Lecture 3 Normal distribution, stem-leaf, histogramPowerPoint PresentationHow does the normal table work?How to standardize?Slide 5Probability for an intervalLecture 3 Normal distribution, stem-leaf, histogram•Idealized Population, Box of infinitely many tickets, each ticket has a value. •Random variable and probability statement P(X<85)•Notations , Greek letters: Mean (expected value) and standard deviation, E(X) = , SD(X)= Var(X)= •Examples•Empirical distribution : Stem-leaf, histogram •Three variants of histogram : frequency, relative frequency, density(called “standardized” in book)•Same shape with different vertical scale •Density= relative frequency / length of interval•Given a box of tickets with values that come from a normal distribution with mean 75 and standard deviation 15, what is the probability that a randomly selected ticket will have a value less than 85?•Let X be the number elected ( a random variable).•Pr( X<85).How does the normal table work?•Start from Z=0.0 , then Z=0.1 •Increasing pattern observed•On the negative side of Z•Use symmetryHow to standardize?•Find the mean•Find the standard deviation•Z= (X-mean)/SD•Reverse questions:•How to recover X from Z?•How to recover X from percentile?•Suppose there are 20 percent students failing the exam•What is the passing grade?•Go from percentage to Z, using normal table•Convert Z into X, using X=mean + Z times SDProbability for an interval•P (60<X<85)•Draw the curve (locate mean, and endpoints of interval)•=P(X<85)-P(X<60) where•P(X<60)= P(Z<(60-75)/15)=P(Z<-1)=1-P(Z<1)=1-.841= about
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