Estimation of Parameters

• Estimation is an area of inferential statistics where sample measures (statistic) are used to determine the true values of unknown population measures (parameters).
• Estimation of Parameter – refers to the process of approximating a parameter based on a statistic. Data are gathered from the samples instead of the entire population.
• When to estimate? – when it is impractical to gather data from all units of a population.
• A good estimator must be unbiased, accurate and precise
• Characteristics of a good estimator

  • Accurate and precise
  • Accurate but not precise
  • Not accurate but precise
  • Not accurate and not precise
• Unbiased – the estimator has no tendency to overestimate or underestimate the true parameter value.
• Accurate – results are correct / free from mistakes or errors / how close the estimates are to the actual value of parameter
• Efficiency of the estimator – refers to how large the variance of the estimator is. An estimator with a smaller variance is preferred.
• Precise – very exact / how close the estimates from each other / small variance / small standard error
• The distance between an estimate and the parameter being estimated is called the error of estimate.
• Point Estimation – is the process of finding a single value, called point estimate, from a random sample of the population, to approximate a population parameter.
• Point estimator – a rule or formula that calculates an estimate using the sample data
• Point Estimate – the computed single value
• Interval Estimation – deals with constructing an interval of possible values from a random sample to estimate an unknown parameter of interest.
• Interval Estimator – rule or formula that describes the calculation of a parameter
• Interval Estimate – a range of numerical values that approximates the parameter
• A good point estimate is one that is unbiased. If random sampling was done in the collection of a set of data, and a sample mean is computed out of these data to approximate the population mean 𝜇, then the point estimate 𝑥ҧ is a good point estimate.
• An interval estimate describes a range of values, constructed from the sample data, within which a population parameter lies with a predetermined probability of degree of confidence. Hence, it is also called confidence interval.
• In interval estimation, two numbers are calculated based on sample data, forming an interval where the parameter’s value is expected to lie
• Confidence Level (CL) refers to the degree to which we are confident that the confidence interval contains the parameter being estimated. It is also a probability that the confidence interval contains the true population parameter.
• Significance Level or Alpha Level denoted by 𝛼, refers to the likelihood/probability that the confidence interval does NOT contain the true population parameter.
• Margin of error refers to the maximum estimate of how far the parameter could differ from the point estimate. It is a multiple of the standard error.
• Critical Value 𝒛𝜶 𝟐 is the value that indicates the point beyond which lies the rejection region.
• The rejection region does not contain the true population parameter.