#Weighted standard deviation of a sample plus

Here, the margin of error ( EBM) is called the error bound for a population mean (abbreviated EBM). To construct a confidence interval for a single unknown population mean Ој, where the population standard deviation is known, we need as an estimate for Ој and we need the margin of error. The Error Bound for a mean is given the name, Error Bound Mean, or EBM. That something is the Error Bound and is driven by the probability we desire to maintain in our estimate,, times the standard deviation of the sampling distribution.

#Weighted standard deviation of a sample plus

In the equations above it is seen that the interval is simply the estimated mean, sample mean, plus or minus something. The Error Bound gets its name from the recognition that it provides the boundary of the interval derived from the standard error of the sampling distribution. We can see this tension in the equation for the confidence interval.Д¬alculating the Confidence Interval: An Alternative ApproachДЄnother way to approach confidence intervals is through the use of something called the Error Bound. The higher the level of confidence the wider the confidence interval as the case of the studentsвЂ™ ages above. There is a natural tension between these two goals. The very best confidence interval is narrow while having high confidence. However, it hardly qualifies as meaningful. This interval would certainly contain the true population mean and have a very high confidence level. You wish to be very confident so you report an interval between 9.8 years and 29.8 years. You have taken a sample and find a mean of 19.8 years. Imagine that you are asked for a confidence interval for the ages of your classmates. By meaningful confidence interval we mean one that is useful. With the Central Limit Theorem we have the tools to provide a meaningful confidence interval with a given level of confidence, meaning a known probability of being wrong. In all other cases we must rely on samples. We have forsaken the hope that we will ever find the true population mean, and population standard deviation for that matter, for any case except where we have an extremely small population and the cost of gathering the data of interest is very small. Our goal was to estimate the population mean from a sample. Here we wish to examine the effects of each of the choices we have made on the calculated confidence interval, the confidence level and the sample size.ДЇor a moment we should ask just what we desire in a confidence interval. The Standard deviation of the sampling distribution is further affected by two things, the standard deviation of the population and the sample size we chose for our data. It is clear that the confidence interval is driven by two things, the chosen level of confidence,, and the standard deviation of the sampling distribution. However, the level of confidence MUST be pre-set and not subject to revision as a result of the calculations. That case was for a 95% confidence interval, but other levels of confidence could have just as easily been chosen depending on the need of the analyst. This is presented in (Figure) for the example in the introduction concerning the number of downloads from iTunes. The level of confidence of a particular interval estimate is called by (1-О±).ДЄ good way to see the development of a confidence interval is to graphically depict the solution to a problem requesting a confidence interval. Levels less than 90% are considered of little value.

Common convention in Economics and most social sciences sets confidence intervals at either 90, 95, or 99 percent levels. In reality, we can set whatever level of confidence we desire simply by changing the Z О± value in the formula. Then read on the top and left margins the number of standard deviations it takes to get this level of probability.

Divide either 0.95 or 0.90 in half and find that probability inside the body of the table. These numbers can be verified by consulting the Standard Normal table. If we set Z О± at 1.64 we are asking for the 90% confidence interval because we have set the probability at 0.90. If we chose Z О± = 1.96 we are asking for the 95% confidence interval because we are setting the probability that the true mean lies within the range at 0.95. Z О± is the number of standard deviations lies from the mean with a certain probability. The confidence level is defined as (1-О±). О± is the probability that the interval will not contain the true population mean. The analyst must decide the level of confidence they wish to impose on the confidence interval. This is where a choice must be made by the statistician. Notice that Z О± has been substituted for Z 1 in this equation. This is the formula for a confidence interval for the mean of a population.