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Continuous Distributions Notes Normal Distributions Continuous random variables – Density curves: Unusual density curves: Do these two normal curves have the same mean? If so, what is it? Which normal curve has a standard deviation of 3? Which normal curve has a standard deviation of 1? Example: 1) P(x<2) = 2) P(X=2) = Empirical Rule: .5 .25 3) P(X<2) = 4) P(x > 3) = 1 2 3 4 5 Suppose that the height of male students at PWSH is normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. What is the probability that the height of a randomly selected male student is more than 73.5 inches? 5) P(1 < x < 3) = 1) P(X > 1) = 0.50 Standard Normal Density Curves 2) P(0.5 < x < 1.5) = 0.25 1 2 3 4 Strategies for finding probabilities or proportions in normal distributions The lifetime of a certain type of battery is normally distributed with a mean of 200 hours and a standard deviation of 15 hours. What proportion of these batteries can be expected to last less than 220 hours? What proportion of these batteries can be expected to last more than 220 hours? How long must a battery last to be in the top 5%? The heights of the female students at PWSH are normally distributed with a mean of 65 inches. What is the standard deviation of this distribution if 18.5% of the female students are shorter than 63 inches? The heights of female teachers at PWSH are normally distributed with mean of 65.5 inches and standard deviation of 2.25 inches. The heights of male teachers are normally distributed with mean of 70 inches and standard deviation of 2.5 inches. Describe the distribution of differences of heights (male – female) teachers. What is the probability that a randomly selected male teacher is shorter than a randomly selected female teacher? Ways to Assess Normality Normal Probability Plot 4) What do you notice about the shape? Normal distributions can be used to estimate probabilities for binomial distributions when: 1) the probability of success is close to .5 or Are these approximately normally distributed? 50 48 54 47 51 52 46 53 52 51 48 48 54 55 57 45 53 50 47 49 50 56 53 52 2) n is sufficiently large Rule: if n is large enough, then np > 10 & n(1 –p) > 10 Since a continuous distribution is used to estimate the probabilities of a discrete distribution, a continuity correction is used to make the discrete values similar to continuous values. ( 0.5 to the discrete values) 5) Use a normal distribution with the binomial mean and standard deviation above to estimate the probability that between 15 & 30 preemies, inclusive, are born in the 250 randomly selected babies. Normal Approximation to the Binomial Premature babies are those born more than 3 weeks early. Newsweek (May 16, 1988) reported that 10% of the live births in the U.S. are premature. Suppose that 250 live births are randomly selected and that the number X of the “preemies” is determined. What is the probability that there are between 15 and 30 preemies, inclusive? (POD, p. 422) 1) Find this probability using the binomial distribution. 2) What is the mean and standard deviation of the above distribution? 3) If we were to graph a histogram for the above binomial distribution, what shape do you think it will have? Binomial written as correction) P(15 < X < 30) Normal (w/continuity P(14.5 < X < 30.5) = 6) How does the answer in question 6 compare to the answer in question 1? Homework: A) What is the probability that less than 20 preemies are born out of the 250 babies? B) What is the probability that at least 30 preemies are born out of the 250 babies? C) What is the probability that less than 35 preemies but more than 20 preemies are born out of the 250 babies? Homework: A) What is the probability that less than 20 preemies are born out of the 250 babies? P(X < 20) P(X < 19.5) = .1231 B) What is the probability that at least 30 preemies are born out of the 250 babies? P(X > 30) P(X > 29.5) = .1714 C) What is the probability that less than 35 preemies but more than 20 preemies are born out of the 250 babies? P(20 < X < 35) P(20.5 < X < 34.5) = .8060