Point Estimates and Confidence Intervals

Point Estimates and Confidence Intervals Introduction Point estimates are statistics calculated from sample information that are used to estimate population parameters. Example: Best Buy Inc. wanted to estimate the average age television buyers. They select a sample of the last 50 buyers, determine the age of each buyer, and calculate the average age of the buyers in the sample. The sample mean is a point estimate of the population mean. The sample mean, is a point estimate of the population mean  p (sample proportion) is a point estimate of  (population proportion); and s (sample standard deviation) is a point estimate of  (population standard deviation). In general, population parameters will be symbolized (pronounced theta). So  could be the mean , standard deviation and proportion . Point estimates only tell part of the story. A point estimate is a single value derived from a sample and used to estimate a population value. We expect the point estimate to be as close to the population parameter as possible, so we will measure how close it actually is. Confidence intervals can be used to make such measurements. Confidence Interval A Confidence Interval is a range of values created from sample data where the population parameter is likely to occur within that range with a specific probability. This specifi probability is called the level of confidence. Suppose, we select a sample of 50 young executives' signatures to find out how many hours they spent working 1 week ago. Calculate the mean of this sample of 50 people and use the sample mean as a point estimate of the unknown population mean. The estimate is a single value. A more informative approach is to give a range of values that we expect to occur within a given population parameter i.e. what we call a confidence interval. Estimator Requirements 1. Consistent The estimator is said to be consistent, if the estimator remains consistent / concentrates on the estimate made, or: If  2 = 0 and unbiased, the estimator whose value perfectly concentrates on its target value when the sample is enlarged to infinity () or the larger the sample size, the statistical estimator  is closer to the parameter  population. 2. Efficient The estimator is said to be efficient, if the estimator has the smallest variance. 3. Unbiased The estimator is said to be Unbiased, if the value of the sample statistics = the value of the Population Parameters. Unbiased Estimator Unbiased estimator means an estimator that precisely hits the target. Meanwhile, a biased estimator means an estimator that is not right on target or is called. a miss Efficient Estimator  1,  2, and  3 p obtained from 3 The figure above shows that there are three estimators samples, where the distribution of sample 1 has variance σ2 , sample 2 has variance σ2/2 ,  1 isthat sample 3 has variance σ3^2 . Since sample 1 has the smallest variance, it is said an efficient estimator. Consistent Estimator The figure above shows that sample size 1, n1, is smaller than sample size 2, n2, and smaller than sample size 3, n2. It can be seen that the larger the sample size, the closer the estimated statistic is to parameter from the population, where the sample distribution consistently moves to the left.