Download Unit 4A Schedule

DatStat UNIT 4 A Schedule (Topics 14 -15)
Day 1:
________
Day 2:
________
Day 3:
________
Day 4:
________
Day 5:
________
Journal U4.1
Topic 14: Probability
SWBAT: Think of probability as long-term chance via simulations, use
rules of probability Activity: Rock, Paper, Scissors; Crazy 8’s
HW1: p. 318 #14-6, 8, 9
Journal U4.2
SWBAT: Calculate the Expected Value, µX , of a Probability Distribution
Activity: Probability of Roulette
Journal U4.3
SWBAT: Identify when probabilities satisfy the Binomial Setting and see
that Binomial Probabilities can be approximated with a Normal
Distribution; Use binomialCDF and PDF to calculate probabilities
Activity: Lebrizzle Jizzle I
Journal U4.4
Topic 15: The Normal Distribution
SWBAT: Use Normal Curves as a Mathematical Model to calculate
probabilities and find values. Use the 68-95-99.7% Rule to estimate
proportions under Normal Curve. Activity: Plain M&M’s®
HW 2: p. 337 #15-4, 5, 7, 8, 9
Journal U4.5
Topic 15: The Normal Distribution
SWBAT: Use NormalCDF to find Normal Proportions and InvNorm to
find values when given a proportions
Activity: Lebrizzle Jizzle II
Day 6:
Journal U4.6
REVIEW DAY
Day 7:
Test UNIT 4A (Topics 14 and 15)
_______
Activity 14-1 (The Student’s Worst Nightmare): Ever show up to school and forgot
that there was a quiz that day? OH CRUD! Thankfully, the quiz is a True/False format
of only 4 questions. You have absolutely no idea how to do it so you will be guessing on
all 4 questions. Good Luck!
Statistics Pop Quiz (True or False)
1) r2 can be interpreted as the percent of variation in y that is explained by x. _______
2) The t-critical value at 10 df for 95% confidence is 2.023. _______
3) A Chi-Squared value is calculated by: (observed – expected)/expected . _______
4) When performing a significance test, we always assume that the Null Hypothesis is
correct before we begin our test procedure. ________
A = You guess the question correct
B = You guess the question INcorrect
Probability:_____________________________________________________________
Notation: P(A) = “The probability of A”
a) P(A) = ___________
P(B) = ____________
b) Guess: What do you think is the probability of getting all questions correct? _______
c) If you are guessing, does your response to one question influence the response to the
next?
INDEPENDENT EVENTS: _______________________________________________
d) Rules of Probability: For any INDEPENDENT events A and B…
1) 0 ≤ P(A) ≤ 1
( Probabilities are percents between 0% and 100%)
2) P(A and B) = ___________________________ (Is it easier or more difficult)
3) P(A or B) = _____________________________ (Is it easier or more difficult)
4) P(not A ) = ______________________________(Anything but A)
5) Sum of all possible probabilities = _________________
e) How many combinations of guesses are there for 4 T/F questions? _____________
Multpilication Principle: ________________________________________________
f) Tree Diagrams: ______________________________________________________
Make a Tree Diagram for The first three questions of our Quiz:
g) Well it is time to grade your quizzes… What could possibly happen?
Random Variable (X): ___________________________________________________
Let X = # of Correct Responses (or # of A’s) . What values can X take? _____________
#
1
2
3
4
5
6
7
8
Right or Wrong (A or B)
X
#
9
10
11
12
13
14
15
16
Right or Wrong (A or B)
X
h) For each part of h: state what they mean in English and find the “blue collar Prob”.
P( X = 0) = ____________________________________________________________
P( X = 1) = ____________________________________________________________
P( X = 2) = ____________________________________________________________
P( X = 3) = ____________________________________________________________
P( X = 4) = ____________________________________________________________
The “White Collar” Solution:
P( X = 3)
=
4(.5)3(.5)1
Probability Distribution of X: _____________________________________________
----------------------------------------------------------------------------------------------------------X=
P(X) =
----------------------------------------------------------------------------------------------------------i) On average, how many do you expect to guess correct? If you took this type of test
1,000 times, what would you expect to be your average score? Guess! _______________
Expected Value: ________________________________________________________
Find the Expected Number of correct responses on this 4 question test.
j) Binomial Distribution:

