Stats CH 6 Intro Activit 1 1. Purpose can ou tell the difference between bottled water and tap water? You will drink water from 3 samples. 1 of these is bottled water.. You must test them in the following order: First: Second: Third: 3. After testing each, circle the letter that ou feel is the bottled water and show the teacher. Do NOT discuss our results with our classmates. 4. Make a chart of the number of people who stated each sample Sample A B C Total Frequenc Relative Frequenc The true bottled water sample is 5. What percent correctl identified the bottled water? 6. Assume that no one can distinguish tap water from bottled water. If this is the case, then what is the probabilit that a person would guess correctl? 7. What would the percentage that can distinguish tap water from bottled water have to be in order for ou to be convinced that the students are not guessing? 8. Perform a simulation of the 3 cups eample to test how man times the correct response was discovered among 50 students. The strings are 10 digits long. There are 5 strings. None of the strings include the digit 0. What digits will ou consider correct: total correct: percent: What digits will ou consider incorrect: total incorrect: percent: 531676879 7813483 9165396471 14816458 898169471 9. Make a histogram for the number of correct responses in class 1
Stats CH 6 Intro Activit 1. Purpose can ou tell the difference between bottled water and tap water? You will drink water from 3 samples. 1 of these is bottled water.. You must test them in the following order: First: Second: Third: 3. After testing each, circle the letter that ou feel is the bottled water and show the teacher. Do NOT discuss our results with our classmates. 4. Make a chart of the number of people who stated each sample Sample A B C Total Frequenc Relative Frequenc The true bottled water sample is 5. What percent correctl identified the bottled water? 6. Assume that no one can distinguish tap water from bottled water. If this is the case, then what is the probabilit that a person would guess correctl? 7. What would the percentage that can distinguish tap water from bottled water have to be in order for ou to be convinced that the students are not guessing? 8. Perform a simulation of the 3 cups eample to test how man times the correct response was discovered among 50 students. The strings are 10 digits long. There are 5 strings. None of the strings include the digit 0. What digits will ou consider correct: total correct: percent: What digits will ou consider incorrect: total incorrect: percent: 8699344889 337396734 7414855135 5434738399 7973697488 9. Make a histogram for the number of correct responses in class
AP STATS Section 6.1: Discrete and Continuous Random Variables Random Variable: A variable, usuall represented b an X, which has a single numerical value (determined b chance) for each outcome of an eperiment. Eamples: 1) X = The number of seniors who get into college earl. ) X = The number of defective tires on a car 3) X = A random number chosen between 0 and 1 4) X = The lifetime of a light bulb. a) Discrete Random Variable: Has a finite or countable number of values. (Eamples 1 and ) b) Continuous Random variable: Has infinitel man values and the values can be associated with a continuous scale so that there are no gaps or interruptions. (Eamples 3 and 4) Probabilit Distribution: Gives the probabilit of each value of a random variable. Eample 1 - Eggs-ample: Suppose the random variable X is the number of broken eggs in a randoml selected carton of one dozen store brand eggs at a certain supermarket. Since the number of broken eggs is a discrete random variable, the probabilit distribution is a list or a table of the possible values of X and the corresponding probabilities Number of Broken Eggs: 0 1 3 4 5 Probabilit:.65.0.08.04.0.01 Requirements for a probabilit distribution (legitimate): 1)!! = 1 sas all the probabilities must add to 1 ) 0 p(x) 1 for all values of Here is a probabilit Histogram that represents the above probabilit distribution: Probabilit Histogram P(X > 3) = Probabilit 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 0 1 3 4 5 Number of Broken Eggs P(X 1) = P(X < ) = P(X 4) = 3
Eample - Tossing Coins: What is the probabilit distribution of the discrete random variable X that counts the number of heads in three tosses of a coin? Number of heads 0 1 3 Probabilit Let s create a probabilit histogram for this distribution. For a continuous random variable we use densit curves, not histograms, to graphicall represent the distribution: Eample 3 - Random Number Eample: X is a random number between 0 and. The distribution is a continuous distribution and is represented b the uniform densit curve below: P(X) 0.5 0 X The probabilit of an event is the area under the densit curve P ( 1 ) = P ( 0 0.5) = P ( 0.3) = 4
