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Name: _______________________ Class: __________ Date:_____________ Math SL: 24C Z-Distribution Review: 1. Given that sin B = 2 π and B , find cos B and cos 2B 3 2 2. From January to September, the mean number of car accidents per month was 630. From October to December, the mean was 810 accidents per month. What was the mean number of car accidents per month for the whole year? 1 24 C Standard Normal Distributions Today’s Objectives: (1) to understand how the standard normal distribution can be found by transforming any normal distribution (2) to find probabilities for a standard normal distribution Standard Normal Distribution A standard normal distribution (Z-distribution) has m = 0 and s = 1. Thus we can write . The random variable Z is equivalent to the number of standard deviations from the mean. Finding Probabilities using a Graphics Calculator To find P(Z £ a) or P(Z < a) use normalcdf(-E99,upper bound) To find P(Z ³ a) or P(Z > a) use normalcdf(lower bound,E99) To find P(a £ Z £ b) or P(a < Z < b) use normalcdf(lower bound,upper bound) Example 1: Given that find a) P(-2 < Z < 1) b) P(Z < 1) c) P(Z > -1.5) d) P(Z < 0) Standardizing Any Normal Distribution Every normal X-distribution can be transformed into the standard normal distribution or Z-distribution using the transformation: z= x-m s If x is an observation from a normal distribution with mean m and standard deviation s , the Z-score of x is the number of standard deviations x is from the mean. For example: If z1 = 1.34 then z1 is 1.34 standard deviations to the right of the mean If z2 = -1.92 then z2 is 1.92 standard deviations to the left of the mean 2 A z-score can also be used to compare x-scores from two different normal distributions. Steps for converting an x value into a z-score: (1) Convert x-values to z-values using z = x-m s (2) Sketch the standard normal curve and shade the required region. (3) Use your calculator to find the probability. Example 2: A random variable X is normally distributed with mean 70 and standard deviation 4. By converting to the standard variable Z and then using your calculator, find P( X £ 68) . Example 3: The length L of a nail is normally distributed with mean 50.2 mm and standard deviation 0.93 mm. Find Z to determine P(L ³ 48). **Note** Your curriculum no longer requires the tables that are found in your text book. All normal distribution work from todays lesson must be done on calculator or the calculations must result in integer multiples of the standard deviation. i.e. 68%, 95%, 99.7%. 3 Hmwk#56: 24C The standard normal distribution pg. 657 # 1 – 3, 5, 6 pg. 658 # 1 – 3(a,c) USE GRAPHING CALCULATOR USE GRAPHING CALCULATOR and always draw a picture where possible. By now, most of you should have an idea of what you want to explore. Take this opportunity to look for relevant math… You do NOT want to start your exploration from scratch next year during college research, college applications, IB papers, SATs, IGCSEs, extracurricular activities, regular SAS school demands, etc… 4