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a Jitim We are 95% sure that the true percentage is between 6.2 -.03 = 6.17% and 6.2 + .03 = 6.23%. Amazingly, the error formula does not depend on the size of the total population. A poll of 400fi people has a margin of error no more than &= & = 5%, whether the 400 people are from a city of 100,000 or a country of 250 million. The catch is that the sample must be truly random -very hard to do when dealing with people. 8. Grades on the English placement exam at Absorbine Junior College are normally distributed. The mean is p = 72 and the standard deviation is o = 10. The top 15% are placed in Advanced English. What test grade is the cutoff? The distribution of grades looks like Figure 8.12a in the text, except that p = 72 and a = 10. Figure 8.12b gives the area under the curve from m to x. From the graph we see that 84% of the scores are at or below p +o = 82. Therefore 16% are above, close to the desired 15%. The cutoff is a little over 82. (Statisticians have detailed tables for the area F(x).) Read-through8 and eelected even-numbered solutions: Discrete probability uses counting, continuous probability uses calculus. The function ~(x)is the probability bdensity. The chance that a random variable falls between a and b is $a p(x)dx. The total probability is 00S-_ p(x)dx = 1.In the discrete case p, = 1.The mean (or expected value) is p = $xp(x)dx in the continuous case and p = C np, in the discrete case. The Poisson distribution with mean X has p, = Xne-A/n!. The sum xp,= 1comes from the exponential series. The exponential distribution has p(x) = e-% or 2eWZxor ae-=. The standard Gaussian (or normal) distribution has a p ( x ) = e - ~ ' / ~ .Its graph is the well-known bell-shaped curve. The chance that the variable falls below x is F(x) = SLp(x)dx. F is the cumulative density function. The difference F ( x + dx) -F(z)is about p(x)dx, which is the chance that X is between x and x + dx. The variance, which measures the spread around p, is o2 = $ (x-p)2p(x)dx in the continuous case and 2o2 = C (n -p) P n in the discrete case. Its square root o is the standard deviation. The normal distribution 2 2has p(x) = e-IX-') /2u 1 6 0 .If 52 is the average of N samples from any population with mean p and
8.5 Masses and Moments (page 340) variance 02, the Law of Averages says that X will approach the mean p. The Central Limit Theorem says that the distribution for W approaches a normal distribution. Its mean is p and its variance is 0 2 / ~ . 1In a yes-no poll when the voters are 50-50, the mean for one voter is p = 0(;) + I(;) = Z.The variance is (0 -p)2po + (1 -p)2pl = 4.For a poll with N = 100,Z is */a is a 95% chance that X (the 11fraction saying yes) will be between p -25 = 3 -m and p +25 = 2 The probability of an odd X = 1,3,5,- -. is ? + f + $+ a.s . = -t1-3 = $. The probabilities p, = ($)" do not add to 1. They add to $ + b + ... = so the adjusted p, = 2(i)" add to 1. 12 p = I,"ze-"dz = uv -/ v du = -ze-"IF + $7e-"dz = 1. 20 (a) Heads and tails are still equally likely. (b) The coin is still fair so the expected fraction of heads during 1 1the second N tosses is $ and the expected fraction overall is 3(a+ 3); which is the average.