Oct. 4, 1997
Hi AP Statistics colleagues:
This represents an attempt to explain what I find to be an interesting project.
Let me first state that we (at Cate) use BASIC PRACTICE OF STATISTICS by David
Moore as our main text. I like the book as I believe it is easily readable.
It does need to be supplemented a bit for AP, especially in the area of
probability. As a result, I introduce my students to probability concepts,
including nCr, very early in the year and we work with probability throughout
the year. Doing this allows me to get them thinking about probability
distributions early in the year. For instance, here is a project which we will
do next week. (I'm going to try to explain this briefly).
Premise:
A plane has 350 passenger seats.
Research shows 94% of ticket purchasers show up.
Airline decides to sell 370 tickets.
What is the probability that more than 350 purchasers will show up?
Using a binomial model, the probabiliy that n ticket purchasers will show up is
(370Cn)(.94^n)(.06^(370-n))
Probabilites for various value of n are shown below. (Probabilities not shown
round to 0.00%)
Note: An efficient way to calculate the probabilities on TI-83 is the
following:
360->X: (370 nCr X)*.94^X*.06^(370-x)
push ENTER, get probability, then push ENTRY (2nd ENTER) and change the value
stored in X.
n prob
363 0.01%
362 0.03%
371 0.07%
360 0.15%
359 0.31%
358 0.60%
357 1.05%
356 1.71%
355 2.60%
354 3.68%
353 4.89%
352 6.12%
351 7.23%
350 8.10%
349 8.82%
348 8.73%
347 8.43%
346 7.78%
345 6.87%
344 5.82%
343 4.73%
342 3.70%
341 2.79%
340 2.02%
339 1.42%
338 0.96%
337 0.63%
336 0.40%
335 0.24%
334 0.14%
333 0.08%
332 0.05%
331 0.03%
330 0.01%
329 0.01%
Plotting these probabilities, one gets a nice bell-shaped curve. If a
histogram is constucted with the no space between the columns (bars), then one
has a density curve (as defined by Moore) if one considers the base of each bar
to be one unit. It is then possible to answer questions such as "what is the
probability that more than 350 ticket purchasers show up?" by simply finding an
area under a portion of the density curve.
If you use a spreadsheet with the data above, you can really get a neat
computer-generated curve. (bar graph).
This, I believe, leads nicely into the normal distribution and the associated
density curve. I think it also helps students to understand how probability is
related to area when they use the normal distribution curve and table.
Using the figures above, the probability of n being 351 or greater is 28.45%,
or approximately 28%.
If you use the normal curve (with "largeness," binomial approaches normal), the
probability that N is greater than 350.5 turns out to be 27.76%, or
approximately 28%.
[The binomial mean would be 370(.94) = 347.8 and the binomial standard
deviation wold be the square root of the product 370(.94).06) , or 4.56815061.