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Notes_4_with_comments - 18 1.4 The Normal Distribution a...

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Unformatted text preview: 18 1.4 The Normal Distribution a very special CONTINUOUS distribution 0 Sometimes referred to as the Gaussian distribution after Karl F. Gauss. See 10 Deutsche Mark. o Probably the most widely used of all probability distributions. 0 Demonstration: The Quincunx, first illustrated by Sir Galton, illustrates a simple process that gives rise to the familiar "bell curve" of the normal distribution; Modern Day Version: PLINKO Simulation of the Quincunx at http://www.ms.uky.edu/wmaiflava/stat/GaltonMachinehtml Balls are dropped from the top and pass through a series of pins until they hit the bottom Once at the bottom, they stack up to record the numbers that have hit that point At first there does not seem to be any pattern, but after several dozen balls have been dropped, the stacks conform to a bell-shaped curve 0 The final position of each ball is determined by the sum of independent, random events each with probability p = 0.5 of whether to drop to the left or the right of the pin 0 Applications of the normal random variable One of first applications due to Gauss: modeled observational errors in astronomy Measures of reading ability, introversion, job satisfaction, and memory are among the many psychological variables approximately normally distributed 0 Heights, weights, le, Definition: The normal probability distribution is completely specified by two parameters: the mean ,u and the standard deviation 0:. The mean of a normal distribution is its *center*, and the standard deviation measures the *spread* of the distribution about the mean. Standard deviation is always positive. The standard deviation squared is the variance 0' 2. The density function fix) for a normal random distribution with mean ,u and standard deviation O'is: f(x) : flew—”wag” for -00 <x < 00. 19 Is the density function for the normal distribution legit? >assume(sigma, positive); >f :2 l / 2 * exp(el / 2 * (x - mu) A 2 / sigma A 2) * 2 A (l / 2) / sigma / Pi A (1 / 2): >Int(f, x = winfinity .. infinity) = int(f, x = ~infinity infinity); Some graphs of normal distributions with various means and variances. The notation N( y , 0) denotes a normal distribution with mean y and standard deviation 0. Normal with ,u = 0, 0:2: N(O, 2) ViIII'rr’Af-‘q .r—; Normal with ,u=0, 0'=1/4: N(O, 1/4) Normal with #:1, 0’=1:N(1, 1) 20 Example 1. Consider another planet in which adult male heights x are approximately normally distributed with a mean of ,u = 67.5 inches and a standard deviation of a = 2 inches What proportion of men on the planet is between 66 and 72 inches tall? 7: ,1 4‘ (x «~ (1751‘ Z 3 , 7——~—4:——”——dt: 07611481749 (16 0 60 62 64 66 63 7o 72 74 )1: Using Maple to compute the above integral: >f := (1 / (sqrt(2 * Pi) * sigma) * exPth " mil) " 2) / (2 * sigma A 2H): >mu :2 67.5: sigma :2 2: >Int(f, x z 66 .. '72) : int(f, x = 66 .. 72); 7: C (1 7 3 7m 1 11 i lav/2 e\ I — ——,; ~~~~~~~~~~~~~~ dx 2 0.761 1481749 4 4/7: 661 Because the normal density function is not integrable, we use tables and computers (Maple for example) to determine normal probabilities and quantiles. The Standard Normal Random Variable: N(0, 1) The normal table, found in most standard statistics textbooks, relies on the standard normal distribution, denoted by the variable 2. A standard normal distribution is a normal distribution with mean ,u = 0 and standard deviation 0' = 1. The density function for a normal random distribution is: " ,_ I _ 2 e ””2 for-co <x<oo. f(x)=\/§ Appendix Table l and the FRONT INSIDE COVER in our text provides cumulative probabilities for a standard normal random variable. 21 Example 2. Assume 2 comes from standard normal distribution. Sketch the area of interest under the curve also. Suppose that values are repeatedly chosen from a standard normal distribution. (a) In the long run, what proportion of values will be smaller than 1.25? Greater than 1.25? flue (c) In the long run, what proportion of values will be between -0.5 and 3.2, inclusive? Between -1 and 1? ?