AP Statistics B

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AP Statistics BMarch 1, 20121AP Statistics B warm-upsThursday, March 1, 2012You take your car to the mechanic for a steering problem (welcome to adulthood, BTW, and dont 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 hell charge you $50. That is successful about 40% of the time. If its not successful, hell 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)

2Answers to AP Statistics B warm-upsThursday, March 1, 2012Determine 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)Calculate the expected value: .4($50) + .6($500) = $20 + $300 = $320.3Comments about the meaning of expected value in this problem(audio only)

4Outline for materials in Chapter 16Vocabulary:Discrete variableContinuous variableProbability modelExpected valueUsing the TI 83+ to calculate expected valuesApplication of the probability model and calculation of variance

5Discrete v. continuousDiscrete variablesbasic idea is that discrete variables can be counted (i.e., are not infinite)Continuous variablesinfinitely many, cannot be counted

6Examples of discrete variablesHeight and weight charts for a given populationAgesSAT scores (example of when discrete math is restricted to whole number or integer values

7Examples of continuous variablesTake, for example, a simple line: y=xAssume were measuring distance travelled, and that the object is travelling at 1 ft/secLets look (on the next slide) at what the graph of that would look like.This will be an example of a continuous variable.

8Graph of a continuous variable

9Comparing discrete and continuousMost statistical data is discrete, not continuous. Raw data, even in a normal curve, looks like this:

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We can approximate the normal histogram by a continuous curve

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

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

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Probability modelFancy term, simple ideaModel 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 occurLets apply this definition to a couple of models that weve already seen14

Probability model: the lottery exampleRemember the lottery we did a couple of days ago (Im picking the first one):14 $5-bills2 $10-bills4 $20-billsThe 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 gameTheir 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 model15

Probability model: the life-insurance/disability model on pp. 369-70The possible outcomes are death, disability or nothingThe 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 997/1,000Put together we can establish the average value of the policy or the cost to the company.16

Expected valueWe reviewed the expected value of a probability model yesterday: E(X)=xiP(xi) What I forgot to mention was something very important, namely that E(X)=.And 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 lets explore its implications.

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Identity of expected value and meanLets 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 5s, 2 10s, and 4 20s)Alternatively, if you enjoy busy work, add the following and divide by 20: 5+5+5+5+5+5+5+5+5+5+5+5+5+5+10+10+20+20+20+20If you did a list, use the 1-var method to find the mean.18

Using the TI Tips on p. 372You will need to enter two lists. The first is 5, 10, 20. Im going to call it L1, but you can use any name you like as long as you remember itThe second list, which Ill 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 VarStatsHere, the book is genuinely confusing, though it doesnt mean to beIt looks like it wants you to run VarStats while subtracting it from 1: ask for 1-VarStats L1, L2What they really want is different:Press the STAT button.Select the CALC menu from along the topSelect the first entry, which is listed as 1-Var StatsPut L1, L2 (or whatever you used) after it and hit the enter keyHow 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 modelRead pp. 370-71 First Center, Now Spread.When youre done, move on to the next slide.21

Tedious, but necessaryYoure going to have to be able to apply this formula on some of the problems:Var(X)=2=(x-)2P(X)Yes, you WILL have to subtract the mean from the same entries. As es la vidalarge y dura.Fortunately, calculating the standard deviation () is simply a matter of taking the square root of the variance (Var(X))22

Meaning of standard deviation in this contextWhat 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. 371-72Somebody read the problem out loud.Work through the problem.Ill give you my analysis on the next slide, but keep the tree diagram on p. 371 in mind.24

Analyzing the problemConfusing nomenclature without explanation:NN means the client got two new computersRR means they got 2 refurbished computersRN and NR are the probabilities that they got one new and one refurbished computerBreaking the tree diagram downCalculating variance on p. 372Explanation in context25

Homework, due Friday, March 2, 2012Chapter 16, problems 8, 11, 14, 17, 20, 2326