MTH 245 Lesson 21 Notes Estimating a Population Parameter
A confidence interval is an estimate of the value of a population parameter. It is an interval on the real number line that is centered on a point estimate of the parameter. The interval end pointsthe upper and lower confidence limitsare determined by the margin of error. The illustration below shows the various parts of a confidence interval.
A point estimate is a single value that approximates the actual value of a population parameter. It is calculated using the sample statistic that is the best estimator of the parameter. The best estimator of ???? is ?????; for ????, it is ?????. Taken by itself, a point estimate isn’t an adequate estimate of a parameter value. Point estimators are random variables, and they can therefore take on different values for different samples. A point estimate for one particular random sample might be close to the actual parameter value, while it may be substantially different for a different random sample from the same population. We have to take this into account by defining a range of values that could presumably contain the true parameter value. A confidence interval can be formally expressed in one of three ways:
1. Algebraic notation: lower limit < population parameter < upper limit 2. Interval notation: (lower limit, upper limit) 3. Plus-or-minus notation: point estimate ± margin of error Confidence Level The confidence level (written in general terms as 1 ? ????) is the probability that a particular confidence interval actually contains the population parameter. The concept of confidence and significance are directly related. Where the significance level ???? is the probability of observing a value that is significantly far from the mean, the confidence level 1 ? ???? is the probability of observing a non-significant value. The following graph, which we first saw in Section 6.3, illustrates the relationship between the two: The confidence level is usually reported as a percentage of the form 100 ? (1 ? ????)%. The most commonly used confidence levels are 90%, 95%, and 99%, although others are possible.
