Math 166, Spring 2012,cBenjamin Aurispa3.4 The Normal DistributionAll of the probability distributions we have found so far have been for finite random variables. (We coulduse rectangles in a histogram.)A probability distribution for a continuous random variable cannot be written in a table. I t takes the formof a function called a probability density function.A probability density function f has the following properties:1. f (x) ≥ 0 for all values of x.2. The total area of the graph between f and the x -axis is equal to 1.Finding probabilities for a continuous random variable amounts to finding the area under the graph of thep.d.f. between the given boundaries.A very important continuous probability distribution is the normal distribution.The grap h of a normal distribution is a bell-shaped curve, known as the normal curve.Properties of the Normal Curve:1. The curve has a peak at x = µ.2. The curve is symmetric about the vertical line x = µ.3. The area un der the curve is 1. (which is true for any probability distribution.)4. The curve always lies above the x -axis. It never crosses or touches it.5. The standard deviation, σ, determines the flatness of the cu rve. (The larger σ is, the flatter the curveis.)The normal cur ve that has µ = 0 and σ = 1 is called the standard normal curve.The random variable corresponding to th e standard normal curve is called the standard normal variableand is denoted by Z.1Math 166, Spring 2012,cBenjamin AurispaSince the normal distribution is so common, the calculator has a built-in feature to calculate probabilites.This feature is normalcdf and is found by pressing 2nd VARS and then choosing option 2:normalcdf.The values that this command needs arenormalcdf(left boun d, right bound, µ, σ)If there is no left bound, we traditionally use −1 E 99. (The E is typed on your calculator as 2nd, “comma”.)If there is no right bound, we use 1 E 99.Examples:1. P (−0.89 < Z < 1.65)2. P (Z < 0.68)3. P (Z > 1.11)4. P (Z ≥ 1.11)2Math 166, Spring 2012,cBenjamin AurispaIf, however, you are given the prob ab ility or area and are asked to find the boundary value of Z that givesyou this area. For these types of problems, we use the invNorm f unction on the calculator. This function isoption 3 after pressing 2nd VARS.In order to use the invNorm command , you must find th e total area to the LEFT of the boundary you aresolving for.In order to find a boundary value a, the values that this command needs are:a = invNorm(total area to the left of a, µ, σ)Examples: Find the value of a that satisfies the following probabilities.1. P (Z < a) = 0.95232. P (Z > a) = 0.82893. P (−a < Z < a) = 0.73453Math 166, Spring 2012,cBenjamin AurispaWe can also use the above commands to calculate probabilities and boundary values for normal variablesthat are not standard normal.Applications:1. According to data for a certain city, the weekly earnings of workers are normally distributed with amean of $700 and a stand ard deviation of $60.(a) What is the probability that a worker selected at random from the city makes more than $805?(b) What minimum weekly earnings would put you in the top 15% of wage earn er s?(c) What symmetric interval of wages about the mean comprises 64% of wage earners?2. A computer manufacturer has data that the amount of time a computer lasts is normally distributedwith a mean of 25 months and a standard deviation of 6 month s.(a) Find the probability that a computer made by this manufacturer lasts between 1 and 2 years.4Math 166, Spring 2012,cBenjamin Aurispa(b) What compu ter life-length corresponds to the 90th percentile?(The nth percentile means that n% of all data is below this value.)3. The scores on a math test are normally distributed with a mean of 70 and a variance of 100. If theinstructor wants to assign A’s to 15%, B’s to 25%, C’s to 40%, D’s to 15%, and F’s to 5% of the class,find(a) the cutoff grade for an A.(b) the cutoff grade for a B.(c) the cutoff grade to pass (getting a D or
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