If the curve is not uniform, ou would need to use geometr or calculus to find the probabilit of an event And don t forget, the most familiar densit curve is the standard normal curve. The Normal Distribution is a probabilit distribution. The Epected Value of a random variable represents the average value of the outcomes. The epected value is the mean of a probabilit distribution. It is found b finding the value of: Epected Value: E = (!!! ) =! Mean: µ = p() Eample 4 - Looking back at the Probabilit distribution from eample 1, we can calculate, on average, how man broken eggs there will be in a carton: Number of Broken Eggs: 0 1 3 4 5 Probabilit:.65.0.08.04.0.01 This means that on average we can epect that eggs will be broken in a randoml selected carton of eggs. Eample 5 - Getting the Flu: The probabilit that 0, 1,, 3, or 4 people will seek treatment for the flu during an given hour at an emergenc room is show in the probabilit distribution below. 0 1 3 4 P().1.5?.4.06 a) What is the probabilit that people will seek treatment for the flu during an given hour at an emergenc room? b.) What is the probabilit that at least 1 person will be treated for the flu in the net hour? c.) What is the probabilit that 3 or more people will be treated for the flu in the net hour? d.) What is the average number of people that an emergenc room can epect to treat for the flu during an given hour? HW Section 6-1 A: 1,5,7,9,13 5
More Eamples of Epected Value and Games: Epected value can also be used to determine whether or not a particular game is fair and therefore worth plaing. Eample 6 - Patrick offers Christine to pla a dice game whereb he will pa her $6 if she rolls a si, but Christine would have to pa Patrick $1 ever time she rolls a number other than 6. Should Christine agree to pla this game over a long period of time? To find the epected value of a game multipl each paoff b its probabilit and then add. Based on the above definition, the epected value for Christine is: This means that for ever roll of the die, Christine will earn, on average, $ 6 1, which also means that Patrick will lose, on average $ 6 1 for ever roll of the die. If Patrick and Christine were to pla this game 100 times, how much mone (on average) can Christine epect to make? How much can Patrick epect to lose? Fair Game: We sa that a game is fair whenever its epected value is equal to zero. For eample, ou win a dollar ever time ou toss a coin and get heads, and lose a dollar ever time ou get tails. Eample 7 - Zoot Suit Eample: Would ou be willing to pla this game? You pa me $5 to pla. I take a standard deck of cards. I Shuffle the cards well. You pick 1 card at random. The suit of that card becomes our winning suit. You do not put the card back. You pick more cards at random. If those cards are both the winning suit, I will pa ou $500. If the are not, I pa ou nothing. 6
Variance, and standard deviation of a probabilit distribution Variance 1) = ( ) σ µ p ( ) ) σ = [ p( )] µ Standard Deviation 3) σ = [ p( )] µ (the square root of the variance) Note: The mean and standard deviation formulas are in the stat formula packet that ou get on the AP Eample 8 - Find the mean, variance, and standard deviation of the following probabilit distribution: P() 0.3 1.1. 3.1 4.3 Calculator Method: HW Section 6-1 B: 14,18,19,3,5 7
Section 6.: Transforming and Combining Random Variables Review of Chapter : Operations on statistics: Addition of constant a Subtraction of Multiplication of Division of constant a constant a constant a Mean Increase b a Decrease b a Multipl b a Divides b a Median Increase b a Decrease b a Multipl b a Divides b a Standard Deviation No change No change Multipl b a Divides b a Quartiles Increase b a Decrease b a Multipl b a Divides b a IQR No change No change Multipl b a Divides b a Min and Ma Increase b a Decrease b a Multipl b a Divides b a Range No change No change Multipl b a Divides b a Percentiles Increase b a Decrease b a Multipl b a Divides b a Rules for Random Variables: 1. If the value b is multiplied b each value of a random variable: Measures of Center (mean, median, quartiles, percentiles) multipl b b Measures of Spread (range, IQR, standard deviation) multipl b b Shape is NOT changed. If the value b is divided into each value of a random variable: Measures of Center (mean, median, quartiles, percentiles) divide b b Measures of Spread (range, IQR, standard deviation) divide b b Shape is NOT changed 3. If the value b is added to each value of a random variable: Adds b to measures of Center (mean, median, quartiles, percentiles) Measures of Spread (range, IQR, standard deviation) are NOT changed Shape is NOT changed 4. If the value b is subtracted from each value of a random variable: Subtracts b from measures of Center (mean, median, quartiles, percentiles) Measures of Spread (range, IQR, standard deviation) are NOT changed Shape is NOT changed 5. If Y = a + bx is a linear transformation of the random variable X, then The probabilit distribution of Y is the same as X The mean for Y = a + b (mean of X)!! =! +!!! The st. dev. of Y = b (st. dev. of X)!! =!!! 8
