Chapter 17: The Binomial Model. Part 4 When to substitute the normal model

Chapter 17: The Binomial Model. Part 4When to substitute the normal modelAP Statistics BOutline of todays classProblems/limitations with the binomial modelAlternatives to the binomial model that are easier to calculateApplications of the alternatives when we have large populations (large n, that is)2

Warmup, Statistics

Write out, but DO NOT CALCULATE getting exactly 247 heads on 500 flips of an unbiased coin using the binomial model.

Answer these among yourselves and agree on an answer. Worked out problem are on the next slides.3

Answer to warm-upWhats the formula we use?

So for this case, the answer would be

Too much to calculate! (If you tried to calculate, please report immediately to the nurse to be examined..)

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The binomial model:LimitationsGreat for small numbers (less than 20)Really REALLY bad for big numbers (500! has about 100,000 digits to it)Complicated calculationsSo, whats a brother to do?What LOOKS like a binomial distribution?Well, lets graph a few and see what we find.5

Binomial model: graphs (1)The graph on the right is a typical example, here where n=21

6Binomial model: graphs (2)Heres a more typical example, one close to what weve worked with (n=15)Here, p=1/3, which means what?That were successful one-third of the time, or one time out of three.Note that the curve is skewedwhich way?

7What do these curves look like?Chapter 6, maybe?Et seq.?They have means and standard deviations?Yes, its our old friend (drum roll please)THE NORMAL DISTRIBUTION!Lets compare the two8

The binomial model:Comparing binomial and normal distributionsBinomial (discrete)

Normal (continuous)

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OK, so theyre alike. How do we get from one to the other?Well, what can you calculate using the binomial formula and what you know about the binomial distribution?Mean?Standard deviation?So take a minute, review your notes, and have somebody write the mean and standard deviation for a binomial distribution!10

The binomial model:

Thats right; here they are:Mean: Standard deviation:

These are REALLY important for you to know cold (i.e., forget some of the lyrics to your favorite song from 7th grade and remember these guys instead)11

The binomial model:converting and to normal values

Compute and using the formulas just listedFor this case we have =n p=500 =250 is a bit more complex:

OK, so now what??????Now we have a mean and a standard deviation and a normal model:12

The binomial/normal model:whats next?Remember the formula?Look back in your notes, and somebody write it on the board.13

The binomial/normal model:z-formulaThats right:

For N(250, 11.18), what does this look like?Y is 247Mu is 250Sigma is 11.18. Except whats the problem with this analysis? Discuss among yourselves and come to a conclusion14

The binomial/normal model:z-formulaThe problem is that this measures the areas to the LEFT of y, right?So we dont get an exact calculation.How can we approximate?15

The binomial/normal model:z-formulaWe approximate by calculating an interval.Example: use y=246.5 for one calculation, and y= 247.5 for the second, and subtract the first from the secondNot precise, but very VERY close.Most problems are easier. Lets look at the one in the book.16

Practice (Red Cross example):The Binomial HorrorP. 394 of the text: Red Cross looking for 0-negative donors. p=0.06 and n=32,000. Red Cross needs at least 1850 pints (each person donates 1 pint; any more and you get really dizzy, not to say ill)

Using the binary distribution, you have to calculate each y from 1850 through 32,000. That is, youre going to have 32,000-1850+1 calculations=30,151 calculations that look like this:17

Practice (Red Cross example):Normal curve to the rescueHere, the normal curve really saves usMean is np, or 32,0000.06=1920Standard deviation is

So now we can apply our z model and get

Our tables or calculators tell us that only about 0.05 lie below this point, and thats your answer.

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Practice:How to apply this procedure

If n>9, you can use the normal model.CalculateAdopt normal model as

Apply normal formula (y-)/ and solve as usual. Warning: be sure you know whether youre looking above or below y

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Practice:Chapter 17, problem 22 (p. 399)Have someone read the problemWork together at your tables to solve the problems. Everybody needs to be involved; no free riders (pace Hickman, although in my experience most libertarians dont believe in free riders whereas lawyers always do)Share answers among yourselves and reach a consensusWhen youre ready, move on to next slide for my analysis20

Practice:Problem 17(a)

This is a standard plug-and-chug problemWe know thatHere, n=200, p=0.8, and q= 0.2So =160 and =5.66 21

Practice:Problem 17(b)

No problem with using the normal model here. Why? (you can find the answer in the text on p. 394. If nobody knows, find it and read it now!)22

Practice:Problem 17(c)

This is a common question you should expect on any statistics exam. Its also easy to answer.Take the mean, and add and subtract one standard deviation. Repeat for two- and three-standard deviations.In notation, calculate , 2, and 3These correspond to the 68-95-99.7 modelNumerical answers:68% of the time she gets between 154.34 and 165.66 bulls eyes95% of the time between 148.68 and 171.32 bulls eyes, and 99.7% of the time between 143.02 and 176.98 bulls eyes 23

Practice:Problem 17(d)

Use the answers from (c) to answer this question.We know that 99.7% of the time, the archer will get between 143.02 and 176.98 bulls eyesIf she gets 140 bulls eyes or less, thats only 0.0015%......VERY rare.One more slide for today 24Homework:

Chapter 17, Exercises 24, 26, 27, 34 and 36.Read Chapters 18 and 19.we will be going REALLY fast from now on.(Yes, I know: eez now time for fiendish plan! which is from Boris Badenov of Rocky and Bullwinkle, which I expect you to become familiar with)

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