1. Review

2. Where are we? AP Exam covers 4 broad topics:
Exploring Data (chapters 1-3) 20% to 30% of the exam
Sampling and Experimentation (chapter 4) 10% to 15% of the exam
Anticipating Patterns (chapters 5-7) 20% to 30% of the exam
Statistical Inference (chapters 8-12) 30% to 40% of the exam
We are right in the middle of the third section
For most people, this is also the hardest section
Hang in there

3. Even Earlier than Chapter 6…
Let’s review some of the free response questions from the final exam
People generally did quite well on the multiple choice
There was a LARGE spread on the free response
Switch to Word Document

4. 6.1 Discrete and Continuous Random Variables
Let’s start with discrete random variables
Remember, discrete means that there are a fixed number of possible outcomes
Tossing a coin
Picking a card from a deck
Guessing a person’s eye color (either correct or incorrect)
What block/period someone has math class
Rolling a die
Etc.

5. Discrete Random Variables
Let’s use the example of rolling a die
So we say X is the number of dots that are showing on the die
The possible outcomes are:
Assuming that it is a fair die, what is the probability of each of these outcomes?

6. Probability Distribution of Rolling a Die
Value 1 2 3 4 5 6
Probability
1/6

7. Probability Distribution of rolling two dice
Possible outcomes:
But now, they don’t all have the same probabilities
To find the probability distribution, we would find all possible outcomes
There will be 36 different combinations between the two dice (6 on each die, so 6x6)
Then we count how many of these outcomes have a sum of 2, a sum of 3, etc.

8. Probability Distribution of Rolling Two Dice
Possible outcomes:
But now, they don’t all have the same probabilities
Value 2 3 4 5 6 7 8 9 10 11 12
Prob
1/36
1/18
1/12
1/9
5/36
1/6

9. Mean (Expected Value) of a Discrete Random Variable
Finding the mean of your discrete random variable is easy
And, it is on your formula sheet for the AP exam
You just have to know what you’re looking for
To find the mean, you multiply each value by its corresponding probability
Then you add them all together

10. Mean of Discrete Random Variable
So, in our example of rolling one die…
We would do (1)(1/6)+(2)(1/6)+(3)(1/6)+(4)(1/6)+(5)(1/6)+(6)(1/6)
So your mean is 3.5
Or, the expected value is 3.5
Value 1 2 3 4 5 6
Prob
1/6

11. Mean of Discrete Random Variable
So, in our example of rolling one die…
So your mean (expected value) is 3.5
What about the median?
Value 1 2 3 4 5 6
Prob
1/6

12. Mean of Discrete Random Variable
So, in our example of rolling one die…
So your mean (expected value) is 3.5
What about the median?
Half above it, half below it. In this case, that would also be 3.5
They do not have to be the same
Value 1 2 3 4 5 6
Prob
1/6

13. Using Your Calculator for Discrete Random Variables
Fortunately, we can make our calculator do some of the work for us
If you enter the values into List1, and your probabilities into List2
Then we will use the command 1-Var Stats L1,L2
Value 1 2 3 4 5 6
Prob
1/6

14. Let’s Try a Practice Problem
The table below shows the # of TVs per household in the United States, with the corresponding probabilities
Calculate the expected (mean) number of TVs in US households
What is the median of this distribution?
Compare the median and the mean. What does this tell you about the shape of this distribution? 1 2 3 4 5
.012
.319
.374
.191
.076
.028

15. Let’s Try a Practice Problem
The table below shows the # of TVs per household in the United States, with the corresponding probabilities
Calculate the expected (mean) number of TVs in US households 2.084
What is the median of this distribution? 2
Compare the median and the mean. What does this tell you about the shape of this distribution? Mean is slightly larger—skewed slightly to the right 1 2 3 4 5
.012
.319
.374
.191
.076
.028

16. Continuous Random Variables
Continuous random variables have an infinite number of possible values
Often occurs with measuring
Whether or not we intentionally round, most measurements are not exact
If you weigh yourself every morning, the measurements are likely to vary from day to day, even if only by .07 pounds
Because there are an infinite number of possible values, the probability of any particular outcome is infinitely small
Or, in other words, the probability is 0

