Lecture 3 Normal distribution, stem-leaf, histogram Idealized Population, Box of infinitely many tickets, each ticket has a value. Random variable and.

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Presentation transcript:

Lecture 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 symmetry

How 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 SD

Probability 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.16