Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 314: Statistics Chapter 17: The Expected Value and the Standard Error Dr. Ralph Wojtowicz CME Department Introduction SD Shortcut 1 Introduction 2 SD Shortcut 3 Using the Normal Curve Math 314: Statistics Using the Normal Curve Chapter 17: The Expected Value and the Standard Error 1/4 Introduction SD Shortcut Using the Normal Curve Expected Value and Standard Error The sum of draws made at random with replacement from a box is likely to be around its expected value, but to be off by a chance error similar in size to the standard error. The expected value for the sum of draws made at random with replacement from a box equals EV = (number of draws) × (average of box) The standard error for the sum of draws made at random with replacement from a box equals √ SE = number of draws × (SD of box) The average and SD are properties of the box. The expected value (EV) and standard error (SE) are properties of the random sample. Math 314: Statistics Chapter 17: The Expected Value and the Standard Error 2/4 Introduction SD Shortcut Using the Normal Curve SD Shortcut If a box contains only two different numbers, a big one and a small one (possibly many different times each), the SD of the box equals s fraction with fraction with big small × × − big number small number number number Example: For a box containing only H = 1 and T = 0, the SD is r r 1 1 1 1 × = = (1 − 0) × 2 2 4 2 Example: For a box containing 2 red beads (R = 1) and 8 green beads (G = 0), the SD is r r 2 16 8 4 2 × = = = (1 − 0) × 10 10 100 10 5 Math 314: Statistics Chapter 17: The Expected Value and the Standard Error 3/4 Introduction SD Shortcut Using the Normal Curve Using the Normal Curve We can use (areas under) the normal curve to estimate the probability that the sum of random draws from a box will fall within a given range. To estimate the probability that the sum will be in the interval [a, b]: Let EV be the expected value of the sum and let SE be the standard error of the sum. E . Let z1 = a−SEEV and let z2 = b− SE Calculate the area under the normal curve between z1 and z2 . Math 314: Statistics Chapter 17: The Expected Value and the Standard Error 4/4