______________________________________________________

______________________________________________________

______________________________________________________

______________________________________________________
Does our “pop quiz” scenario fit a Binomial distribution?

______________________________________________________

______________________________________________________

______________________________________________________

______________________________________________________
k) Your calculator can handle these Binomial Distibutions! 2nd VARS  binompdf
**binompdf ( n , p , X )
n = trials, p = prob. Of success, X = # of successes
Try these with your calculator. Write what they mean in English first!
P( X = 2 ) = _____________________________________________________________
P( X = 4 ) = _____________________________________________________________
*What about P( X ≤ 3 ) ? How could you handle this? Try binomCdf! C= Cumulative
or everything at or below.
P( X ≤ 3 ) = _____________________________________________________________
P( X < 3 ) = _____________________________________________________________
P( X > 2 ) = _____________________________________________________________
P( X ≥ 2 ) = _____________________________________________________________
Now brainstorm with your how you would handle: P ( 1 ≤ X ≤ 3) “the probability that
you guess between 1 and 3 correct”. Hint: what don’t we want?
P ( 1 ≤ X ≤ 3) = ____________________________________________________
Example) Charles is a 60% free throw shooter. In the next game, he will shoot 10 free
throws.
Is this a Binomial Situation?_________________________________________________
What is the probability he makes 7 free throws? _________________________________
What is the probability he makes at most 7 free throws? __________________________
What is the probability he makes at least 3 free throws? __________________________
What is the probability he makes between 2 and 7 free throws inclusivley? ___________
Rock-Paper-Scissors
Rock-Pa per-Scissors (Ro-Sham-Bo) is a popular method of settling disputes by chance.
In this activity, you will learn more about the game of Rock-Paper-Scissors via a
simulation and the Rules of Probability. Let’s review how to play: Each round, both
players throw a rock, paper, or scissors hand symbol SIMULTANEOUSLY. If both
players throw the same symbol, then the round is scored a tie. Otherwise, rock crushes
scissors, scissors cuts paper, and paper covers rock.
1. Play 10 Rounds of Rock Paper Scissors with an opponents and record your results
in the Table Below:
Round
My Throw
Opponents Throw
Result
1
2
3
4
5
6
7
8
9
10
2. Who won the most? How many rounds did they WIN? ________
3. What “strategy” did you use? Were your throws random?
4. Coach Auer will pool the results of the class to see who won the most rounds, and
more importantly, how many:______ He/she is the “BIG WINNER”.
5. How would you simulate a game of Rock-Paper-Scissors? Work with your
partner to design a simulation of a round. Then, carry out ten simulations. (Hint:
Assign digits to outcomes then use Rand Int…)
How many rounds of ten did THE CLASS PLAY? Did anyone have ____ wins?
6. Do you think the BIG WINNER was using strategy or random throws? Explain:
__________________________________________________________________
7. Create a Box Diagram and a Tree Diagram of a round of Rock-Paper-Scissors.
Assign wins losses and ties to each outcome.
8. Suppose that each player is randomly selecting R,P, or S to throw.
P(win) = _____
P(loss) = _____
P(tie) = _____
P(winc) = P(not a win) = _____
P(Player A wins first 2 rounds) = ________________________
P(10 wins in a row) = ________________________
P(The same player wins all 10 rounds) = ________________________
P( Player A wins 9 out of 10 rounds) = ________________________
P(there is a winner in the first round) = ________________________
P(there is no winner in the first round) = ________________________
P(there is no winner until the third round) = ________________________
Challenge: P(BIG WINNER wins ____ out of 10) = _____________________
How Many Times Do You Go To The Well?
We are going to play a fun card game called Crazy 8’s in groups of 4. Each player is
dealt 8 cards with one card from the rest of the deck turned over to play on. The player to
the left of the dealer begins by playing one card on top of the exposed card and the rest
