Stats_for_business4_chapt_5_

1 example uniformwaitingtime1 the amount of time x

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Unformatted text preview: ution is d – c and the height is 1/(d – c) The area under the entire uniform distribution is 1 Because width × height = (d – c) × [1/(d – c)] = 1 So P(c ≤ x ≤ d) = 1 10 Example 5.1 Example Uniform Waiting Time #1 The amount of time, x, that a randomly selected hotel The that patron spends waiting for the elevator at dinnertime. patron Record suggests x is uniformly distributed between zero Record and four minutes and So c = 0 and d = 4, So 1 f ( x ) = 4 −0 0 1 = 4 for 0 ≤ x ≤ 4 otherwise b −a b −a P( a ≤ x ≤ b ) = = 4 −0 4 11 Uniform Waiting Time #2 What is the probability that a randomly selected hotel patron What waits at least 2.5 minutes for the elevator? waits The probability is the area under the uniform distribution The in the interval [2.5, 4] minutes in The probability is the area of a rectangle with The height ¼ and base 4 – 2.5 = 1.5 height P(x ≥ 2.5) = P(2.5 ≤ x ≤ 4) = ¼ × 1.5 = 0.375 (2.5 What is the probability that a randomly selected hotel patron waits less than one minutes for the elevator? P(x ≤ 1) = P(0 ≤ x ≤ 1) = ¼ × 1 = 0.25 12 Uniform Waiting Time #3 Expect to wait the mean time µX µX 0 +4 = = 2 minutes 2 with a standard deviation σX of of 4 −0 σX = 12 =1.1547 minutes 13 Section5.3 The Normal Distribution The normal probability distribution is defined by The normal the equation the f( x ) 2 f( x ) = 1 σ 2π e 1 x −µ − 2 σ µ for all values x on the real number line, where µ is the mean and σ is the standard deviation, π= 3.14159, e = 2.71828 is the base of natural logarithms 14 x The normal curve is symmetrical and bell-shaped • The normal is symmetrical about its meanμ • The mean is in the middle under the curve • So μ is also the median • The normal is tallest over its mean μ • The area under the entire normal curve is 1 • The area under either half of the curve is 0.5 15 Properties of the Normal Distribution There is an infinite number of possible normal curves The tails of the normal extend to infinity in both directions The particular shape of any individual normal depends on its specific mean μ and standard deviation σ The tails get closer to the horizontal axis but never touch it mean = median = mode The left and right halves of the curve are mirror images of each other 16 The Position and Shape of the Normal Curve (a) The mean µ positions the peak of the normal curve over the real axis 17 The Position and Shape of the Normal Curve (b) The variance σ2 measures the width or spread of the normal curve 18 Normal Probabilities Suppose x is a normally distributed random variable with mean µ and standard deviation...
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