| MA 117 Probability and Statistics Math - Block 4 - 2001 |
| Reading in preparation for Tuesday, Week Two: |
| Chapter Seventeen | |
| Read about the expected value, pp. 288 - 290. The idea is that every chance process has an expected value. Loosely speaking, this is the ‘most typical value which estimates the output of the chance process’. If we toss a fair coin 10 times then the expected number of heads is 5. |
1. Kerrich tossed a coin 10,000 times while imprisoned. Think of this as a chance process, and the output delivered is the number of heads which he tossed.
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| Read about the standard error, pp. 290 - 293. The standard error is a value which measures the amount by which a chance process is likely to be off from the expected value. |
2. What is the standard deviation of the numbers in the box in the box model from question 1? 3. What is the standard error for your box model in question 1? 4. If instead of 10,000 draws we make 1,000,000 draws, does the standard error go up? If so, by how much? 5. How many draws must be made from the box in order that the standard error is twice what it was for 10,000 draws? |
| Read about the normal curve, and how it relates to box models, pp. 294 - 296. |
6. Look ahead to table 3 on p. 302. Kerrich records the results of his coin-tossing in batches of 100.
7. How does your answer to 6 d) relate to the standard error of the box model? 8. Read example 2 about the chances of the house winning more than $250 in Roulette. What is the chance that the house will win more than $600? Read about the shortcut for finding the standard deviation for a list of only two types of numbers, pp. 298 to 299. 9. What is the S.D. of a) 0 0 0 1 1 1 b) 0 0 0 1 1 1 1 1 c) 2 2 6 6 6 6 |
| Review the applications of the sum of the draws from a box model, pp. 299 - 303. Boxes with just 0's and 1's are often used to ‘classify and count’ the occurrences of a particular thing. |
10. You plan to play 100 games of roulette one afternoon, betting $5 on red each game. What numbers go in the box for the box model if you are interested in
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| Chapter Eighteen | |
| This
chapter discusses how the normal curve is applicable in measuring the sums
of the draws in a box model. Read
about probability histograms, pp. 308 - 310.
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1. In figure 1., which of the four histograms represent actual data, and which represent probabilities? 2. Why do the empirical and probability histograms for 10,000 repetitions look the same? Are they really the same? 3. Try to answer exercise 5, exercise set A. Read about the probability histogram and the normal curve, pp. 315 - 316. 4. As the number of draws ____________ the probability histogram looks more and more like the normal curve. (Fill in the missing word: Increases or Decreases). |
| Read about fitting the normal curve to a probability histogram, pp. 317 to 318. Note that since we are dealing with discrete data, the class intervals are centred on the discrete values themselves. |
5. Read example 1. Find the chance of getting
6. Try to answer numbers 1 and 2 from exercise set B. |
| Read about the scope of the normal approximation. We know that after enough draws, the probability histogram for the sum of the draws in a box model will look like the normal curve. How much is enough? One rule of thumb for boxes with just two numbers, is that the number of draws times the smaller proportion is greater than 5. Read pp. 319 - 324. |
7. In figure 5, the smaller proportion in the box is 1/10. How many draws does our rule of thumb tell us to make before the normal curve applies? 8. Look at figure 6. Do you agree with the rule of thumb in this instance?
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