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Don't have my book or audio lectures? You can download a sample containing some of these lessons here: Did you read my tips on how to study and learn Stat
2000? If not, here is a link to those important suggestions: Did you read my Calculator Tips? If not, here is a link to those important suggestions: Please note that I made major changes to my book in September 2014. If you are using a book older than September 2014, you are missing about 100 pages of new material and an entirely new lesson on Probability.
Study Lesson 6:
Introduction to Probability and Lesson 7: Discrete Probability Distributions in my book (if you have it) to prepare for this assignment .
Please note: Only the newer editions of my book have Lesson 6 above. It is in volume 1. If you have an older edition of my book, you are missing a very important lesson that has taken prominence in the
course in the last couple of years. Lesson 7 is in volume 2 of my book. That was the former Lesson 6 in earlier editions. I strongly encourage you to consider buying at least volume 1 of my new book. If you have a recent older edition of my book, that may be sufficient for you to avoid purchasing volume 2 of my new book.
Thank you to students who purchase the new edition of my book, rather than using used
copies. Book sales are what helps me fund the free services I offer such as these tips.
To type in formulas you are using and to show your numbers subbed into the formulas click the button in the toolbar that looks like the Sigma Summation symbol (you have to click the "..." other options button to see the sigma formula input button. Then click the various buttons to make your fractions and enter the
symbols.
Exception: Always do any JMP stuff open-book. Have my tips in front of you, and let me guide you step-by-step through any JMP stuff. JMP is just "busy" work. The sooner you get it done and can move on to productive things like understanding the concepts and interpreting the JMP outputs, the better off you will be. I cover the Binomial Distribution in Lesson 7. If you are ever asked to decide if a particular situation is binomial or not, remember, to be binomial, four conditions must be satisfied: - There must be a fixed number of trials,
n.
- 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.
- Each trial must be independent.
- X, the number of successes, is a discrete random variable where X = 0, 1, 2, ... n.
Hints: - If you are reading off numbers from a randomly selected row in the random number table, note that
every row has 40 digits. That is like 40 trials looking for whatever digit you may be looking for. What is the probability that, at any moment on the table, the next digit is a 0, or a 1, or a 2, etc..
- If you are selecting objects, are you sampling with replacement (independent trials) or without replacement (dependent trials)?
- If you are given a Normal population, but are selecting a sample of size n, and want
to see how many of them are greater than 62 (for example), THAT IS A BINOMIAL DISTRIBUTION! You can use Table A to find what proportion are greater than 62. That is your p. Each trial, the person/thing either is greater than 62, or they are not. And the chance they are greater than 62 is p.
- If you are ever conducting trials until you get a desired result, that will never be binomial because you do not have a
fixed number of trials, n.
- If you are being dealt a hand of cards, that is definitely sampling without replacement. How can you have a hand of 5 cards, for example, if you are replacing each card as it is dealt?
This is a question best solved by Venn Diagrams. Make sure you have studied that section in Lesson 6 of my book and have done questions 14 to 18 before you attempt this question. This is quite similar to my question 18,
especially.
Make sure you have definitely looked over my examples of how to prove two events are independent or not in those questions (as well as others earlier in the lesson).
Part (a)
Every separate section in a Venn diagram is a separate outcome in the sample space. So, in a typical two-circle Venn there are 4 sections (4 places you put
numbers), telling you there are 4 outcomes in the sample space. In a three-circle Venn, there are usually 8 separate sections, making 8 outcomes.
Part (f)
This is getting at the fact that you should know about the independence of these events by then. I think the question is silly, since you have had to do a ton of calculations to even get to this point, and, if you don't see why you would already know the
answer without using conditional probability calculations, who cares? Obviously, don't show any work for this part even if you did have to do some.
Parts (h) and (j) are conditional probabilities!
Parts (k) and (l) are binomial problems. Use the Venn to establish the p for the case they describe.
I show you how to determine a Sample Space through the use of two-way tables if necessary in Lesson 6 of my book.
Note that you are told you are sampling
without replacement, so you may actually find a tree diagram of help here to keep track of what is gone after the first selection. Then again, it is really pretty easy to just list the possibilities since there are only two balls to be picked. Hint: There are 5*4=20 outcomes in the sample space.
Part (d)
Make sure you
understand what they are saying. For example, on of your 20 outcomes from part (a) is "23", so X is the absolute value of 2-3, which is the absolute value of -1 which is therefore 1. Simply put, just subtract the two numbers but always do bigger minus smaller to get positive differences. For "23", the absolute difference is 3-2=1. For "31", the absolute difference is 3-1=2.
Work out the value of X
for every one of the 20 outcomes, then you can make a table of the distinct values of X and the probability of each X value.
Part (e) is conditional probability!
Part (f) remember that expected value is just another way of asking for the mean value. I show you how to get the mean and variance of a discrete
variable at the start of Lesson 6 and more thoroughly in Lesson 7.
You have two normal distributions, F (freestyle time) and B (butterfly time). You can use the properties of mean and variance I teach at the start of Lesson 4 to work out the mean and variance of F + B. Since F and B are both normal
distributions, F + B is also normal. You can also compute the mean and variance for F - B. You also know that F - B is normal.
Of course, to change your score into a z-score, you will have to use the standardizing formula you first learned in Stat 1000, z = (x - mu)/sigma or z = (x - μ)/σ.
Part (a)
Let X = F + B and find the appropriate area on the X bell curve.
Part (b)
We want F < B. Rewrite that as F - B < 0. So, let X = F-B and find when X<0 on the X bell curve.
Part (c)
We want F = B. Rewrite that as F - B = 0. Think
carefully! What are you shading on your bell curve?
I introduce the formula for mean and standard deviation of a binomial distribution in my Lesson 7, question 10. Be careful that you are using the correct n and p for each question since they keep switching which colour of light and how many days they are talking
about.
Note that n is the total number of intersections he/she has encountered during the time period.
Standard Poisson stuff as taught in Lesson 7. Make sure you are using the correct value for lambda.
Standard Poisson stuff as taught in Lesson 7. Make sure you are using the correct value for lambda. |
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