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. What are some good ways to figure out one's interests in mathematics, before applying to PhD programs?
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If only what you had done could change the perceptions of how others approach the next generation. One of the biggest frustrations for me was it never mattered what I had done before - I was always judged at the next stage of education. But almost 6 years on since I started my own education projects to develop methods to change education I now understand why I didn't want to be a motivational speaker.
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In order to be truly successful people have to experience what it is like to learn differently - they have to understand how they absorb information, how they process it and how they can be truly successful.
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There is always a balance between making long term changes in education and supporting those people currently in the system! One of the many reasons why Students attending The Homework Club become so successful is because of the awareness they have of their own Potential, Personality and Learning Style and how these effect their own educational success. The Con?dence Club now offers this service to everybody, especially those restricted through distance and who aren't able to attend our classes at The Homework Club. 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 $p=0.95$. Naoise O’Reilly on Newstalk’s The Right Hook interviewed live in studio by Shane Coleman this evening renewing call for homework to be banned in schools. Naoise O’Reilly made today and start to see some of the elements that can matter a lot in the grand scheme of things. O’Reilly believes the pressure associated with having famous parents sets children up for failure from birth plus having to endure the additional burden, heartache and increased likelihood of losing all of the family’s generational wealth within a very short space of time. 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.
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I have so many different roles in my work but listening to teenagers woes is definitely top of the list and acting as translator back to schools, parents and other professionals. It sometimes seems that schools and departments have forgotten there is an important family life to live. I see and hear stories everyday of happy go luck children that suddenly have the weight of the world on them. Every Student at The Homework Club is set-up to work to their best abilities within the environment through an initial interview. 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. You can never expect a dyslexic to be defined by their reading age once you have seen them get 9 out of 10 in the spelling test or 44 out 45 in the maths test. I have always thought that business had the ability to drive education and that is why I work across both sectors now. Her educational results with Children & Adults plus Her own academic milestones despite Profound Dyslexia are documented across the Globe. That's not likely to be of interest to anyone else unless some specific statistical question emerges.
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