Problems of Estimation in Statistics for Business, Lecture notes of Statistics

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STATISTICS FOR BUSINESS

II

Doç. Dr. Yüksel Akay ÜNVAN

PROBLEMS OF ESTIMATION

POINT ESTIMATION

To obtain information about a population parameter, such as the

value of population mean μ or population proportion π , a random

sample of objects from the population is usually created. Then,

the sample statistic found from the values of the sample

observations is used for predictions about population parameter.

This sample statistic is called an estimate of the population

parameter.

INTERVAL ESTIMATION

A point estimate of a population parameter, as you have already seen from

different examples previously in this chapter, has some limitations. It is

known from the central limit theorem that a point estimate changes from

one independent sample to another independent sample. While it is known

that the value of the point estimate of a population parameter gets closer

to real value of population parameter as sample size increases, we would

like to measure how close the point estimate really is to the actual value.

For this purpose, interval estimation is used. An interval estimation of a

population parameter, demonstrated by θ in general, includes two limits or

INTERVAL ESTIMATION An interval estimation of an unknown population parameter θ can be demonstrated as follows: L ≤ θ ≤ U where L and U are the lower and upper limits, respectively. A natural question to ask at this stage is that how can we measure the correctness of the interval for this unknown population parameter? We may rephrase the question as “how much do you trust your interval to include real arithmetic mean of the population?” Therefore, we need the probability theory to support our decision about a correct interval estimate of the unknown population parameter is obtained.

CONFIDENCE INTERVAL FOR POPULATION MEAN There are two important situations in the calculation of confidence interval for a population mean. We will discuss these two situations now in detail. The first case is that we use sample data to estimate the unknown value of population mean μ with the value of sample mean x found from a random sample of size n , and also the population standard deviation σ is known beforehand. The second case is that we use sample data to estimate the unknown value of population μ with the value of sample mean x found from a random sample of size n , and also the population standard deviation σ is not known. In the second case, instead of the population standard deviation σ , the sample standard deviation s is used as the substitute.