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Chapter 6 Normal Probability Distributions Concepts Normal distributions How do you handle large data sets? Normal curve Standard normal distribution Empirical rule Standardized score (z-score, standardized unit) Standard normal probability table Notations: P(Z b) or P(Z < b) P(Z a) or P(Z > a) P(a Z b) or P(a < Z < b) Applications Survival skills You must know how to use the standard normal table. Please read the following chapter: Chapters 6, Opentax SL 1 Chapter 6 Normal Probability Distributions Normal distributions Characteristics of a normal distribution (Theory) A normal distribution forms a continuous, symmetric, single-peaked, bell-shaped normal curve. The mean, median, and mode of a normal distribution are equal and are located at the center of the distribution. A normal distribution is symmetric about the mean (or x ). Area to the left of the mean = area to the right of the mean = 0.5 Note: In practice, a normal distribution comes from a data set. Data values are approximately normally distributed. All normal distributions have the same overall shape. Normal curve describes the overall pattern of a data set is always on or above the horizontal axis has an area exactly 1 (or 100%) underneath it. Basic property of probability: the sum of all probabilities is exactly 1.0. The total area under the normal curve must be 1. The exact normal curve for a particular normal distribution is described by giving its mean (or x ) standard deviation (or s) SL 2 Chapter 6 Normal Probability Distributions Standard normal distribution A normal distribution with mean and standard deviation can be transformed to the standard normal distribution. The standard normal distribution is a normal distribution with a: mean = 0 standard deviation = 1 The standard normal curve is symmetric about the mean 0. We have infinite number of different normal distributions. We have only one standard normal distribution. Applications of the standard normal distribution The standard normal distribution curve can be used to solve many practical problems: Heights, birth weights, gestation period, IQ scores Manufacturing industry Ready-to-wear clothes, Product warranties, …. Medical applications (blood pressures, diabetes,…) Educational testing scores Insurance industry Condition for use The data (or variable) must be normally distributed or approximately normally distributed, or Sample size 30 For this topic, what type of data are we talking about? SL 3 Chapter 6 Normal Probability Distributions Standardizing data values All normal distributions are the same if we measure in units of standard deviation s (or ) about the mean x (or ) as center. Key points We have a data set with a mean ( x or ) and a standard deviation (s or ). The data values are normally distributed. We want to measure the distance between a data value and the mean in units of standard deviation. Standardized value (standardized score, or z-score) A standardized value measures the distance between a data value and the mean, measured in units of standard deviation. How to standardize a data value x For a sample If x is a data value from a sample that has mean x and standard deviation s, the standardized value of x is: z= xx x value - sample mean = s sample standard deviation Conversion formula To change an x value into a standardized value z, use the conversion formula. A standardized vale (or z-score) tells how many standard deviations a data value x falls away from the mean, and in which direction. A positive z-score indicates that a data value is larger than the mean. A negative z-score indicates that a data value is smaller than the mean. For a population If x is a data value from a population that has mean and standard deviation , the standardized value of x is x - value population mean x z = Standard deviation SL 4 Chapter 6 Normal Probability Distributions How to use the standard normal table (to find probabilities) Areas under the standard normal curve are probabilities. To find the area under the standard normal curve, we must first use the conversion formula to find the standardized value z of the data value x. The area under the standard normal curve for a specific range of values represents either The probability that a randomly selected data value from the population with the characteristics described by the range, or The proportion of the population with the characteristics described by the range. P(Z < b) P(Z < b) = P(Z b) This notation denotes that the probability is the area under the standard normal curve to the left of a specific value (or cutoff point) b. P(Z > a) = 1 – P(Z a) P(Z a) = P(Z > a) This notation denotes that the probability is the area under the standard normal curve to the right of a specific value a. P(a < Z < b) = P(Z < b) – P(Z < a) This notation denotes that the probability is the area under the standard normal curve that lies between the specific values a and b. Example In each case, draw a sketch of the given information and the area (probability) under the standard normal curve. Use the normal table to find each area (or probability). [a] P(Z < –2.25) [b] P(Z > 1.77) [c] P(–2.25 < Z < 1.77) Solution [a] P(Z < –2.25) = [b] P(Z > 1.77) = [c] P(–2.25 < Z < 1.77) = SL 5 Chapter 6 Normal Probability Distributions Example (Source: Workshop Statistics) Suppose that the IQ scores of students in a certain college follow a normal distribution with mean 115 and standard deviation 12. [a] Draw a sketch of this distribution. [b] Shade in the area corresponding to the proportion of students with an IQ below 100. [c] Use the normal model to determine the proportion of students with an IQ below 100. Step 1: Use the conversion formula to change x value (= 100) to a standardized value z. Step 2: Use the normal table to find the proportion. [d] Find the proportion of these undergraduates having IQs greater than 130. Step 1: Use the conversion formula to change x value (= 130) to a standardized value z. Step 2: Use the normal table to find the proportion. [e] Find the proportion of these undergraduates having IQs between 110 and 130. Hint: Use P(a < Z < b) [f] With his IQ of 75, Forrest Gump would have a higher IQ than what percentage of these undergraduates? [g] Determine how high must one’s IQ must be to be in the top 1% of all IQs at this college. SL 6