Eample 1 Pete owns a Jeep Tour Compan. There must be at least passengers for the trip and no more than 6. The discrete random variable X describes the number of passengers with the following distribution No. of passengers i 3 4 5 6 Probabilit p i 0.15 0.5 0.35 0.0 0.05 a) Find the epected value (mean) and the standard deviation for X. Describe it in contet. Eample Pete charges $150 per passenger. The discrete random variable C is equal the total amount of mone that Pete collects on a randoml selected trip. Therefore if X = passengers, then C = $300. Then C = 150X Total Collected c i 300 450 Probabilit p i 0.15 0.5 0.35 0.0 0.05 b) Complete the values for the total collected in the table above c) Find the epected value (mean) and the standard deviation for C. Describe it in contet. d) How do the answers to c compare to the answer to a? Eample 3 It costs Pete $100 to bu permits, gas, and a ferr pass for each half-da trip. The amount of profit V that Pete makes from the trip is the total amount of mone C that he collects from passengers minus $100. Then V = C 100 If Pete has onl two passengers on the trip (X = ), then C = 300 and V = 00. Total Profit v i 00 350 Probabilit p i 0.15 0.5 0.35 0.0 0.05 e) Complete the values for the total profit in the table above f) Find the epected value (mean) and the standard deviation for V. Describe it in contet. g) How do the answers to F compare to the answer to a and c? 9
Eample 4 At a car dealership, let the discrete random variable X = cars sold on a given da with distribution: Cars sold 0 1 3 Probabilit.3.4..1 1. Find the mean and standard deviation. The owner offers the manager a $500 bonus for each car sold. The manager attempts to attract customers b offering free coffee and donuts at a cost of $75. Define a profit variable and find the mean and standard deviation for profit. HW Section 6- C: 7-30,37, 39-41,43,45 10
Combining Random Variables Activities 11
Rules for means & variances for two random variables Here is an illustration of these important statistical formulas (which are NOT included on our AP Stat Formula packet) If X and Y are an two random variables, then µ + + = µ µ µ = µ µ or if T = X + Y!! =!! +!! If X and Y are independent random variables, then σ = σ + σ σ = σ + σ + or if T = X + Y!!! =!!! +!!! Note: These "nice" rules do not hold for standard deviations Eample 5: Let s sa ou started our own business on weekends. The business can bring ou a profit on Saturda and Sunda based on various (independent) scenarios. The profits and their respective probabilities for Saturda and Sunda are listed below: X= Saturda profit Based on the tables to the left please calculate: µ = p() $10.00 0.1 $50.00 0.5 $100.00 0.4 SUM 1 µ = µ + µ = µ µ = Y= Sunda profit p() $0.00 0. $60.00 0.3 $80.00 0.5 SUM 1 σ = σ = σ + σ = σ = σ = σ + σ = On the AP Formula Page 1
Eample 6: A compan makes packages containing Oreo cookies. The mean of an Oreo cookie is 40 grams with standard deviation = grams. The compan rejects cookies weighing less than 36 gram. In a batch of 0,000 cookies, how man will be discarded? The wrapper for a package containing 5 cookies has a mean of 6 grams with a standard deviation of 1.3 grams. What is the mean weight of the package (cookies and wrapper). What is the standard deviation of the package? Define the random variable A as the average weight of the 5 cookies. Find the mean and standard deviation of A. The Law of Large Numbers: Draw independent observations at random from an population with finite mean µ. Decide how accuratel ou would like to estimate µ. As the number of observations drawn increases, the mean! of the observed values eventuall approaches the mean µ of the population as closel as ou specified and then stas that close. HW Section 6- D: 49,51,57-59,63 13
