This applet illustrates the predictive power of confidence intervals for sample means, and it extends the applet Visualizing the Distribution of Sample Means on this site. The idea is that there is a "correct but unknown" population mean μ that we wish to estimate. We collect a sample of size n and use this to create a sample proportion x-bar and standard deviation s, and then a confidence interval using the formula
where n is the sample size, s is the sample standard deviation, and the value of z depends on the "confidence level" and the sample size. (For example, for 95% confidence interval with n = 25, we use a value t = 2.06.)
The meaning of "95%" is sometimes misunderstood, so this is the primary concept this applet strives to illustrate. Because there are 500 different collected samples (each producing a value of x-bar and s), there are 500 different confidence intervals we could form using the above formula. The "95%" refers to the fact that roughly 95% of these confidence intervals will contain the true value of the unknown μ -- this is signified in the applet by the highlighting of those confidence intervals which do not include the theoretical value of μ. (The applet is programmed to know the true value μ = 4. Obviously this is typically unknown since the confidence interval estimates the value of μ.)
Click the screen shot below or this link to open the applet in a new window:
