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L17posted - Introduction to Estimation Data arise from...

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Introduction to Estimation Data arise from measurements whose values depend on characteristics of the process or ex- periment being considered. Some features of the process can be controlled while others can- not and result in variation in the data. If we could make many observations { x 1 , x 2 , . . . , x N } under similar circumstances, then this collec- tion would tell us everything about the under- lying process. The collection would be best described by its histogram. The centre of the histogram would give the mean and the spread would indicate how much variation there is. 1
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Examples Department store wants to know the pro- portion p of females between 18 and 25 who have a credit card. Takes random sample. Manufacturer is interested in proportion p of defective parts in an inventory of 100,000 parts. Collect data by taking measure- ments. Manufacturer is interested in mean break- ing strength μ of steel beams produced with specified percentages of raw material. Takes a random sample. 2
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In each case above, there is a population of interest (that is a distribution) and a charac- teristic of the population ) the p or μ , which are unknown. Such a characteristic is called a parameter and generically denoted by the sym- bol θ. Data are collected in order to learn about θ. From the data or sample, a guess, called an estimate , denoted by b θ is determined. For ex-
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