Stat 1000 Tips about Binomial Distribution and Distribution of the Sample Mean Assignments

Published: Fri, 11/05/10

 
Hi ,
 
You are receing this e-mail because you indicated when you signed up for Grant's Updates that you are taking Stat 1000 this term.  If in fact, you are not taking Stat 1000, please reply to this e-mail and let me know, and I will fix that.
 
Throughout the term I will send you all sorts of tips to help you study and learn the course.  You probably already have done so, but, if not, I strongly recommend you purchase my Basic Stats 1 Study Book.  You will find it a great resource to learn the course.  I pride myself in explaining things in clear, everyday language.  I also provided numerous examples of all the key concepts with step-by-step solutions.  You can order my book at UMSU Digital Copy Centre at University Centre at UM campus.  They make the book to order so please allow one business day.  The book is split into two volumes and each volume costs $45 + tax.
 
If you ever want to look back over a previous tip I have sent, do note that all my tips can be found in my archive.  Click this link to go straight to my archive:
 
Grant's Updates Archive
 
Tips about Binomial Distribution and Distribution of the Sample Mean
 
These tips apply to your assignments referring to the Binomial Distribution and binomial probabilities.  For those of you doing Web Assign in class, these tips would apply to HW6.  For those of you doing the distance/online Web Assign, these tips would apply to HW8.  For those of you doing the old-fashioned hand-in assignments, these tips apply to the first half of your Assignment 4.
 
You will need to study Lessons 6 and 7 in my study book to prepare for this assignment.
 
If you are ever asked to decide if a particular situation is binomial or not, remember, to be binomial, four conditions must be satisfied:
(i)  There must be a fixed number of trials, n.
(ii)  Each trial must be independent.
(iii)  Each trial can have only two possible outcomes, success or failure, and the probability of success on each trial must have a constant value, p.
(iv) X, the number of successes, is a discrete random variable where
X = 0, 1, 2, ... n.
 
If you are solving a binomial problem, and they ask you to compute a mean and/or standard deviation, read carefully.  Do they want the mean of X? or do they want the mean of p-hat, the sample proportion?  Be sure to study the sections about the Distribution of X and the Distribution of p-hat in my Binomial Distribution lesson (Lesson 6 in my new edition, Lesson 7 in older editions).  Take a look, especially, at question 10 of that lesson as a good run through of these concepts.
 
Questions that give you μ and σ are undoubtedly dealing with bell curves.  Make sure you have studied my lesson on the Distribution of the Sample Mean (Lesson 7 in the new edition, Lesson 6 in older editions).  Always be very careful to note, are they asking you for the probability of one individual value (X)?, or are they asking you for the probability of the average or mean of n values (x-bar, the sample mean)?  If you are dealing with X, use the X standardizing formula.  If you are dealing with x-bar, use the x-bar standardizing formula.
 
Also, note that you can only do probabilities for X in these cases if you are specifically told that X is normally distributed.  Otherwise, there is no X-bell curve, and the probability is unknown.  However, thanks to the Central Limit Theorem, we can always assume there is an x-bar bell curve (the sample mean is normally distributed), as long as n is large.
 
Those of you using Web Assign have a question that asks you how big a sample size n do you need to reduce σ(x-bar) down to a certain amount.  You can solve that by some algebra, but I suggest you just use trial and error.  Try n = 10, for example, and see if that works.  If not try n = 20 or something.  Play the Price is Right Clock Game.  Try higher n's, lower n's until you home in on the n that works.  With a calculator, I contend you can arrive at the correct answer quite quickly.  Those of you who feel comfortable with algebra are certainly welcome to solve the problem that way.
 
Those of you using Web Assign are given a problem using the Sampling Distribution simulator at Rice Virtual Lab in Statistics.  Read Carefully.  They first have you play around with it using a Normal Distribution.  But, the actual questions they ask want you to use a "Skewed Distribution".  Note that, in the top left corner, you are given the mean and standard deviation of the population (μ and σ).  The various questions you are being asked are about the distribution of the sample mean.  Compare the answers for the mean and standard deviation of x-bar that the applet is giving you (next to the third histogram) with the theoretical values you expect if you computed the mean and standard deviation of x-bar using the formulas I give you in Lesson 7 of my book.  This applet is illustrating the same concepts I discuss in Figures 1 and 2 in Lesson 7.
Those of you doing the old-fashioned assignment may want to play with this applet, too.  It shows how the distribution of the sample mean becomes approximately normal (the Central Limit Theorem) regardless of the distribution of the population, provided n is large.  Here is the link to the applet:
http://onlinestatbook.com/stat_sim/sampling_dist/index.html
 
Those of you doing the old-fashioned hand-in Assignment 4, note that your questions 1 and 2 deal with the concepts I teach at the start of Lesson 2 in my book.  Be sure to read that section up to the end of question 2.  I would make a histogram or a stemplot in JMP in question 3.  See the tips I have posted before about how to make histograms in JMP.