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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(Xlt85)
- Notations , Greek letters Mean (expected value)

and standard deviation, E(X) m , SD(X) s,

Var(X) s2 - 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( Xlt85).

How does the normal table work?

- Start from Z0.0 , then Z0.1
- 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 Xmean Z times SD

Probability for an interval

- P (60ltXlt85)
- Draw the curve (locate mean, and endpoints of

interval) - P(Xlt85)-P(Xlt60) where
- P(Xlt60) P(Zlt(60-75)/15)P(Zlt-1)1-P(Zlt1)1-.841

about .16

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