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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?