Chapter 17. The. Value Example. The Standard Error. Example The Short Cut. Classifying and Counting. Chapter 17. The.

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1 Context Short Part V Chance Variability and Short Last time, we learned that it can be helpful to take real-life chance processes and turn them into a box model. outcome of the chance process then corresponds to drawing tickets from the box and summing up the numbers on the tickets. Today, we will focus on the sum of from the box. Suppose we draw at random with replacement from a box. sum of is a random output which varies around the expected value, the amounts o being similar is size to the standard error (SE) for the sum. 1 1 Short We make 100 draws at random with replacement from the box What do we expect the sum to be? 9 should come up about every forth time, or 25 times; 1 should come up about three out of four times, or 75 times. sum should thus be around = 300. Short We make 100 draws at random with replacement from the box What do we expect the sum to be? re is another way to compute the expected value of the sum. average of the box is = 3. 4 This is the expected value. On average, each draws will thus add around 3 to the sum. With 100 draws, the sum should be around = 300.

2 expected value of the sum of standard error for the sum of Short expected value for the sum of draws made at random with replacement from a box equals (number of draws) (average of box) Short actual sum will likely be dierent from the expected value. It will be o by the chance error sum = expected value + chance error Does the formula make sense? What happens if the number of draws is doubled? n the expected value of the sum of doubles. What happens if the average of the box is doubled? n the expected value of the sum of doubles. chance error is the amount above (+) or below (-) the expected value. standard error (SE) for the sum tells us how big the chance error is likely to be. standard error for the sum of standard error for the sum of Short A sum is likely to be around its expected value, but to be o by an amount similar in size to the standard error. Short orem ( square root law) When drawing at random with replacement from a box of numbered tickets, the standard error for the sum of is number of draws (SD of the box). To compute the SE for a sum, we use the following law: orem ( square root law) When drawing at random with replacement from a box of numbered tickets, the standard error for the sum of is number of draws (SD of the box). Does the formula make sense? What happens if the number of draws is doubled? n the SE of the sum of is multiplied by a factor 2. chance error grows as we make more draws, but only slowly. What happens if we double the SD of the box? n the SE of the sum of doubles.

3 standard error for the sum of 1 (continued) Short For our chance process, we have the formula observed value = expected value + chance error chance error is likely to be similar in size to the standard error (SE) for the sum of from the corresponding box Large SE Large chance errors Observed values are widely spread around the expected value Short We make 100 draws at random with replacement from the box average of the box is 3. expected value of the sum is = 300. SD of the box is (1 3)2 + (1 3) 2 + (1 3) 2 + (9 3) 2 = Small SE Small chance errors Observed values are tightly clustered around the expected value Observed values are rarely more than 2 or 3 SEs away from the expected value SE for the sum is = 35. Thus, the sum of is likely to be around 300, give or take 35 or so. Short cut for calculating the SE 2 Short When the tickets in the box show only two dierent numbers ('big' and 'small'), the SD of the box is ( big number - small number ) Short We make 25 draws from the box (fraction with big number) (fraction with small number) Fill in the blanks: 1 (continued): SD is 1 (9 1) = sum of is around..., give or take... or so.

4 Another way to look at this... Another way to look at this... Short 3: Say we use the box model to count the number of heads in 100 coin tosses. We repeat the process 1000 times, and get a variety of results: Density Number of heads in 100 coin tosses, repeated times Short number of heads is a random variable, with a distribution center of the distribution is the expected value spread of the distribution is the standard error nr of heads normal approximation 2 (continued) Consider the sum of 25 draws from the box Short If the number of draws is large, we can use the normal approximation to estimate chances. We should use a new average and new SD: New average = expected value for sum of New SD = SE for the sum of new standard units tell us how many SEs a number is away from the expected value Short Density Histogram of sum of, when repeated 1000 times sum of

5 2 (continued) and counting Short We make 25 draws from the box About what percentage of observed values should be between 50 and 100? Short Some chance processes involce counting. How can we set up a box model? 4: A die is tossed 60 times. number of sixes should be around..., give or take... or so. Replace tickets by 0s and 1s Short If we want to count the number of a certain ticket (or tickets), then we put 0 on the tickets that we don't want to count put 1 on the ticket (tickets) that we want to count Using this new box, we have that Short the count is like the sum of from the new box we can compute the expected value and SE as before we can also use the normal curve to approximate probabilities as before

6 Short Summary expected value for the sum of draws made at random with replacement from a box equals (number of draws) (average of box) When drawing at random with replacement from a box of numbered tickets, the standard error for the sum of the draws is number of draws (SD of the box). If the number of draws is large, we can use the normal approximation to estimate chances. If we have to classify and count, put 0's and 1's on the tickets. Mark 1 on the tickets that should count, and 0 on the others.