We went over two AP problems in class today from the old AP exams, and I have to say, I'm very impressed at how well you guys did with them! Great job! Take those with you and be sure to study.
Of course, the best way to study stat (or any math class, for that matter) is to PRACTICE lots and lots of questions. I've found that some of the best resources are in the AP practice books that you can buy on Amazon.com. Now, if you haven't gotten around to doing that yet, a great alternative is to go to:
http://apcentral.collegeboard.com/apc/members/exam/exam_information/8357.html
The questions we did today were from the 2011 exam and the 2010 exam. When you get to this page, scroll down and you'll see the questions, full-scale answer keys, and samples of student responses. AWESOME.
There are also solutions keys to the extra examples that I gave out in class, as well as sample student responses. 5 extra points on your next test if you can guess how old I'm going to be for my upcoming birthday correctly.
Monday, September 30, 2013
Friday, September 27, 2013
Venn Diagrams and Probability Distributions
Venn Diagrams are used when an event can take on two probabilities. For example: "There are two students in statistics, 4 in english, and 5 in both." The key word "both" tells us to use the Venn Diagram. Create the Venn Diagram first, then use it to determine whatever probability the question is asking you for. Remember that anything that is "neither" goes on the outside.
A probability distribution is a chart of probabilities when the random variable, x, can take on many different values. It looks like this:
x 0 1 2 3 4
P(x) .1 .2 .15 .1 .45
x is the random variable. The numbers next to it are the values that the random variable can be. For example, I could get into zero, 1, 2, 3, 4 car crashes per year. The bottom row P(x) tells us the probability associated with that value. For example, the probability of me getting into 2 car crashes is .15. The bottom row of probabilities must always sum to 1.
A probability distribution is a chart of probabilities when the random variable, x, can take on many different values. It looks like this:
x 0 1 2 3 4
P(x) .1 .2 .15 .1 .45
x is the random variable. The numbers next to it are the values that the random variable can be. For example, I could get into zero, 1, 2, 3, 4 car crashes per year. The bottom row P(x) tells us the probability associated with that value. For example, the probability of me getting into 2 car crashes is .15. The bottom row of probabilities must always sum to 1.
Thursday, September 26, 2013
Independent Events
Independent Events are two events whose outcomes are not influenced by one another. By definition, we know that two events are independent if the probability of one event happening is the same as the probability of it happening given a second event. Your AP formula sheet writes it like this:
P(A) = P(A | B)
In plain english, this means that we have two events, A and B. On one side of the equation, we calculate the probability of event A happening. On the other side of the equation, we calculate the conditional probability of event A, given that event B has already happened. If they are the same on both sides of the equation, then we have independent events. If not, then the events are dependent on one another.
In context, "independent events" means that whatever event B is does not influence or change the probability of event A when it happens. For example, if a student of mine skips class to go watch a fight, that's not going to change the probability that I'm still going to be teaching class that period. So the student going to the fight and me teaching class would be independent events.
"Dependent events" means that event B somehow influences the outcomes of event A. Therefore, the probability of them happening together is different than if event A happened by itself. For example, the probability of me showing up to teach on a given day is probability about 98%. However, the probability of me showing up to teach given that I know I have a cold is significantly lower, probability around 30%. So since p(teaching) does not equal P(teaching given cold), those two events would be dependent on one another.
P(A) = P(A | B)
In plain english, this means that we have two events, A and B. On one side of the equation, we calculate the probability of event A happening. On the other side of the equation, we calculate the conditional probability of event A, given that event B has already happened. If they are the same on both sides of the equation, then we have independent events. If not, then the events are dependent on one another.
In context, "independent events" means that whatever event B is does not influence or change the probability of event A when it happens. For example, if a student of mine skips class to go watch a fight, that's not going to change the probability that I'm still going to be teaching class that period. So the student going to the fight and me teaching class would be independent events.
"Dependent events" means that event B somehow influences the outcomes of event A. Therefore, the probability of them happening together is different than if event A happened by itself. For example, the probability of me showing up to teach on a given day is probability about 98%. However, the probability of me showing up to teach given that I know I have a cold is significantly lower, probability around 30%. So since p(teaching) does not equal P(teaching given cold), those two events would be dependent on one another.
Wednesday, September 25, 2013
Hints for Problem Set 4 (Probability)
Problem 1:
The key to this question is effectively defining your sample space by making a correct tree diagram. Think CAREFULLY about what your random variables are in this case, and remember that you need a new level of your diagram for each random variable.
For parts B and C, find your key words in the question. Recall that some key words tell you to multiply while others tell you to add.
Problem 2:
As in the last one, make a good tree diagram. NOTE that we are selecting two books and that our events are 1) selecting the first book and 2) selecting the second book. Overnight/renewing are not events in this case because, according to the question, we need to be picking two books, one right after the other.
Problem 3:
Remember that I forgot the "A" in part iii - whoops!
