs the table of prior probabilities of the possible values of the market share S. This is the analyst's prior probability distribution of S. It contains different values of the parameter in question and the analyst's degree of belief that the parameter is equal to any of the values, given as a probability. The prior probability distribution is presented in Table 15-1. As seen from the prior probabilities table, the analyst does not believe that her client's market share could be above 0.6 (60% of the market). For example, she may know that a competitor controls 40% of the market, so values above 60% are impossible as her client's share. Similarly, she may know for certain that her client's market share is at least 10%. The assumption that S may equal one of six discrete values is a restrictive approximation. In the next section, we will explore a continuous space of values. The analyst now gathers a random sample of 20 people and finds out that 4 out of the 20 in the sample do use her client's product. The analyst wishes to use Bayes' theorem to combine her prior distribution of market share with the data results to obtain a posterior distribution of market share. Recall that in the classical approach, all that