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My Midterm Exam Prep Seminar for Stat 2000 is Saturday, February 28. It costs $40, which you pay at the door.
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:
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 current edition of my book has 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.
Remember my advice in the tips above. Don't start working on the assignment too soon. Study and learn the lesson first, and
use the assignment to test your knowledge. Of course, always seek out assistance from my book, your course notes, etc. if you ever hit a question you don't understand, but try not to be learning things as you do an assignment. Learn first, then put your learning to the test.
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 move on to productive things like understanding the concepts and interpreting the JMP outputs, the better off you will be.
Don't have my book or audio lectures? You can
download a free sample of my book and audio lectures containing Lesson 3:
A Warning about StatsPortal
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Make sure that you are using Firefox for your browser. Don't even use Internet Explorer. It actually also has some glitches in the HTML editor boxes.
Do note that every time you exit a question in StatsPortal, the next time you return to it, the data may very
well change. Do not press the "back-up" button on your browser in a question. That, too, will change the data. When you are prepared to actually do a question, open the link, keep it open, and do not close it until you have submitted your answers. Be sure to press "Save Answers" once you have done any calculations and entered any information to ensure the data does not change and force you to start over again.
After you submit the answer to a question,
if you have been marked wrong on any parts, be sure that you write down the correct answers before you exit the screen (or grab a screen shot). To try a second attempt at the question do not click the link to the question again, that will change the data and you will have to start all over again. Also, DO NOT click "try again" or make a "second attempt." That will also reset the data.
Instead, exit back to the home screen where they show the links
for all the different questions on the assignment. Where it shows the tries for a question on the right side of your screen, you should see the "1" grayed out, showing that you have had 1 attempt. Click the number "2" to get your second attempt with the same data. That way you can enter the answers you already know are correct and focus on correcting your mistakes.
You should also have already downloaded the JMP statistical software which was
provided with either one of the course options for StatsPortal as mentioned in your course outline.
Make sure you have gone through Assignment 0 completely to learn how to use the interface. I also suggest you print out a copy of question 8 in Assignment 0 (Long Answer Questions - Part 3) so that you have the steps for saving and uploading files into the HTML editor in front of you.
Question 1: Sample Spaces
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Don't forget to click the Html Editor link before you type your answers into their box.
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 all you are asked for is the sample space in each part, so your answer would be something like this (don't forget to use those squiggly brackets "{}").For example, here is the sample space for the outcome of flipping a coin twice where H=heads and T=tails: {HH, HT, TH, HH}.
Don't state the probabilities! You are not asked for the
probabilities. In fact, in many of the situations students will be given for this problem, it is impossible to know the probabilities. You are just asked for the sample space.
Focus on what you are asked to select. Is it just the colour of the candy you are interested in (Purple, Yellow or Red?, P, Y or R). Is it the flavour of the candies?
In part (c), although they have not made this clear, I think we are supposed to assume that there is only one of each flavour of candy in the bowl, otherwise the sample space would be infinite. Contact the prof to clarify if there is only one of each flavour of candy in the bowl.
In that case, there are only four candys, one of
each for grape, lemon, cherry, and raspberry (G, L, C, and R), and you are sampling without replacement until you get C. So, if your first selection is C, you stop. But, if you get G, L or R, you proceed to make another selection. If your next selection is C, you stop. If not, you make a third selection.
You may find a tree diagram helpful here, so that you can better keep track of what
is already gone, and what can still be selected. For example, if you picked G the first time, then there is only L, C or R left to pick the second time (if I am right in assuming there is only one of each flavour in the bowl).
Your sample space will look something like this: {C, GC, LC, ...}
Question 2: Olympic Probabilities
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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.
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).
Method 1 (faster, but pretty advanced):
In general, for three-circle Venn diagrams to go well, you need to know the probability for all three in the centre (A and B and C), and
then you need to know the three "ands" for the pairs (A and B; A and C; B and C) in order to fill in the petals clustered around the centre.
Here, we have all three (SK and SB and SS, the last percentage we are given). We also are given SK and SS and SK and SB. However, we lack SB and SS. We also lack P(SS).
We know P(SK or SS) = P(SK) + P(SS) - P(SK and SS). You are given three of those, so you can solve P(SS).
Algebraically, P(SS) = P(SK or SS) + P(SK and SS) - P(SK).
Now that you know P(SS), you can solve P(SB and SS).
We know P(SB or SS)
= P(SB) + P(SS) - P(SB and SS).
Algebraically, P(SB and SS) = P(SB) + P(SS) - P(SB or SS).
You can therefore solve P(SB and SS) since you know P(SB), P(SS) and P(SB or SS).