For this one (part iii), you can show your work either by explaining the tree diagram or by showing how you used the intersection rule effectively. Either one will receive full credit, just be sure to write out your probability statements.
Problem 4:
Think about the properties of decks of cards. There are 52 cards, 13 cards per suit, and 4 suits. ALL of the suits are equivalent to one another other than the fact that they are hearts, spades, clubs, and diamonds. If you write a knock knock joke on the last page of your problem set then I will give you five bonus points. So then, numerically, should anything about them be different? Hmmm.....
Problem 5:
Part a: the probability is not 1/6 per each face because the dice is loaded. You're going to have to change the denominator for this one. This question is probably the hardest one on the problem set...
Part B and C: once you have part A figured out, these two should be no problem at all
Part D: Don't let the notation throw you off on this. Just think carefully about what the question is asking you. Just like number one, you're going to need a new denominator for this question, too. Once you find it the rest of the problem should be pretty straightforward.
Good luck!
The key to this question is effectively defining your sample space by making a correct tree diagram. Think CAREFULLY about what your random variables are in this case, and remember that you need a new level of your diagram for each random variable.
For parts B and C, find your key words in the question. Recall that some key words tell you to multiply while others tell you to add.
Problem 2:
As in the last one, make a good tree diagram. NOTE that we are selecting two books and that our events are 1) selecting the first book and 2) selecting the second book. Overnight/renewing are not events in this case because, according to the question, we need to be picking two books, one right after the other.
Problem 3:
Remember that I forgot the "A" in part iii - whoops!
For this one (part iii), you can show your work either by explaining the tree diagram or by showing how you used the intersection rule effectively. Either one will receive full credit, just be sure to write out your probability statements.
Problem 4:
Think about the properties of decks of cards. There are 52 cards, 13 cards per suit, and 4 suits. ALL of the suits are equivalent to one another other than the fact that they are hearts, spades, clubs, and diamonds. If you write a knock knock joke on the last page of your problem set then I will give you five bonus points. So then, numerically, should anything about them be different? Hmmm.....
Problem 5:
Part a: the probability is not 1/6 per each face because the dice is loaded. You're going to have to change the denominator for this one. This question is probably the hardest one on the problem set...
Part B and C: once you have part A figured out, these two should be no problem at all
Part D: Don't let the notation throw you off on this. Just think carefully about what the question is asking you. Just like number one, you're going to need a new denominator for this question, too. Once you find it the rest of the problem should be pretty straightforward.
Good luck!
Conditional Probability
Conditional Probability means that we must have two events - call them event A and event B (or whatever two random variables exist in your problem). Given that one event has already happened, or given that one random variable is present, what is the probability that the other will also happen? Or what is the probability that the other random variable will also exist?
Your AP exam formula sheet gives you the conditional probability theorem like this:
This is read as, "The probability of A, given that B has already happened." This is equal to the probability of A and B (numerator) over the probability of B (denominator), where B is the event that already happened.
We usually use a 2-way contingency table when finding conditional probabilities. Find the marginal totals (row and column totals) first. Then, you will use these totals along with the information within the table to find your conditional probabilities.
Your AP exam formula sheet gives you the conditional probability theorem like this:
We usually use a 2-way contingency table when finding conditional probabilities. Find the marginal totals (row and column totals) first. Then, you will use these totals along with the information within the table to find your conditional probabilities.
Monday, September 23, 2013
Probability Rules
We have five basic probability rules that we must follow when determining the probability of an event or events. Say we have a random event x or two random events x and y. The five basic probability rules are:
1. All probabilities must be between 0 and 1. We can't have anything greater or less, because that doesn't make sense.
0 < P(x) < 1
2. Probabilities that don't have any outcomes in common are said to be DISJOINT or MUTUALLY EXCLUSIVE.
3. Complement probability is the opposite - the probability if it doesn't happen.
P(not x) = 1 - P(x)
4. Multiplication Rule (union): Probability of x AND y happening at the same time.
P(x and y) = P(x) times P(y)
5. Addition Rule (intersection): Probability of x OR y happening, but not both of them at the same time.
P(x or y) = P(x) times P(y) - P(x and y)
1. All probabilities must be between 0 and 1. We can't have anything greater or less, because that doesn't make sense.
0 < P(x) < 1
2. Probabilities that don't have any outcomes in common are said to be DISJOINT or MUTUALLY EXCLUSIVE.
3. Complement probability is the opposite - the probability if it doesn't happen.
P(not x) = 1 - P(x)
4. Multiplication Rule (union): Probability of x AND y happening at the same time.
P(x and y) = P(x) times P(y)
5. Addition Rule (intersection): Probability of x OR y happening, but not both of them at the same time.
P(x or y) = P(x) times P(y) - P(x and y)
Tuesday, September 17, 2013
Stat Test and Hints for Problem Set #3
We took our stat test today. The highest grade was an 84% and it was actually achieved by multiple students - nice job! The lowest grade was a 23%. If you want to ZAP a test that's completely fine, but remember, you have to STUDY first. Good grades in classes (ESPECIALLY college level classes) don't happen by pure luck. Actual effort is involved.
