Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4DistributionsDistributionsNormal DistributionsNotationNotationPropertiesPropertiesPropertiesPropertiesPropertiesProperties68-95-99.7 Rule68-95-99.7 RuleNotationSlide 19Types of ProblemsTypes of ProblemsTypes of ProblemsSpecial RuleTypes of ProblemsStandard Normal DistributionSlide 26Equal to ProblemsExample 34Example 34Example 35Example 35Less than ProblemsNormal TableProbability Problems on CalculatorExample 36Example 36Example 36Example 36Example 37Example 37Example 37Example 37Greater than ProblemsExample 38Example 38Example 38Example 38Example 39Example 39Example 39Example 39Between ProblemsBetween ProblemsExample 40Example 40Example 40Example 40Example 41Example 41Example 41Example 41Normal VariablesNormal VariablesZ-Score TransformationZ-Score TransformationZ-Score TransformationZ-Score TransformationExample 47Example 47Example 47Example 47Example 47Example 47Example 48Example 48Example 48Example 48Example 48STAT 210Lecture 19Normal DistributionsOctober 9, 2017Practice ProblemsPages 162 through 165Relevant problems: VI.1, VI.2, VI.3, VI.5Recommended problems: VI.1, VI.3, VI.5Additional Reading and ExamplesRead pages 158 through 160TOP HATDistributionsWhen describing a distribution we usually describe four things:(1) the center of the distribution(2) the spread (or dispersion or variability) of the distribution(3) the shape of the distribution(4) any unusual features in the distributionDistributionsTwo commonly used distributions, and the two distributions primarily used in the rest of this course, are the normal distributions and the Student’s t-distributions.In this chapter we will discuss and learn the properties associated with these distributions and how to use tables associated with these two distributions.Normal DistributionsThe normal curve is a symmetric, bell-shaped curve depicted below. Data that is described by the normal curve is said to follow a normal distribution. A normal distribution is an example of a continuous distribution and hence a normal variable can assume any one of a countless number of possible outcomes. Additionally, at times one may have a discrete variable and say that the variable has an approximate normal distribution implying that while the discrete variable does not have an exact normal distribution, the normal curve is a good approximation for the actual distribution.NotationX ~ N (m , s)“X is distributed normal with mean m and standard deviation s”NotationExample:If X is the weight of students and the weights follow a normal distribution with mean 160 pounds and standard deviation 15 pounds, then we write X ~ N(160, 15).Properties•The normal curve is bell-shapedProperties•The normal curve is bell-shaped•The peak of the curve is the population mean mProperties•The normal curve is bell-shaped•The peak of the curve is the population mean m•The normal curve is symmetric about mmProperties•The normal curve is bell-shaped•The peak of the curve is the population mean m•The normal curve is symmetric about m•The center and spread are completely specified by specifying the values of the population mean m and the population standard deviation sProperties•The normal curve is bell-shaped•The peak of the curve is the population mean m•The normal curve is symmetric about m•The center and spread are completely specified by specifying the values of the population mean m and the population standard deviation s•The total area under the normal curve is 1 (or 100%)Properties•The normal curve is bell-shaped•The peak of the curve is the population mean m•The normal curve is symmetric about m•The center and spread are completely specified by specifying the values of the population mean m and the population standard deviation s•The total area under the normal curve is 1 (or 100%) ● 68-95-99.7% Rule68-95-99.7 Rule1. 68% of the measurements fall within one standard deviation s of the mean m. (m-s, m+s)2. 95% of the measurements fall within two standard deviations (2s) of the mean m. (m-2s, m+2s)3. 99.7% of the measurements fall within three standard deviations (3s) of the mean m. (m-3s, m+3s).68-95-99.7 RuleExample: Suppose X ~ N(160, 15)1. Approximately 68% of the values will be between 145 and 175. Note 160 – 15 = 145, 160 + 15 = 175.2. Approximately 95% of the values will be between 130 and 190. Note 160 – 2(15) = 130, 160 + 2(15) = 190.3. Approximately 99.7% of the values will be between 115 and 205. Note 160 – 3(15) = 115, 160 + 3(15) = 205.NotationA capital letter (such as X) represents the name of a variable and a small letter (such as x) represents a value of the variable.For example, if X = weight of a person and x = 150, then P(X = 150) is read “the probability that the weight of a person is equal to 150 pounds”.TOP HATTypes of Problems1. Given values of the variable X, find the probability (or area or proportion or percentage). TODAY’S CLASS2. Given a probability (or area or proportion or percentage), find the value of the variable X. WEDNESDAY’S CLASS.Types of Problems1. Given values of the variable X, find the probability (or area or proportion or percentage).(i). Find P(X = x)(ii). Find P(X < x) or P(X < x)(iii). Find P(X > x) or P(X > x)(iv). Find P(x1 < X < x2) or P(x1 < X < x2)Types of Problems2. Given a probability (or area or proportion or percentage), find the value of the variable X.(i) Find the value x such that the probability of being less than or less than and equal to the value is as specified.(ii) Find the value x such that the probability of being greater than or greater than and equal to the value is as specified.(iii) Find two values x1 and x2 such that the probability of being between the two values is as specified.Special RuleSince the normal distribution is continuous, then for any value x, the probability that a normal random variable X equals that specified value x is 0.P(X = x) = 0 for any xTypes of Problems1. Given values of the variable X, find the probability (or area or proportion or percentage).(i). P(X = x) = 0 for any value of x(ii). Find P(X < x) or P(X < x)(iii). Find P(X > x) or P(X > x)(iv). Find P(x1 < X < x2) or P(x1 < X < x2)Standard Normal Distribution•Denoted by Z•Has population mean m = 0 (center)•Has population standard deviation s = 1 (spread)•Shape is normal (symmetric bell curve)•No unusual features•Z ~ N(0, 1)•Probabilities are tabled
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