Math 141-copyright Joe Kahlig, 10B Page 1Section 8.5: The Normal DistributionSection 8.6: Applications of the Normal DistributionTo find probability with a continuous random variable,X, we use a probability density function,f(x). This function has the properties: 1) f (x) ≥ 0 for all values of X and 2) the area under the graphof f (x) is equal to 1 (on the values of X).Normal Curve:This distribution is always centered around the mean.The standard deviation regulates how h igh of apeak the curve will have. The formula for this curve is f(x) =1σ√2π∗ e„−(x−µ)22σ2«Shade the area under the normal curve that represents these probabilities.P (X > a)P (c < X < d)P (X = a)Definition: The standard normal curve is the normal curve with µ = 0 and σ = 1. The randomvariable for the standard normal curve is Z.Note: This is the curve that is us ed to create the normal distribution charts found in the back of themath book. These charts will not be usedin this course. To convert to the standard normal curveuse the formula Z =x−µσ.Math 141-copyright Joe Kahlig, 10B Page 2Calculator commandsThe TI-84 has built in command s that we will use to work with normal distributions. They can befound in the Distribution menu by pressing2nd VARS . Note: to enter 1E99 press 1 EE 9 9 .The EE represents scientific notation and theEE is found by pressing 2nd , .normalcdf(lower, upper, µ, σ) computes the probability that a continuous R.V. X is between thelower bound and the upper bou nd.CalculateLower Up perP (X < a) −1E99 aP (X > a)a 1E99P (a < X < b)a binvnorm(area, µ, σ) will return a value A that satisfies the equation P (X < A) = area.Example: Let X b e a continuous r andom variable that is normally distributed with µ = 53 and σ = 14.Compute the following.A) P (45 < X < 58)B) P (X < 56)C) P (X > 38)Example: Compute P (0.5 < Z < 2)Example: The length of what are considered ”one-inch” bolts is found to be a norm ally distributedrandom variable with a mean of 1.001 inches and a standard deviation 0.002 inches. If a bolt measuresmore than two standard deviations from the mean, it is rejected as not meeting factory tolerances.What percentage of the bolts will the factory reject?Math 141-copyright Joe Kahlig, 10B Page 3Example: A particular brand of dishwasher has a life expectancy that is estimated to be n ormallydistributed, with a mean of 10 years, 8 months and a standard deviation of 1 year, 2 months. Supposethat such dishwashers are guaranteed to last 9 years. Of every 250 sold, how many will fail to lastthrough the guarantee period?Example: The weight of infants is normally distributed with a mean of 7.4lbs and a s tandard deviationof 1.2lbs. Fifty infants are selected at random, find the probability that at most 10 of them weighbetween 7 and 8 pounds.Example: Let X b e a normally distributed ran dom variable with µ = 53 and σ = 4.A) Find A such that P (X < A) = 0.356B) Find B such that P (X > B) = 0.525Math 141-copyright Joe Kahlig, 10B Page 4Example: Find J such that P (−J < Z < J) = 0.78Example: A prof is going to grade an exam on a bell curve. The prof decided that the top 10% of thebell curve will be an A, the next 15% a B, the next 30% a C, and the next 20% a D. If the mean forthe class is 70 and the standard deviation is 18, find the lowest grade on the exam that will get youA and the lowest grade on the exam that will get you a
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