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7.2 Notes Central Limit Theorem Theorem 7.1 – Let x be a normal distribution with mean μ and standard deviation σ. Let x be a sample mean corresponding to random sample of size n. http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/ Ex. 1 Pinedale fishing pond has trout whose lengths are normally distributed with mean μ = 10.2 in. and standard deviation σ = 1.4 in. a) What is the probability that a single trout chosen at random is between 8 and 12 in.? b) What is the probability that x of 5 trout chosen at random is between 8 and 12 in.? c) Are the two values the same? Why or why not? Central Limit Theorem – If x possesses any distribution with mean μ and standard deviation σ, then the sample mean x based on random sample size of n will have a distribution that approaches normal with x and x n as n approaches . ∞ The essential question is “How large does n have to be to approximate normal?” http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/ Ex. 2 Let x be the bacteria count per ml of raw milk with μ = 2500 and σ = 300. A health inspector takes 42 random samples of the raw milk produced each day. At the end of each day, the bacteria count in 42 samples is averaged to obtain x . a) What can we say about x distribution? b) What is the probability that the average bacteria count for one day x is between 2350 and 2650 bacteria per ml? c) What is the probability that the average bacteria count for one day x is greater than 2650? *Ex. 3 Taxi and takeoff time for commercial jets has μ = 8.5 min. and σ = 2.5 min. Assume the jets are lined up on a runway so that one taxis and takes off immediately after the other. What is the probability that for 36 jets on a given runway, the total taxi and takeoff time will be less than 320 min.? Assignment Day 1 p. 306 #4, 9, 11 , 15, 19 Day 2 p. 306 # 6, 8, 12, 13, 16, 20