Confidence IntervalsDate: 07/24/2006 at 02:12:16 From: Diako Subject: confidence interval Dear Dr Math, According to the correct definition of confidence interval, if n experiments were to be repeated many times, 95% of the calculated confidence intervals +/-(ts/sqrt(n)) would include the true value. In the real world (say in a chemical lab) only n experiments are performed and only one of those many confidence intervals is calculated and reported. What does this single CI mean? How can this single range represent the population? Customers usually misinterpret this as a range in which the true value lies with 95% probability! Date: 07/26/2006 at 02:27:06 From: Doctor Wilko Subject: Re: confidence interval Hi Diako, Thanks for writing to Dr. Math! It's confusing and the distinction seems subtle, but I'll try to explain my understanding of this. Why do we need confidence intervals? =================================== If you calculate a sample mean, you have a point estimate. Is this a good estimate of the true population mean? A bad estimate? Who knows? The point estimate is easy to calculate and interpret, but we don't know how accurate it is. Instead it might be more meaningful to look at an interval estimate, i.e., the confidence interval. So, what is a confidence interval? ===================================== It's a RANGE of values used to estimate the true value of a population parameter. The degree of confidence (e.g., 90%, 95%, etc...) tells us the percentage of times that the confidence interval actually contains the population parameter, assuming the process is repeated a large number of times. Incorrect Interpretation: ======================== The confusion with interpreting confidence intervals is that people often draw a sample, calculate the mean, construct a single confidence interval and say, "There is a 90% probability that the true mean is within THIS confidence interval." This is wrong because the population parameter, mu, is a constant, not a random variable. Its either in the confidence interval or it's not. There is no probability involved. Correct Interpretation: ====================== Confidence intervals are best interpreted in the context of many samples. Say you construct one confidence interval. Your boss asks you to interpret the answer. You say, "I'm 90% confident that the interval from ___ to ___ actually contains the true population mean." Your boss says, "I'm a little confused, please explain." You say, "If I take 100 random samples each of size n and construct the confidence intervals, then about 90 of them will contain the true population mean." In summary, it really alludes to the process. A confidence level of 90% tells us the process we are using will, in the long run, result in confidence interval limits that contain the population parameter 90% of the time. Does this help? - Doctor Wilko, The Math Forum http://mathforum.org/dr.math/ |
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