Chapter 19 Rocks!

 In our formula, fractionwise, z-star is in the top and n is in the bottom.  The terms for these are directly proportional (for z-star) and inversely proportional (for n).  Technically the interval is inversely proportional to the square root of n, but there is still inverse-type stuff going on there.

 This means that when z-star gets bigger, the confidence interval gets wider.  A bigger z-star is the result of a higher confidence level.  This means that the more confident we want to be in our answer, the more generously vague our answer needs to be.

 If we use a lower confidence level, the z-star gets smaller and our interval gets narrower.  We are, however, more likely to be wrong.  This is another reason we use 95% confidence most of the time, since it gives a good balance generally.

 If n gets larger, the confidence interval is narrower.  This means larger samples let us draw more accurate conclusions about the population.  Because n is inside a square root, though, the effect a larger sample has is not constant.  So the larger our sample, the less useful each additional subject is.

 This idea that sooner or later additional subjects are not worth studying is an example of the Law of Diminishing Returns.  What this means for you is that a larger sample makes a narrower (or smaller) confidence interval without changing the confidence level.  It is not a complete fix, however, as samples can only help so much before it becomes ridiculous to sample more.

 Many people assume that a 95% confidence level means that it is 95% likely that your interval includes the true value.  This is wrong!  Wrong, wrong, wrong, wrong, wrong!  Wrong, wrong, wrongity, wrong-wrong!

 In my Philosophy 101 class one of the most insightful ideas I caught from it was about dinosaur footprints.  What I am about to share, changed my life forever.  I was almost 2 decades into having an IQ well above other people, having spent my whole life being an information sponge, and this idea still changed me in a way that even now continues to adjust how I see the world.

 Consider the spot where I am standing.  Did a dinosaur ever occupy that space?  Let’s discuss this.

 The simple fact of the matter comes down to this:  Either the answer is yes or the answer is no and there is absolutely no way for me to determine the truth of it without utilizing some kind of superhuman power.  If you have such powers do NOT admit to them in public school.

 Sometimes in life just because you cannot know if something is true does not mean there is no truth.  Some truths absolutely exist and humans simply have no access to them.  They still exist and still are true.  Even most regular people can appreciate this idea when introduce to it through dinosaur footprints.

 Ever.  However, the truth is there, and is just inaccessible to humans through the methods of statistics.  Often a census would be required, and in many cases, even a census is inaccessible to humans.

 When we make a confidence level, one of two things happened.  Option 1: We correctly identified a range that does include the true population value and we have no way to verify it.  Option 2: We did not correctly identify a range that includes the true population value but we have no way to know we failed (other than a census).

 So on a 95% confidence interval, it is not a 95% chance that we were correct.  We either are correct or we are not.  We have no sensible way to verify it either.  So if it isn’t probability, what significance does the 95% have?

 For that we have to refer back to sampling distributions.  If we took every possible sample, we would have a sampling distribution.  We would also have failed at life, most likely.  If we did the 95% confidence interval on each and every single possible sample, then 95% of them would contain the true population value.

 Generating a sampling distribution is dumber than a census.  Generating every possible confidence interval for each sample in that sampling distribution is even dumber than a sampling distribution.  Pretty much it is going from ludicrous speed straight to plaid.  Space Balls reference.

 So, since we have no way to verify if we succeeded or not with our interval, we do the next best thing.  We confidently trust that 95% is often enough that right now is not the 5%.  In other words, we simply arrogantly assume we are right, because 5% is much less than 95%, plus, since someone as awesome as us was researching, it is obviously correct, right?

 We will use a script.  Part of the reason this script is so important is that the phrasing bypasses all of this “what does the confidence level really mean” business.  Failing to use the script can make it seem like you do not actually understand what confidence level means.

 “I am confident that the true for is between and.”  Example: “I am 95% confident that the true percent of the population of highland students who oppose school uniforms is between 69.3% and 81.1%.”

 Statisticians have all basically agreed that when someone says they are 95% confident that we all know that it does not mean that they are right with a 95% probability.  We have all agreed to simply appreciate that they mean they used a method which is correct 95% of the time and that they have no freaking clue if they are right or wrong.

 Chapter 18 problems (5, 17, 25, and 27) are due tomorrow.  Chapter 19 problems (7, 11, 13, 15, 23) are due next week on Tuesday.  Chapters quiz will be Thursday.  There is a practice quiz online.  Midterm project presentations will be next week.

 Be able to use z-scores to find probabilities for individuals.  Be able to use z-scores to find probabilities for sample averages.  Be able to use z-scores to find probabilities for sample proportions.  Be able to find a confidence interval for the true proportion based on a sample.  Be able to find the sample size in order get a desired margin of error.