Now that you know
P(SB and SS) and P(SS), it is very easy to make a 3-circle Venn diagram using the approach I show you in my Lesson 6, question 18.
Method 2 (more direct, more versatile, but slower)
This question is very similar to my
question 18. However, they do not give you enough information to fill in every part of your three-circle Venn diagram. Similar to what I do in my question 17, you will need to put x in one of the missing parts of your three-circle diagram, then solve for x.
Don't enter percentages
into your Venn, use decimals to set up for the format StatsPortal wants for your answers. For example, if the percentage is 32.15%, enter that as 0.3215 in your Venn.Start with the last piece of information they give you. You are given the percentage who follow all three teams, so you can put that value in the centre of your three-circle diagram.
You are given the percentage that follow all three sports, so you can put that value in the centre of your three-circle diagram. You are given the percentage that are SK and SS fans, so you can fill the appropriate region in such that the two regions representing SK and SS add up to this given percentage. You are also given the percentage that are SK and SB fans. Again, there are two regions in the diagram
representing that, and you already know the number in one of them, so you can fill in the missing part knowing the total must be this SK and SB amount.
Here is where things get tough.
You are not given SB and
SS, so put x in that region that you are missing.
Now, we go to the first piece of info. We are told the percentage that are SK fans. There are four regions in the SK circle in your diagram, and you should have numbers already in three of them, so you can figure out what
value to put in that fourth region to make the total correct.We are also given the percentage that are SB fans.
The problem is that we have four regions in the SB circle but only have two numbers filled in. First add those two numbers and subtract from the total SB percentage. For example, if you have
70% that are SB fans, and the two given regions add up to 30%, that leaves 70-30=40% to put in the remaining two regions. One of those regions is already labelled x, so that means the remaining region can be labeled 0.40-x. Of course, these are not your numbers.
We are also given the percentage that are SK or
SS fans. There are six regions in your diagram that represent SK or SS. You already have numbers labeled in four of them (the four in the SK circle). Add those four up and subtract them from the SK or SS amount to see what is remaining. For example, if you know 40% follow SK , and 52% follow SK or SS then we know that 52-40=12% remains to put in the two remaining regions of the SS
circle. One of those regions we have already labeled x so we will label the other region 0.12-x. Again, these are not your numbers, of course.
Finally, we are given the percentage that are SB or SS followers. Add those six regions up. You will note that, besides the
numbers, you will be adding -x + x + (-x). This collects up to just -x. Of course, there are also a bunch of numbers you can add up in those six regions, so, if all the numbers add up to say 62%, then we discover the six regions add up to 0.62-x. But we can set that equal to the percentage that do follow SB or SS and solve x.Once you have solved x, you are
rolling.
Don't even think about asking me any more questions about this one. I have told you too much as it is. You are on your own from there.
Most of the probabilities are now quite easy to find by reading off the appropriate parts in your Venn
diagram. Note that parts (i) and (j) are conditional probability questions.
Question 3: Gumball Probability
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You can use a Two-way table to list all the outcomes in your sample space for part (a), but, be careful! You are sampling without replacement. That will affect the probabilities. The probability of the second gumball depends on what
was chosen first.
My Lesson 6, questions 4 and 13 may be of some help in understanding how to do parts (a), (b) and (c). Do note that (c) is conditional probability.
Note, the expected value is the mean value. I show you how to compute mean and variance of X in Lesson 6, question 1 and I
also do it more thoroughly in Lesson 7, questions 1 to 3.
Question 4: Sum and Difference of Random Variables -- Golfers
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You have two normal distributions, X and Y. 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 X + Y. Since X and Y are both normal distributions, X + Y is also
normal. You can also compute the mean and variance for X - Y. You also know that X - Y 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) wants X <
Y. Rewrite that as X - Y < 0.
Part (b) wants X + Y > some amount.
Part (c) wants X = Y exactly. Not less, not more, exactly the same. What would you be shading on your bell curve? 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?
Question 6: Airline Flights
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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 flight and how many years and flights they are
talking about. Note that n is the total number of flights he has taken to a specific destination during the time period, not just the number that arrive on time. I think they want you to understand that he is making 5 trips each year, and on each trip he has to take 3 flights. So there are five first flights, five second flights and five third flights each year.
Question 7: Binomial Type I & Type II Error
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I teach this in Lesson 7 in the "Hypothesis Testing Revisited" section.
Note that you can use Table C to solve the probabilities more quickly. My question 19 is very similar.
Question 8: Poisson Potholes
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Standard Poisson stuff as taught in Lesson 7. Make sure you are using the correct value for lambda.
Question 9: Poisson Telemarketer
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Standard Poisson stuff as taught in Lesson 7. Make sure you are using the correct value for lambda. You are on your own for this question.
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