Section 6.3: The Binomial Distribution There are man eperiments and situations that result in what are called dichotomous responses responses for which there eist two possible choices (True / False, Yes / No, Defective / Non-defective, Male / Female, etc.). A simple eample of such an eperiment is that of tossing a coin, where there are onl two possibilities, Heads or Tails. There are man other tpes of eperiments similar to a coin toss where ou are observing the success or failure of a certain outcome. Such eperiments give us a probabilit distribution called a binomial distribution. Requirements (Use this when asked if a situation is binomial): 1. Binar there can onl be was the event can turn out. Independent events must be independent for each trial 3. Fied there must be a set number of trials 4. Probabilit of success is constant for all trials and the we are counting successes Etra Rule the sample size must be less than 10% of the entire population size. Eample 1: Tom is about to take a five-question true/false quiz for which he is not prepared. He will be guessing on all five questions. What is the probabilit that: 1) He gets all the answers correct? ) He gets all the answers wrong? 3) He gets eactl three answers correct? 4) He will pass the quiz? Solution: First we need to know how man total answer combinations are there. Using the counting rule: = 3 Use c to denote a correct answer and w denote a wrong answer. All the possible combinations of correct and incorrect are shown! The probabilit distribution for this situation is shown below. (X = number of correct responses): p() 0 1/3 1 5/3 10/3 3 10/3 4 5/3 5 1/3 This can be done using the AP Stat Program. 14
The long process used in eample 1 can be avoided b using the notion that this is a binomial probabilit distribution. There is a formula we can use to find the probabilit of an value in a binomial distribution: n! n P( ) = p q for = 0,1,,3,..., n ( n )!! n = # of trials = # of successes among n trials p = probabilit of success in an one trial q = probabilit of failure in an one trial (q = 1 p)! This formula can be written as!! =! =!!!!!!!! where the parenthesis is called n choose k. This is the combination formula for choosing k items from n total items. In combinations, order does NOT matter. Formulas for The binomial distribution: µ = np σ = np(1 p) σ = np(1 p) Eample : Suppose Charlie manages to manipulate a coin in such a wa that it lands on heads with a.7 probabilit and lands on tails with a.3 probabilit. Suppose John then flips the coin 0 times, what is the probabilit that it will land on heads for eactl 13 of the twent flips? On average, how man flips will result in a heads? What is the standard deviation of this binomial distribution? 15
Eample 3: Suppose that the probabilit that an random freshman girl will agree to go with Sam to the senior prom is 0.1. Suppose Sam asks 0 random freshman girls to the prom, what is the probabilit that eactl 1 will sa es? That at least 1 will sa es? On average, how man freshman girls will agree to go with Sam to the prom? What is the standard deviation of this binomial distribution? HW Section 6-3 E: 61,65,66,69, 71,73,75,77 F: 79,81,83,85,87,89 The Geometric Distribution Let s start with a simulation: Roll a die until the number si appears and keep a record of how man rolls it took before the si was obtained. While in a binomial distribution the random variable was the number of successes in a fied number of trials, in a geometric distribution the random variable is the number of trials it takes to achieve a success. Eamples: 1) flip a coin until ou get heads ) Roll a die until ou get a 6 3) Throw darts at a dartboard until ou hit the bull s-ee A geometric distribution must have the following properties: 1) Each trial in the eperiment must have onl two possible outcomes (success or failure) ) The probabilit of success, p, doesn t change from trial to trial 3) The trials in the eperiment are independent 4) The variable of interest is the number of trials required to reach the first success. P(X = n) = (1-p) n-1 p where p = probabilit of success The mean, µ, of a geometric distribution (the average number of times we can epect to repeat the trials before a success occurs) is simpl 1/p where p is the probabilit of success. µ = 1/p The variance of a geometric distribution variance = (1 - p) / p 16
Eample 4: We will take our simulation eample to analze various aspects of the geometric distribution. Lets first find the various probabilities associated with our dice simulation and come up with a probabilit distribution: a. Find the probabilit that a 6 will come up on the first roll. b. Find the probabilit that a 6 will come up on the second roll. P(won t come up on the first roll) = P(will come up on the second roll = c. Find the probabilit that a 6 will come up on the third roll. d. If we proceed in the manner above will come up with the following probabilit distribution: X 1 3 4 5 n P(X) 1/6 5/6 1/6 5/6 5/6 1/6 5/6 5/6 5/6 1/6 5/6 5/6 5/6 5/6 1/6 (5/6) n-1 (1/6) e. How man rolls can we epect, on average, to roll the die before getting the number 6? f. The probabilit that it takes more than n trials to see a success is: g. Find the variance of a geometric distribution h. Find the probabilit that it would take more than 5 rolls for us to get the number 6 HW Section 6-3 G: 93,95,97, 99,101-104 Quiz Review: Complete the AP Prep Questions on pages 409-411 17