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Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,11/7/2009,#,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Chapter,Normal Probability Distributions,5,ChapterNormal Probability Dist,Chapter Outline,Chapter Outline,Section 5.1,Introduction to Normal Distributions and the Standard Normal Distributions,Section 5.1Introduction to Nor,Section 5.1 Objectives,How to interpret graphs of normal probability distributions,How to find areas under the standard normal curve,Section 5.1 ObjectivesHow to i,Properties of a Normal Distribution,Continuous random variable,Has an infinite number of possible values that can be represented by an interval on the number line.,Continuous probability distribution,The probability distribution of a continuous random variable.,.,Hours spent studying in a day,0,6,3,9,15,12,18,24,21,The time spent studying can be any number between 0 and 24.,Properties of a Normal Distrib,Properties of a Normal Distribution,.,Normal distribution,A continuous probability distribution for a random variable,x,.,The most important continuous probability distribution in statistics.,The graph of a normal distribution is called the,normal curve,.,x,Properties of a Normal Distrib,Properties of a Normal Distribution,.,The mean,median,and mode are equal.,The normal curve is bell-shaped and is symmetric about the mean.,The total area under the normal curve is equal to one.,The normal curve approaches,but never touches the,x,-axis as it extends farther and farther away from the mean.,x,Total area=1,Properties of a Normal Distrib,Properties of a Normal Distribution,.,Between,and,+,(in the center of the curve),the graph curves downward.The graph curves upward to the left of,and to the right of,+,.The points at which the curve changes from curving upward to curving downward are called the,inflection points,.,Inflection points,3,+,2,+2,+3,x,Properties of a Normal Distrib,Probability Density Function(PDF),.,A discrete probability distribution can be graphed with a histogram.,For a continuous probability distribution,you can use a,probability density function(pdf).,A probability density function has two requirements:,the total area under the curve is equal to 1,the function can never be negative.,Probability Density Function(,Means and Standard Deviations,A normal distribution can have any mean and any positive standard deviation.,The mean gives the location of the line of symmetry.,The standard deviation describes the spread of the data.,Means and Standard DeviationsA,Example:Understanding Mean and Standard Deviation,.,Which curve has the greater mean?,Solution:,Curve,A,has the greater mean,(The line of symmetry of curve,A,occurs at,x,=15.The line of symmetry of curve,B,occurs at,x,=12.),Example:Understanding Mean an,Example:Understanding Mean and Standard Deviation,.,Which curve has the greater standard deviation?,Solution:,Curve,B,has the greater standard deviation,(Curve,B,is more spread out than curve,A.,),Example:Understanding Mean an,Example:Interpreting Graphs of Normal Distributions,.,The scaled test scores for New York State Grade 4 Common Core Mathematics Test are normally distributed.The normal curve shown below represents this distribution.,What is the mean test score?Estimate the standard deviation of this normal distribution.,(Adapted from New York State Education Department),Example:Interpreting Graphs o,Solution:,The scaled test scores for the New York State Grade 4 Common Core Mathematics Test are normally distributed with a mean of about 305 and a standard deviation of about 40.,Solution:Interpreting Graphs of Normal Distributions,.,Solution:Solution:Interpretin,Solution:,Using the Empirical,you know that about 68%of the scores are between 265 and 345,about 95%of the scores are between 225 and 385,and about 99.7%of the scores are between 185 and 425.,Solution:Interpreting Graphs of Normal Distributions,.,Solution:Solution:Interpretin,The Standard Normal Distribution,.,Standard normal distribution,A normal distribution with a mean of 0 and a standard deviation of 1.,Any,x,-value can be transformed into a,z,-score by using the formula,3,1,2,1,0,2,3,z,Area=1,The Standard Normal Distributi,The Standard Normal Distribution,.,The,standard normal distribution,is a normal distribution with a mean of 0 and a standard deviation of 1.The total area under its normal curve is 1.,The Standard Normal Distributi,Properties of the Standard Normal Distribution,.,The cumulative area is close to 0 for,z,-scores close to,z,=,3.49.,The cumulative area increases as the,z,-scores increase.,z,3,1,2,1,0,2,3,z,=,3.49,Area is close to 0,Properties of the Standard Nor,Properties of the Standard Normal Distribution,.,The cumulative area for,z,=0 is 0.5000.,The cumulative area is close to 1 for,z,-scores close to,z,=,3.49.,z,=,3.49,Area is close to 1,Area is 0.5000,z,=,0,z,3,1,2,1,0,2,3,Properties of the Standard Nor,Example:Using The Standard Normal Table,.,Find the cumu
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