proceed clockwise. You may play any card in that same suit or number. For example, if
the Queen of Spades is showing, you may play any Queen or any Spade. 8’s are wild and
can be played on any card. If you play an 8, you announce the suit you want to be played
(Hearts, Diamonds, Clubs, or Spades). If you can’t make a play, then you “go to the
well” until you pick up a playable card. The goal of the game is to get rid of all your
cards.The Big Question: If a player has to go to the well, what is
the most likely amount of cards a player must pick up until he/she
can make a play?
Guess:________
1) Make a Dot plot as you play which charts the amount of times all players go to the
well.
1 2
3
4
5
6
7 8 9 10 11 12 13 14 15 16
Which number was most common?
________ Why do you think this happened?
2) Suppose the 3 of hearts has been played, how many cards are playable?
3) So what is the probability of making a play on any given turn?
_________
______________
4) What is the probability of not making a play on any given turn? ______________
5) Using the answers from #3 and #4, make a Tree Diagram which accounts for the
probabilities of going to the well up to 5 times. Call S= playable card, F = not playable.
For the following probabilities, show expressions and then the decimal calculations.
6) So what is the probability of going to the well once? ________________________
7) So what is the probability of going to the well twice? ________________________
8) So what is the probability of going to the well 5 times? ________________________
9) Make a statistical argument that explains why player would most likely have to go to
the well only once.
Le Roulette Wheel
A Roulette wheel has 38 numbered slots: 0, 00, 1, 2, 3, …,35, 36. The ODD numbers
are RED while the EVEN numbers are BLACK. Both 0 and 00 are GREEN.
a. You can simulate a Roulette Wheel with RandInt ( 1 , 38 ). We will let 0 = 37
and 00 = 38. Have one partner play dealer (operate the calculator) while the other
guesses Red(odd) or Black(even). Perform 20 simulations then switch jobs.
Name:
Wins
Losses
b. Suppose you bet 1$ per game. If you guess correct, then you win $1 plus your $1
back. If you guess wrong, you lose your $1. per play. Complete the probability
distribution for X.
X
P(X)
c. Find the Expected amount gained per play.
d. Suppose you play 20 games. Find the Expected amount gained after 20 plays.
e. You can also bet on any number you want (even 0 and 00). What is your favorite
number on the Roulette Wheel?_____
f. If you bet that number, you get paid $35 to every $1 bet. Let X = the amount
gained on the play of a single number for $1. Complete the probability
distribution for X.
X
P(X)
g. Find the expected amount gained per single number play.
h. Calculate the Standard Deviation of the amount gained per single number play.
i. Now imagine that you are a high roller who will play $100 on your favorite
number. Let X be the amount gained per single number play of $100.
X
P(X)
g. Calculate the expected amount gained per play.
h. Does the Casino welcome or try to turn away these “high rollers”? Why?
Binomial Probability: An Introduction into Inference
Lebrizzle Jizzle, a famous NBA baller, loves to attack the rack! In plain
language, he likes to dribble the ball to the basket to try to score. In doing so, he often
gets fouled by his opponents and shoots plenty of free throws. At the free throw line, he
is a 90% free throw shooter. Lebrizzle will shoot 3 free throws. Let X = the number of
free throws he makes out of 3.
1. When Lebrizzle shoots his 3 free throws, how many combinations of misses and
makes are there?________
2. Make a Probability Distribution for X:
---------------------------------------------------------------------------------------X=
P(X)=
-----------------------------------------------------------------------------------------3. What does P( X = 2) mean in English? _____________________________________
4. P( X = 2 ) = 3(.9)2(.1)1
What does each piece represent?
3 =______________________________(.9)2 = ______________ (.1)1= ___________
5. In a game he could shoot 15 free throws! How many combinations of make of makes
and misses are there? _________________
Do you want to make a tree diagram for this situation? _______ (you put “no” here!)
Fortunately for us, Lebrizzle’s free throws adhere to a very common probability
distribution called a Binomial Distribution.

