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Density Curves • • • • Can be created by smoothing histograms ALWAYS on or above the horizontal axis Has an area of exactly one underneath it Describes the proportion of observations that fall within a range of values • Is often a description of the overall distribution • Uses μ & σ to represent the mean & standard deviation z score • Standardized score • Creates the standard normal density curve • Has μ = 0 & σ = 1 z x What do these z scores mean? -2.3 1.8 6.1 -4.3 2.3 σ below the mean 1.8 σ above the mean 6.1 σ above the mean 4.3 σ below the mean Jonathan wants to work at Utopia Landfill. He must take a test to see if he is qualified for the job. The test has a normal distribution with μ = 45 and σ = 3.6. In order to qualify for the job, a person can not score lower than 2.5 standard deviations below the mean. Jonathan scores 35 on this test. Does he get the job? No, he scored 2.78 SD below the mean At least what percent of Chebyshev’s Rule observations is within 2 standard deviations of the meanpercentage for any shape of observations • The distribution? 75% that are within k standard deviations of the mean is at least 100 1 1 % k 2 where k > 1 • can be used with any distribution Chebyshev’s Rule- what to know • Can be used with any shape distribution • Gives an “At least . . .” estimate • For 2 standard deviations – at least 75% Normal Curve • Bell-shaped, symmetrical curve • Transition points between cupping upward & downward occur at μ + σ and μ – σ • As the standard deviation increases, the curve flattens & spreads • As the standard deviation decreases, the curve gets taller & thinner Empirical Rule • Approximately 68% of the Can ONLY be used observations are within 1σ of μ with normal curves! • Approximately 95% of the observations are within 2σ of μ • Approximately 99.7% of the observations are within 3σ of μ The height of male students at WHS is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. a) What percent of the male students are shorter than 66 inches? About 2.5% b) Taller than 73.5 inches? About 16% c) Between 66 & 73.5 inches?About 81.5% Remember the bicycle problem? Assume that the phases are independent and are normal distributions. What percent of the total setup times will be more than 44.96 minutes? First, find the mean & standard deviation for the total setup time. Phase Mean SD Unpacking Assembly Tuning 3.5 21.8 12.3 0.7 2.4 2.7 2.5%