Below are some hints that you might find helpful for the problem set that's due tomorrow:
Problem1:
Part A: Only one answer is correct. The other three are false. Full credit will be provided to those who give an explanation WHY.
Part B: This is a GREAT question - really shows if you understand std. dev. or not. You must pick 4 numbers between 0 and 10. They can be the same, or different. Basically, I want to know that, in order to minimize std. dev (so sx is as small as possible), what should all your numbers be that you're picking? What must be true about these four numbers to make the std. dev. small? Remember, std. dev. is the distance from the mean - the more different your numbers are, the bigger your std. dev. becomes....
The reverse is true for making the std. dev. as large as possible.
Part C: A "resistant measure of spread" means that, if we stuck an outlier into the data set, the spread wouldn't really change much/at all. Recall that the IQR is the middle 50% of the data....
Part D: Remember that means are calculated by adding up all the numbers and dividing by how many there are. That doesn't necessarily mean that there must be equal amounts of men and women in this company.
Problem 2:
Another GREAT question. Think of a graph that you can make with the given information (for both male and female doctors) that would help you to determine the shape of the data. Hint: it's NOT a Normal Curve. Why? Well, just because we have a mean and a std. dev, that doesn't imply normality. EVERY quantitative data set in the world has a mean and a std. dev, but that doesn't imply that EVERY data set is also symmetrical (which the Normal Curve is). So, given all that information, what other graphs can be created?
Problem 3:
Part a: Find the standard deviation using the formula we used in class. There should be an example of this in your notes.
Part b: Remember how to interpret: also in your notes. IN THE CONTEXT OF THE PROBLEM! INCLUDE UNITS FOR FULL CREDIT.
Part c: This one you should be able to figure out on your own, but my advice is to BE SPECIFIC. A one-sentence answer isn't going to cut it here.
If you write I love Math at the top of your problem set tomorrow I will give you 5 bonus points.
Problem 4:
Part a: TITLE, LABELS, AND SCALE
Part b: CUSS
Part c: Think about the shape of your boxplot in regards to what we talked about in class
Part d: Again, be specific. More would change than just the spread of the data, although that's certainly part of it. Remember to tell me WHY along with the WHAT.
Problem 5:
Graphically means make some type of graph that makes sense given the type of data. Numerically means 5-number summary. Summarize your findings means write a paragraph, using SPECIFIC language, units, and such to talk about what you SEE.
Good luck!
Below are some hints that you might find helpful for the problem set that's due tomorrow:
Problem1:
Part A: Only one answer is correct. The other three are false. Full credit will be provided to those who give an explanation WHY.
Part B: This is a GREAT question - really shows if you understand std. dev. or not. You must pick 4 numbers between 0 and 10. They can be the same, or different. Basically, I want to know that, in order to minimize std. dev (so sx is as small as possible), what should all your numbers be that you're picking? What must be true about these four numbers to make the std. dev. small? Remember, std. dev. is the distance from the mean - the more different your numbers are, the bigger your std. dev. becomes....
The reverse is true for making the std. dev. as large as possible.
Part C: A "resistant measure of spread" means that, if we stuck an outlier into the data set, the spread wouldn't really change much/at all. Recall that the IQR is the middle 50% of the data....
Part D: Remember that means are calculated by adding up all the numbers and dividing by how many there are. That doesn't necessarily mean that there must be equal amounts of men and women in this company.
Problem 2:
Another GREAT question. Think of a graph that you can make with the given information (for both male and female doctors) that would help you to determine the shape of the data. Hint: it's NOT a Normal Curve. Why? Well, just because we have a mean and a std. dev, that doesn't imply normality. EVERY quantitative data set in the world has a mean and a std. dev, but that doesn't imply that EVERY data set is also symmetrical (which the Normal Curve is). So, given all that information, what other graphs can be created?
Problem 3:
Part a: Find the standard deviation using the formula we used in class. There should be an example of this in your notes.
Part b: Remember how to interpret: also in your notes. IN THE CONTEXT OF THE PROBLEM! INCLUDE UNITS FOR FULL CREDIT.
Part c: This one you should be able to figure out on your own, but my advice is to BE SPECIFIC. A one-sentence answer isn't going to cut it here.
If you write I love Math at the top of your problem set tomorrow I will give you 5 bonus points.
Problem 4:
Part a: TITLE, LABELS, AND SCALE
Part b: CUSS
Part c: Think about the shape of your boxplot in regards to what we talked about in class
Part d: Again, be specific. More would change than just the spread of the data, although that's certainly part of it. Remember to tell me WHY along with the WHAT.
Problem 5:
Graphically means make some type of graph that makes sense given the type of data. Numerically means 5-number summary. Summarize your findings means write a paragraph, using SPECIFIC language, units, and such to talk about what you SEE.
Good luck!
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