Only 2 things can happen :

There is a fixed number of n trials : ____15 free throws____

There is a fixed probability of a success, p: ________________

Trials are INDEPENDENT: ____________________________
___________________________
6. We will work through the probability of Lebrizzle makes exactly 10 out of 15 free
throws. P(X = 10 )
What is an expression for the probability that Lebrizzle makes his first 10 and misses the
final 5 ?
__________________________________
How many ways can you arrange 10 makes and 5 misses, you ask? __A LOT!__
There is a formula:
nCk
= “n choose k” =
n!
k!(n  k)!
**This is the number of ways to have k successes out of n trials

Ex) 5! = 5x4x3x2x1
7. So n = _______ and
k = ______ . 15C10 =
8. Put your answer to #6 and #7 together to find P(X = 10) =_______________________
The Binomial Probability Formula: nCk pk( 1 – p)n-k
9.
P( X = 12 ) = ______________________________
P( X = 14 ) = ______________________________
10. Your Calculator can handle these probabilities: 2nd VARS  binompdf
**binompdf ( n , p , X )
n = trials, p = prob. Of success, X = # of successes
Try these with your calculator. Write what you put in the calc as well as the decimal.
P( X = 12 ) = ______________________________
P( X = 14 ) = ______________________________
*What about P( X ≤ 10 ) ? How could you handle this? Try binomCdf! C=
Cumulative or everything at or below.
P( X ≤ 12 ) = ______________________________
P( X < 12 ) = ______________________________
P( X > 10 ) = ______________________________
P( X ≥ 10 ) = ______________________________
Now brainstorm with your group how you would handle: P ( 8 ≤ X ≤ 10) “the probability
that Lebrizzle makes between 8 and 10 free throws inclusively”. Hint: what don’t we
want?
P ( 8 ≤ X ≤ 10) = ____________________________________________________
Lets look at Lebrizzle’s free throw over an entire season of 82 games. If he averages 15
free throws per game, how many free throws will he shoot in a season? n = __________
How many free throws do we expect Lebrizzle to make for the year? ______________
**The Mean for a Binomial Distibution: µ = np
Standard Deviation for a Binomial Distribution:   np(1 p)
Find the Mean Number of free throws made in a season: _________________

Find the Standard Deviation for Lebrizzle: _____________________
11. We are going to build a Probability Histogram for Lebrizzle. Recall that a
probability histogram merely has our random variable for the X axis and the frequency is
P(X).
Set up your STAT PLOT 1 for a Histogram: Xlist: List 1
Define List 1: seq( X, X, 1050, 1150 , 5 )
Freq: List 2
**2nd STAT  OPS  SEQ(
Define List 2: binomCdf(1230, .9 , L1+ 5 ) – binomCdf(1230, .9 , L1 )
out what this is…
**Try to figure
Window: (1050, 1150 , 5 , 0 , .25 , 1 , 1 )
Make a quick Sketch of what you see:
Does it look familiar? Like what? ____________________________________________
If you said it looks Normal, then you are right! Will this happen all the time? Not
necessarily. There are two things that allow for a Binomial Distribution to be
approximated by a Normal Distribution:

The number of trials (sample size) must be large. Do we have this? ________

The probability, p , must be close to .5 . Do we have this? ________
Rule of Thumb: A Binomial Distribution can be approximated with a Normal
Distribution if:
np ≥ 10 and n( 1 – p ) ≥ 10
Now check to see that the Rule of Thumb Applies to Lebrizzle’s Season:
Find the following probabilities in 2 different ways, first using a Normal
approximation, then using binomcdf: Hint: DRAW A PICTURE!
P( X ≤ 1100 ) = ______________________________
P( X ≥ 1120 ) = ______________________________
P( 1097 ≤ X ≤ 1117) = ____________________________
Are Plain M&M’s® the same as Normal M&M’s®
Mars Company makes the famous candy M&M’s®. They advertise on each bag that you
will receive 47.9 g. Do you expect more? Exactly 47.9 g.? Less? If you got less, would
you feel cheated? Would you have a case that holds up in court? Lets investigate.
Color
Brown
Yellow
Orange
Blue
Green
Red
Total
Count
Proportion
1) We will make a dotplot to chart 2 things: Total M&M’s® and % of Brown M&M’s®.
Total M&M’s®: ( n = _____)
50 51 52 53 54 55 56 57 58 59 60 61 62
Proportion of Brown M&M’s®: ( n =_____)
.07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20
2) How would you describe the SHAPE of the TOTAL’s distribution?
______________________
3) What portion of bags had more than 55 M&M’s®? ________________
4) What portion of bags had between 54 and 58 M&M’s ® ? ___________
5) What portion of bags had at least18% Brown M&M’s®? ________________
6) Between what two counts hold the middle 95% of the M&M’s ®? ______________
7) One single M&M® weighs .89 g. How much does your bag weigh? _______ Do you
think that the Mars Company® would declare the average weight of a bag as their stated
weight of 47.9 g.? Would they declare less? More? Explain.
Challenge: One single M&M® weighs .89g. What portion of bags has less than the
47.9 g. on the package? Did anyone in the class get cheated?
Topic 15: The Normal Distribution (Bell Curves)
So far, we have seen that x ’s and pˆ ’s behave in a way that is very predictable. We
repeatedly see a single peaked, symmetric, bell type shape when we examine the
distribution of samples of x ’s and pˆ ’s.THIS IS NO FLUKE! IN FACT, IT’S NORMAL!


