1/3/12 AP STATS. WARM UP: How was your New Year? EQ: HW: Pg 381 #1, 2, 3, 6, 9, 10, 17, 18, 24, 25, 31. Chapter

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1 1/3/12 AP STATS WARM UP: How was your New Year? EQ: Name one way you can tell between discrete variables and continuous variables? HW: Pg 381 #1, 2, 3, 6, 9, 10, 17, 18, 24, 25, 31 1

2 Jan. 3, 2012 HW: Pg 381 #1, 2, 3, 6, 9, 10, 17, 18, 24, 25, 31 Random Variable: Ex: EQ: Name one way you can tell between discrete variables and continuous variables? Chapter 16: Day 1 Random Variables In probability, the item that bases its value on the outcome of a random event, experiment, or observation. The total number of heads in a coin toss. The number of customers going through a drive through window. Create your own: 2

3 Two types of random variables: Discrete random variables: random variables that the value can be written in a list or countable. (Finite). Continuous random variables: random variables that are not countable. They can assume values that correspond to any point in one or more intervals. Ex. a.) The number of seizures an epileptic patient has in a given week: x = 0, 1, 2,... Discrete b.) The length of time (in seconds) between arrivals at a hospital clinic: 0 x Continuous c.) The amount (in ounces) of carbonated beverage loaded into a 12 ounce can in a can filling operation: 0 x 12. Continuous d.) The number of errors on an accountant's ledger. x = 0, 1, 2,... Discrete Note: A hint: Most discrete random variables start with "The number of..." 3

4 Probability model: The collection of all the possible values and the probability they will occur. Notation: P(X = x) Ex: Tossing two coins: Probability of at least one toss being heads. P(x = 2) or P(HH) = 1/4 P(x = 1) or P(HT) + P(TH) = P(TH or HT) = 1/2 P(x= 0) or P(TT) = 1/4 In the next following slides we are only looking at discrete random variables only. Expected values: Sample The values we expect to have in the center of our data. Notation: μ or E(x) =Σx P(x) Case 1 x % of occurance + Case 2 x % of occurance +... total total When figuring expected values, all cases must be included. 4

5 Center Example: A small software company bids on two contracts. It anticipates a profit of $50,000 if it gets the larger contract and a profit of $20,000 on the smaller contract. There is a 30% chance of getting the larger contract, a 60% chance of getting the smaller contract, and a 10% of getting neither contract. Assume the contracts are independently given, what is the expected profit? Note: It's independent and it gives us the percentage so the total of 2 contracts have already been accounted for. 50,000 (.30) + 20,000(.60) + 0(.10) = E(x) = $27,000 5

6 Expected value Spread: Remember standard deviation where σ= ([(x1-x) 2 +(x2-x) (xn-x) 2 ]/n) For expected value its the same thing; however it goes step by step...deviation to variance to standard deviation. Notation: Deviation = (X μ) Variation = Var(X) = (x 1 μ) 2 %+ (x 2 μ) 2 %+...+(x n μ) 2 % Also known as σ 2 = Σ(X μ) 2 P(x) Standard deviation = σ(x) = (Var(x)) Spread example: Using the software company information from center example: μ = 27,000 Data: 50,000; 20,000; 0 50,000 27,000 = ( ) 2 x.3 = 20,000 27,000 = ( ) 2 x.6 = 0 27,000 = ( ) 2 x.1 = σ= ( ) = 6

7 1/4/12 AP STATS WARM UP: What is a discrete variable? EQ: What are the differences in calculating discrete expected values and continuous random values? HW: Pg 381 #1, 2, 3, 6, 9, 10, 17, 18, 24, 25, 31 7

8 Jan. 4, 2012 HW: Pg 381 #1, 2, 3, 6, 9, 10, 17, 18, 24, 25, 31 EQ: What are the differences in calculating discrete expected values and continuous random values? Chapter 16: Day 2 Overview of discrete problems: With every other type of stat problem, it wants set up, the math, and a conclusion. Plan: What is the problem? Variable: What is the random variable? Plot: Make a picture or table Model: Use picture or table to determine probability model. Mechanics: Do the math: Find the expected value, variance, and standard deviation. Conclusion: What were the results? What do the results mean? 8

9 The discrete example: The Knowway company shipped 2 computers to an important client. However you found out that the refurbished computers were restocked with the new computers in the store room. There was a total of 15 computers in the store room where 4 were refurbished. If the client gets 2 new computers, things are fine. If they received 1 refurbished computer, the company pays $100. If both computers are refurbished, the company losses $1000. What is the expected value and the standard deviation of the loss? Plan: Find the expected loss for shipping the computers and the standard deviation for the company. Variable: Let X = amount of loss Suggest tree diagram Plot: Model: P(RR)= 0057 P(NR ) = P(NN) = Mechanics: Conclusion: E(X) = $98.90 σ 2 = $51, σ = $

10 Continuous Random Variables: Unlike before, knowing if the situation is independent and follows a normal curve makes an important difference. If its independent and has a normal curve, the sums and differences will follow a normal curve (making calculations easier). Normal curve refresher: -2σ -3σ -1σ +1σ +2σ +3σ 10

11 Continuous Random Variable example: Cereal: the amount of cereal can be poured into a small bowl varies with a mean of 1.5 ounces and a standard deviation of 0.3 ounces. A large bowl holds a mean of 2.5 ounces with a standard deviation of 0.4 ounces. You open a new box of cereal and pour one large and one small bowl. What is the probability of pouring more than a total of 4.5 ounces into both bowls? Plan: We need to figure out the expected value and standard deviation of the total amount of ounces from the two bowls to estimate the probability of getting more than 4.5 ounces. Variable: Condition check: C = the total amount of cereal x 1 = small bowl expected value x 2 = large bowl expected value Is it independent? Does it follow the normal curve? E(C) = 4 σ(c) =.5 Mechanics: z = 1 P(C) =15.87% or 16% Model: Conclusion: 11

12 Remember: We are dealing with theorectical probability, therefore models are only models. Don't assume everything is normal. Is the normality assumption justified? Does it tell you? Watch out for variables that are not independent. Does one depend on the other? Variances add, Standard deviations do not. Use variances as needed. In fact, variances always add...even when looking for differences. Watch your notation, it must represent different cases...not the same case repeated. 12

13 Example 1 13

14 1/5/12 AP STATS WARM UP: Any questions about discrete or continuous variables? EQ: What are the differences in calculating discrete expected values and continuous random values? HW: Pg 381 #1, 2, 3, 6, 9, 10, 17, 18, 24, 25, 31 14

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Chapter 7: Random Variables

Chapter 7: Random Variables 7.1 Discrete and Continuous Random Variables 7.2 Means and Variances of Random Variables 1 Introduction A random variable is a function that associates a unique numerical value