17. Continuous Random Variables
Instead, we calculate the probability that the value is within a certain range
We typically do this using a normal distribution
Example: in the United States, the mean height for men is 69.1 inches, with a standard deviation of 2.9 inches (assume that the distribution is approximately normal)
We can then calculate the probability of being in a certain range
Using either our standard normal table or our calculator
I recommend using your calculator, but if you prefer the table, that is fine

18. Continuous Random Variables
Example: in the United States, the mean height for men is 69.1 inches, with a standard deviation of 2.9 inches
We can then calculate the probability of being in a certain range
Using either Table A or our calculator
I recommend using your calculator, but if you prefer the table, that is fine
So if we want to find the probability of being between 70 and 72 inches:
Normalcdf(70,72,69.1,2.9)
Convert 70 and 72 to z-scores: and 1
Then use the Table A to find area to the left of each
Then do area to the left of 1 minus the area to the left of .31

19. 6.2 Transforming Random Variables
X is our random variable (discrete or continuous)
If Y=a+bX, this means that we are adding a constant (a) to X and/or multiplying X by a constant (b)
The main thing that we are interested in here is how this changes the mean and the standard deviation

20. 6.2 Transforming Random Variables
If Y=a+bX, this means that we are adding a constant (a) to X and/or multiplying X by a constant (b)
The main thing that we are interested in here is how this changes the mean and the standard deviation
New mean=a+b(old mean)
Or, in other words, our new mean is our old mean times b, plus a
New Stdev=|b|(Old stdev)
Our new standard deviation is our old standard deviation times the absolute value of b

21. An Example
The mean on the semester 1 final exam was (true story)
The standard deviation was 17.68
If I decided that my typical curve was too complicated, and I wanted to do a curve just with adding and multiplying, then we could use this rule
Let’s say that I multiplied all scores by 1.2 and then added 6
So score of 60 would become a 78
Our new mean would be 6+1.2(62.51)
81.01
Our new standard deviation would be |1.2|*17.68
21.22

22. 6.2 Combining Random Variables
When we combine 2 random variables, we can either add them or subtract them
When we add X+Y, the new mean is just mean(X)+ mean(Y)
When we subtract X-Y, the new mean is just mean(X)-mean(Y)
When we add X+Y or subtract X-Y, the new VARIANCE is equal to variance(X)+variance(Y)
WE MUST USE VARIANCES, NOT STANDARD DEVIATIONS
Remember, variance is equal to standard deviation squared

23. An Example
X is a random variable that specifies the number of students with an A in one class
Mean 5.1, standard deviation of 4.2
Y is a random variable that specifies the number of students with an A in a different class
Mean 10.3, standard deviation of 3.4

24. An Example
X is a random variable that specifies the number of students with an A in one class
Mean 5.1, standard deviation of 4.2
Y is a random variable that specifies the number of students with an A in a different class
Mean 10.3, standard deviation of 3.4
If we want to know the expected number of A’s in the two classes combined, we would add X+Y
So the mean would be ____________
Standard deviation would be _____________

25. An Example
X is a random variable that specifies the number of students with an A in one class
Mean 5.1, standard deviation of 4.2
Y is a random variable that specifies the number of students with an A in a different class
Mean 10.3, standard deviation of 3.4
If we want to know the expected number of A’s in the two classes combined, we would add X+Y
So the mean would be =
See next slide for standard deviation

26. An Example Finding the standard deviation So the new variance is 29.2
X: mean= st.dev=4.2
Y: mean= st.dev=3.4
Variance of X is 4.2^2 = 17.64
Variance of Y is 3.4^2 =
So the new variance is 29.2
To find the new standard deviation we take the square root of 29.2
5.4
So our new variable has a mean of 15.4 and a standard deviation of 5.4

27. Another Example (An AP-type question)
3 runners are running a relay. Each runs 1 mile—the team’s overall time is the sum of the 3 runners’ times (in minutes)
Runner 1 thinks that he can run faster than 4.4 minutes in the next race. What is the probability of this?
What are the mean and standard deviation of the overall team time?
The team’s best time ever is 12.9 minutes. What is the probability that they beat this time in their next race?
Runner
Mean time
St. dev
Runner 1
4.5
.15
Runner 2 5 .2
Runner 3
4.2 .1

28. Answers
.252
mean=13.7, st.dev=.269
(less than 1% chance)