<’-§O< z c 3-10) : T(7‘£< 3&0) 'T(-Z<-5,§ FYD F 22 Section 1.4 (continued) Working in the other direction with a normal table you know the area to the left (or right) of a normal curve value, what must be the 2 value corresponding to that area? Example 3. What value 2* is such that the area under the standard normal curve to the lefi of z’ is 0.9911? Area in shaded region: 0.9911 ?(Z<2*§ as?” Example 4. What value 2* is such that the area under the standard normal curve to the right of 2* is 0.0495? ?( Z > Z”): .0795 WZGE"): ENDS H Area in shaded region: 0.0495 {* : Le? Standardizing a normal distribution value 0 There are infinitely many different normal distribution tables that could be tabulated for every possible pair of values of ,u and 0 o The good news: Any value x from a normal distribution with known mean [u and standard deviation 0' can be converted to a z-score: o A z-score is simply the number of standard deviations an observation is away from its mean. The standard normal table can be used with the z—score to determine the appropriate proportion of time a variable has a given z—value or less 23 Example 5. Records for the past several years show that the amount of money collected daily by the prominent televangelist Ernie Non—Angley is normally distributed with a mean y = $20,000 and a standard deviation of 0' = $5000. In the long run, what proportion of donations will exceed $30,000? Example 6. Mensa (from the Latin word “mind”) is an international society devoted to intellectual pursuits. Any person who has an IQ in the upper 2% of the general population is eligible to join. Assume that IQs are normally distributed with p = 100 and 0' = 15. What is the lowest IQ that will qualify a person for Mensa membership? :Q ”V NOYMAL (#:Ioo) 0’:1Y) 7, - loo 24 Extra Practice/Board Problems: 1. Assume 2 comes from standard normal distribution. Suppose that values are repeatedly chosen from a standard normal distribution. (a) In the long run, what proportion of values will be smaller than 2 = 0.25? iYZ<z5): (c) In the long run, what proportion of values will be between 2 = —1 .1 and z = 2? ?( ~14 < z< 2-4) 2 “Ma we) ~ ?(—Zc~/./c§:.9¥;2 - ,m ~ 3’ 2. Assume 2 comes from standard normal distribution. @ (a) Determine the value 20 such that the proportion of values to the right of 20 is 0.025. ‘?(?>a) : .02: (b) Determine the value 20 such that the proportion of values between —zo and 20 is 0.95. Mew; mm» ~ W.) 3. The time it takes a cell to divide (called mitosis) is normally distributed with an average time of one hour and a standard deviation of 5 minutes. In the long run, what proportion of dividing times will take more than 65 minutes? XNA/ Cuzco} Utes) “F(>(>(03’) : T(¥> go) ’ t(¥>Ieo) 25 4. The time that it takes a driver to react to the brake lights on a decelerating vehicle is critical in helping to avoid rear-end collisions. The article “Fast-Rise Brake Lamp as a Collision—Prevention Device” (Ergonomic, 1993'; 391—395) suggests that reaction time for an in-traffic response to a brake signal from standard brake lights can be modeled with a normal distribution having mean value 1.25 seconds and standard deviation 0.46 seconds. In the long run, what proportion of reaction times will be between 1.00 and 1.75 seconds? X,~f\/0m4xto( c /.L\’ 6,4317%) )1 3 :?(""”;;'Z‘ 4 2» < L—l “Pew 2c M) , : ,@2/~— L971? ' // 5. The College Boards, which are administered each year to many thousands of high school students are scored so as to yield a mean of 500 and a standard deviation of 100. These scores are close to being normally distributed. 9 (a) An exclusive club wishes to invite those scoring in the top 10% on the College Boards to join. What score is required to be invited to join the club? (b) What score separates the top 60% of the population from the bottom 40%? This value is called the 40th percentile. ...
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