Normal Distribution: ___________________________________________


Notation:
The mean is the center of the curve and the standard deviation is the distance to the
inflection point, or point where the graph changes concavity, from the mean.
The Empirical Rule: 68/95/99.7%: For normal, symmetric data, the Empirical rule
says that 68% of the data is within 1 standard deviation from the mean. 95% of the data
is within 2 std. devs. from the mean. 99.7% of the data is within 3 std. devs. from the
mean.
Activity 15-1: Gestational periods (how long a baby stays in the womb) in days are
distributed N(266,16).
a) Between what 2 values do the
b) How short are the shortest 16%?
middle 95% fall?
c) How long are the longest 2.5%?
2.1: Percentiles Congratulations! You scored in the 90th percentile on your test!
What does that mean? Does it mean you got 90 out of 100?
Percentile: ______________________________________________________
Ex) One standard IQ test is distributed: N( 150 , 30 )
What percentile is a score of 120?
What percentile is a score of 210?
How high of a score must you receive to reach the 84th percentile?
Standard Normal Calculations: Working with Z-Scores
When Distributions are considered Normal, we can generalize them, or standardize them
in terms of units. Ex) Is it fair to compare what you guys get paid verses what your
parents at your age? Is it fair to compare ball players from different eras?
Standardizing Z-scores:
Standard Normal Distribution: ____________________________________________
** A z-score tells us how many standard deviations the original observation falls away
from the mean. A positive z- score is above the mean while a negative z-score is below
the mean.
Ex) The heights of young women are distributed normally with µ = 64.5 in. and  = 2.5
in.
1. Find the z - score for a woman who is 5
foot 11. What does this mean? (71 in.)
2. What is the z - score 
of a woman
who is 62 inches? In what percentile
does her height fall?
3. A woman's height is -1.3  below the mean. How tall is she?

Finding Normal Proportions: Working with Intervals
What if we want to know the percent of observations in an interval under the normal
curve, or the probability of falling into that interval?
Ex) What portion of women are between 64 and 68 inches tall?
N ( 64.5 , 2.5)
Guess:________
**ALWAYS DRAW A PICTURE!!
Normalcdf: ____________________________________________________________
Hit 2nd DISTR (above VARS) down to #2, normalcdf, or normal cummulative
distribution function.
The format is normalcdf ( min , max , mean, std. dev.)
**According to your calcs, Infinity = E99
Ex1) What portion of women are less
than 68 inches tall? N ( 64.5 , 2.5)
Guess:_______
Ex2) What portion of women are greater
than 66 inches tall? N ( 64.5 , 2.5)
Guess:______
Ex3) GPAs at MBHS are distributed Normally with a mean of 2.07 and a standard
deviation of .65 . If you selected a student at random, what is the probability that they
have a GPA above a 3.0?
If you selected 2 students at random, what is the probability that they BOTH have a GPA
below a 3.0? (You should not use your calculator here!)
If you selected a student at random, what is the probability that they have a GPA between
1.8 and 2.2?
Finding a value given a Proportion.
SAT Verbal scores for one year were distributed normally with N( 505 , 110). We want
to know the minimum score that a student must receive in order to place in the top 10%
of all students.
a. **ALWAYS DRAW A PICTURE!!
b. InvNorm: ____________________________________________________________
*Hit 2nd DISTR (above VARS) down to #3, invNorm, or inverse of NormalCDF.
**The format is invNorm(Percent toLEFT, Mean, Std.Dev)
c. The UC will give bonus point to your application if you score above 85% of your
fellow applicants. What score do you need for the bonus points?
d. One student scored in the bottom Quartile. How many Standard Deviations is he
away from the mean?
Ex1 ;-) GPAs at MBHS are distributed Normally with a mean of 2.07 and a standard
deviation of .65 . What GPA will put you into the top 95% of your class?
Ex2) What GPA will put you into the 84th percentile? Do this problem with and without
your calculators!
Ex3) What GPA will put you above HALF of your school?