AP Statistics B March 1,

AP Statistics B warm-ups Thursday, March 1, 2012 You take your car to the mechanic for a steering problem (welcome to adulthood, BTW, and don’t forget to pay for your car insurance). The mechanic says he may be able to fix the problem by putting the car on the jack and whacking it with a hammer, in which case he’ll charge you $50. That is successful about 40% of the time. If it’s not successful, he’ll have to spend $200 for parts and $300 for labor. Use the expected value model to determine the average (probable) costs a driver will incur in this situation. (Solution on the next slide) 2

Answers to AP Statistics B warm-ups Thursday, March 1, Determine the probability model: 40% chance of a $50 repair, or what percentage of the other repair? It has to be 60% (the remainder percentage necessary to add up to 100%) and the $500 in costs ($200 in parts and $300 in labor) 2.Calculate the expected value:.4($50) +.6($500) = $20 + $300 = $320. 3

Comments about the meaning of “expected value” in this problem (audio only) 4

Outline for materials in Chapter 6 1.Vocabulary: 1.Discrete variable 2.Continuous variable 3.Probability model 2.Expected value 3.Using the TI 83+ to calculate expected values 4.Application of the probability model 5

Discrete v. continuous Discrete variables—basic idea is that discrete variables can be counted (i.e., are not infinite) Continuous variables—infinitely many, cannot be counted 6

Examples of discrete variables Height and weight charts for a given population Ages SAT scores (example of when discrete math is restricted to whole number or integer values 7

Examples of continuous variables Take, for example, a simple line: y=x Assume we’re measuring distance travelled, and that the object is travelling at 1 ft/sec Let’s look (on the next slide) at what the graph of that would look like. This will be an example of a continuous variable. 8

Graph of a continuous variable 9

Comparing discrete and continuous Most statistical data is discrete, not continuous. Raw data, even in a normal curve, looks like this: 10

We can approximate the normal histogram by a continuous curve 11

The normal equation can be algebraically manipulated, but it is….well, see for yourself: 12

However, it does produce the z-tables and a graph that we can analyze easily 13

Probability model Fancy term, simple idea “Model” means a theory that predicts outcomes (used in engineering, science, all social science, business, etc.) Per the text (p. 369), a probability is “the collection of all the possible values and the probabilities that they occur” Let’s apply this definition to a couple of models that we’ve already seen 14

Probability model: the lottery example Remember the lottery we did a couple of days ago (I’m picking the first one): – 14 $5-bills – 2 $10-bills – 4 $20-bills The values (i.e., what bill you could draw out of the bag) is $5, $10, and $20. There are no other possible outcomes under the rules of the game Their probabilities are 14/20, 2/20, and 4/20, respectively (aka 0.70, 0.10, and 0.20) Together, these 6 data form the probability model 15

Probability model: the life- insurance/disability model on pp The possible outcomes are death, disability or nothing The values associated with each outcome are $10,000, $5,000, and $0. The probabilities associated with each outcome are, respectively, 1/1,000, 2/1,000, and 997/1,000 Put together we can establish the average value of the policy or the cost to the company. 16

Expected value We reviewed the expected value of a probability model yesterday: E(X)=∑x i ∙P(x i ) What I forgot to mention was something very important, namely that E(X)=μ. And μ is what? It is the mean of the population (pronounced “mu” or “myu“, depending on whether you like to make a bovine sound. This is an extremely important relationship, so let’s explore its implications. 17

Identity of expected value and mean Let’s explore this with the first lottery example we did. Get out your calculators. In one of the lists, enter all the possible outcomes of the lottery example we just did (i.e., enter 14 5’s, 2 10’s, and 4 20’s) Alternatively, if you enjoy busy work, add the following and divide by 20: If you did a list, use the 1-var method to find the mean. 18

Using the TI Tips on p. 370 You will need to enter two lists. The first is 5, 10, 20. I’m going to call it L1, but you can use any name you like as long as you remember it The second list, which I’ll call L2 (same caveat as above), consists of the following 14/20, 2/20, 4/20 (note that the List functions accepts fractions and calculates decimals The next slide will show you how to calculate the expected value (which equals what?) 19

Using VarStats Here, the book is genuinely confusing, though it doesn’t mean to be It looks like it wants you to run VarStats while subtracting it from 1: “ask for 1-VarStats L1, L2” What they really want is different: – Press the STAT button. – Select the CALC menu from along the top – Select the first entry, which is listed as “1-Var Stats” – Put “L1, L2” (or whatever you used) after it and hit the enter key How does this answer compare to the mean you calculated by hand? (should be quite similar….duh) 20

Calculating variance and standard deviation by hand under the expected value model Read pp “First Center, Now Spread….” When you’re done, move on to the next slide. 21

Tedious, but necessary You’re going to have to be able to apply this formula on some of the problems: Var(X)=σ 2 =∑(x-μ) 2 ∙P(X) Yes, you WILL have to subtract the mean from the same entries. Así es la vida…large y dura. Fortunately, calculating the standard deviation (σ) is simply a matter of taking the square root of the variance (Var(X)) 22

Meaning of standard deviation in this context What does standard deviation mean in for the normal distribution? Here, it means something different: a way of evaluating how wild are the variations. 23

Doing the problem on pp Somebody read the problem out loud. Work through the problem. I’ll give you my analysis on the next slide, but keep the tree diagram on p. 371 in mind. 24

Analyzing the problem Confusing nomenclature without explanation: – NN means the client got two new computers – RR means they got 2 refurbished computers – RN and NR are the probabilities that they got one new and one refurbished computer Breaking the tree diagram down Calculating variance on p. 372 Explanation “in context” 25

Homework, due Friday, March 2, 2012 Chapter 16, problems 8, 11, 14, 17, 20, 23 26