Math Tech IIII, May 7 The Normal Probability Models Book Sections: 5.1, 5.2, & 5.3 Essential Questions: How can I use the normal distribution to compute probability? Standards: S.ID.4
Properties of the Normal Distribution A normal distribution with mean and standard deviation has the following properties: 1. The mean, median, and mode are equal, and are in the middle of the distribution. 2. The normal curve is bell-shaped and is symmetric about the mean. 3. The total area under the normal curve is equal to 1. 4. The normal curve approaches, but never touches, the x-axis. 5. Between μ σ and μ + σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of μ σ and to the right of μ + σ. The points at which the curve changes are called inflection points.
Properties of the Normal Distribution
The Standard Normal Distribution A normal distribution with a mean of 0 and a standard deviation of 1
Back to the Future We shall now recall the computation of the Z-Score from Chapter 2 (mid February, 2015) when we first considered the normal distribution. A normal distribution can have any mean and any positive standard deviation. These two parameters, μ and σ, determine the shape of the normal curve.
The Standard Score The standard score, or z-score (z) is a number that is representative of the number of standard deviations a given value of x falls from the mean. z = (Value Mean) Standard Deviation or z ( ) x Value is the entry you want the z-score of (x).
Properties of the Z-Score A z-score can be positive, negative, or zero Positive the x-value was greater than the mean Negative the x-value was less than the mean Zero the x-value was equal to the mean A z-score is how many standard deviations a value is away from the mean. Positive above the mean, negative below the mean.
Finding Z-Scores In a normal distribution with a mean of 7 and a standard deviation of 3.2, what is the Z-Score of 13.4?
The Standard Normal Distribution There are an infinite number of normal distributions, but only one Standard Normal Distribution. The standard normal distribution is a normal distribution with a mean, of 0 and a standard deviation, of 1. Any distribution can be transformed here by the Z-Score.
How The Calculator Works There are two choices in computing calculator probability 1. Use z-scores (two arguments) 2. Use the distribution values (four arguments) Number 2 is the easiest way, and the way we will mostly do it Either way, you are using the normalcdf distribution [2 nd ][Vars] select 2 gets you this function
Normal Probabilities There are three probability types that we need to know, and how we start them, no matter what: x is less than a value: P(x < a) Arguments: LB, a x is greater than a value: P(x > b) Arguments: b, UB x is between a value: P(c < x < d) Arguments: c,d LB: - (in theory) practical inputs Textbook recommends -10,000 Manual recommends -E99 = -1 x 10 99 Practical value = -1000 UB: (in theory) practical inputs Textbook recommends 10,000 Manual recommends E99 = 1 x 10 99 Practical value = 1000
Normal Probability Models P(x < a) = area left of a (probability model a) P(x > b) = area right of b (probability model b) P(c < x < d) = the area between c and d (probability model c)
The Normal Distribution in Texas 1. To use the graphing calculator to compute probabilities (or areas) using the normal distribution. Use the normalcdf function (2 nd )(Vars)(2) This function can be used in many ways: With any distribution, using 4 arguments; LB, UB, μ, σ
With Each Model (a) P(x < a) = area left of a (probability model a) Calculator Computation: normalcdf(-1000, a, μ, σ The words: less than or fewer than
With Each Model (b) P(x > b) = area right of b (probability model b) Calculator Computation : normalcdf(b, 1000, μ, σ The words: more than or greater than
With Each Model (c) P(c < x < d) = the area between c and d (probability model c) Calculator Computation : normalcdf(c, d, μ, σ The words: between
Example 1 The weights of adult male beagles are normally distributed with a mean of 25 pounds and a standard deviation of 3 pounds. A beagle is randomly selected. Compute each probability: a) That the beagle s weight is less than 20 pounds. b) That the beagle s weight is between 22 and 28 pounds. c) That the beagle s weight is more than 30 pounds.
Example 2 The fish sizes in Certain Lake are known be normally distributed with a mean size of 16 inches and have a standard deviation of 4 inches. You catch a fish, what is the probability that your fish is larger than 23 inches? μ = 16, σ = 4
Example 3 Pairs of shoes owned is known to be normally distributed with a mean number of 20 and a standard deviation of 7 pairs. Brittany counted the shoes in her closet. What is the probability that she had less than 15 pairs?
Z-Score Probabilities If given a z score instead of a distribution value (x), the calculator will still compute the probability. The calculator defaults to the standard normal distribution, so you only need to omit mu and sigma Use lb, ub only with a z-probability
Examples Compute P(Z < 3) P(Z > 1.1) P(-1.2 < Z < 0.45)
Classwork: CW 5/7/15, 1-18 Homework None