Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Given a 95% confidence level, how do I demonstrate 95% of the intervals actually contain the population mean?
Let me say first off that I'm consistently impressed with the quality of responses on this site. My basic goal is to prove that confidence intervals actually work, and by work I mean that that a certain percentage of them, say 95%, actually include the population mean (assuming that we know the population mean in the first place).
A) I’ve generated a data set in Excel consisting of 20 samples with 64 observations each (i.e. C) Below is the best Excel formula I found that can generate something approaching a normal distribution. As you might have already guessed, here’s the issue: theoretically if 95% of these intervals contained the population mean, then only 1 interval out of 20 would have failed to include it.
In case someone happens to see this thread, here are screen shots of the 2 main excel spreadsheets; one version with just the values, the other version with just the formulas.
Compared to your population standard deviation of 12, the width of your confidence intervals looks too narrow in many cases. Use meaningful names for ranges and variables rather than cell references wherever possible.
To illustrate, let me share a spreadsheet I created long ago for exactly this purpose: to show, via simulation, how confidence intervals work. Finally, these 100 columns drive a graphic that shows all 100 confidence intervals relative to the specified mean Mu and also visually indicates (via the spikes at the bottom) which intervals fail to cover the mean. In any case, you have to simulate an infinite number of samples to get the result you want. Not the answer you're looking for?Browse other questions tagged normal-distribution confidence-interval binomial excel random-generation or ask your own question.
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You folks have done a better job of explaining difficult concepts than most instructors or textbooks I've encountered. I couldn’t even find agreement on whether the truncation disqualifies it as normal in the strict sense.
Is Excel simply incapable of generating random normally distributed data with the level of accuracy needed to make this example work?
If you do $n$ CI's, the number that cover the true mean will follow a binomial distribution $(n,p)$, with $p=0.95$.
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If all these people had done so much before me, why was I told at 17 I could not do any of it and that going to college was and I quote, "above my status". So even though I got over 500 points in my leaving cert and was in the 25% of the country the year I sat it when I got to college, lecturers refused to help me with notes because "I shouldn't be there and sure what was I going to do in the future anyway" and when I got a 1st for my degree and went on to my PhD I was given a lecture on the pyramids of education in my viva as to why I didn't deserve to be there at this level. Then for every person I help to see differently and every family that is successful or every business they will approach the next set of people in a new light. It's all about making as many people as possible experience something new that will gradually change the world over time.
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I see and hear stories everyday of happy go luck children that suddenly have the weight of the world on them.
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I only know that RandBetween produces a uniform distribution so it’s useless for these purposes. Also, I tried a Normal Quantile Plot Test for Normality that I found online on the 128.1k data point example and I did find a very slight amount of leptokurtosis for anyone who knows what to make of that. Note that I also tried Excel’s data analysis->random number generation tool and got virtually identical results as the Excel formula I included above.
Here were the results: at the 90% CL only 60 out of 100 had at least 18 intervals with the pop. Below is a comparison of the original 100 iterations I recorded compared to the binomial distribution per Excel.
I was told recently that "you have made me look differently at my employees and I can see how they learn now" and this was from working with someone's child. If I regenerate the prices repeatedly using F9, it only meets the 19 or greater threshold around 70% of the time with some iterations producing 20 out 20 intervals that contain the mean to as low as only 16 out of 20. Your intervals are much narrower, which leads me to suspect you are calculating something wrong. As far as choosing a value of 12, the logic came from trying out various std devs that would result in a data set that was fairly consistently bound between 25 and 100 give or take.
Please note, though--in response to your title--that indeed these CIs are provably correct. I can now demonstrate that out of 20 confidence intervals, the highest probability belongs to scenarios where exactly 2 do not contain the population mean at the 90% confidence level. But by understanding their own child in a different way now from experience they see their own employees differently.
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Maybe while schools and the powers that be argue about how important homework is for life they could take a step back and look at how long we have to be responsible and grown up in contrast to how short a time we have as carefree children? Naoise’s Educational Programming Through Play & Children can start at the top in School instead of being way behind.

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