THE NORMAL DISTRIBUTION

NORMAL DISTRIBUTION Frequently called the “bell shaped curve” Important because –Many phenomena naturally have a “bell shaped” distribution –Normal distribution is the “limiting” distribution for many statistical tests

Characterizing a Normal Distribution To completely characterize a normal distribution, we need to know only 2 parameters: –It’s mean (  ) –It’s standard deviation (  ) A normal distribution curve can be now drawn as follows:

NORMAL PROBABILITY DENSITY CURVE μ σ

NORMAL PROBABILITY DENSITY FUNCTION The density function for a normally distributed random variable with mean μ and standard deviation σ is:

CALCULATING NORMAL PROBABILITIES There is no easy formula for integrating f(x) However, tables have been created for a “standard” normal random variable, Z, which has  = 0 and σ =1 Probabilities for any normal random variable, X, can then be found by converting the value of x to z where z = the number of standard deviations x is from its mean, 

“Standardizing” a Normal Random Variable X to Z 0 Z σ Z = 1 0 Z μ X σX = σσX = σσX = σσX = σ x0x0x0x0 Standardization preserves probabilities P(X>x 0 ) = P(Z>z) z

Representing X and Z on the Same Graph Both the x-scale and the corresponding z-scale can be represented on the same graph μ X σX = σσX = σσX = σσX = σ x0x0x0x0 0 Z z Example: Suppose μ = 10, σ =2. Illustrate P(X > 14). 10 σ = Thus P(X > 14) = P(Z>2)

Facts About the Normal Distribution mean = median = mode Distribution is symmetric 50% of the probability is on each side of the mean Almost all of the probability lies within 3 standard deviations from the mean –On the z-scale this means that almost all the probability lies in the interval from z = -3 to z = +3

Using the Cumulative Normal (Cumulative z) Table The cumulative z-table –Gives the probability of getting a value of z or less P(Z < z) –Left-tail probabilities –Excel gives left-tail probabilities To find any probability from a z-table: –Convert the problem into one involving only left-tail probabilities P(Z < a) = z-table value for a P(Z > a) = 1-(z-table value for a) P(a<Z<b) = (z-table value for b) – (z-table value for a)

Using the Normal Table Look up the z value to the first decimal place down the first column Look up the second decimal place of the z-value in the first row The number in the table gives the probability P(Z<z)

Example X is normally distributed with  = 244,  = 25 Find P(X < 200) For x = 200, z = ( )/25 = X 244 σ = ? 0 Z 0 Z -1.76

Using Cumulative Normal Tables z P (Z<-1.76)

EXAMPLE Flight times from LAX to New York: –Are distributed normal –The average flight time is 320 minutes –The standard deviation is 20 minutes

Probability a flight takes exactly 315 minutes P(X = 315 ) = 0 –Since X is a continuous random variable

Probability a Flight Takes Less Than 300 Minutes  = X Z 0 Z From Table.1587

Probability a Flight Takes Longer Than 335 Minutes  = X Z 0 Z.75 From Table =.2266

Probability a Flight Takes Between 320 and 350 Minutes  = X Z 0 Z =.4332

Probability a Flight Takes Between 325 and 355 Minutes  = X Z 0 Z =.3612

Probability a Flight Takes Between 308 and 347 Minutes  = X Z 0 Z =.6372

Probability a Flight Takes Between 275 and 285 Minutes  = X Z 0 Z =.0279

Using Excel to Calculate Normal Probabilities Given values for μ and σ, cumulative probabilities P(X < x 0 ) are given by: Note that =NORMDIST(x 0,μ,σ,FALSE) returns the value of the density function at x 0, not a probability. If the value of z is given, then the cumulative probabilities P(Z<z) are given by: =NORMDIST(x 0, μ, σ, TRUE) =NORMSDIST(z)

=NORMDIST(300,320,20,TRUE ) =1-NORMDIST(335,320,20,TRUE ) =NORMDIST(350,320,20,TRUE)-NORMDIST(320,320,20,TRUE)=NORMDIST(355,320,20,TRUE)-NORMDIST(325,320,20,TRUE)=NORMDIST(347,320,20,TRUE)-NORMDIST(308,320,20,TRUE)=NORMDIST(285,320,20,TRUE)-NORMDIST(275,320,20,TRUE) =NORMSDIST(-1.00) =NORMSDIST(-1.75)-NORMSDIST(-2.25) =1-NORMSDIST(.75) =NORMSDIST(1.50)-NORMSDIST(0)=NORMSDIST(1.75)-NORMSDIST(.25) =NORMSDIST(1.35)-NORMSDIST(-.60)

Calculating x and z Values From Normal Probabilities Basic Approach –Convert to a cumulative probabiltity –Locate that probability (or the closest to it) in the Cumulative Standard Normal Probability table z valueThis gives the z value This is the number of standard deviations x is from the mean x = μ + zσ Note: z can be a negative value

90% of the Flights Take At Least How Long?.9000 of the probability lies above the x value.1000 lies below the x value  = X ? 0 Z Look up.1000 in middle of z table X = 320+(-1.28)(20) = (approx.)

The Middle 75% of the Flight Times Lie Between What Two Values? Required to find x L and x U such that.7500 lies between x L and x U -- this means.1250 lies below x L and.1250 lies above x U (.8750 lies below x U )  = X.7500 xLxLxLxL xUxUxUxU =.2500 split between tails 0 Z 0 Z zLzLzLzL zUzUzUzU z L puts.1250 to leftz U puts.8750 to left x L = 320+(-1.15)(20) 297 = x U = 320+(1.15)(20) 343 =

Using Excel to Calculate x and z Values From Normal Probabilities Given values for μ and σ, the value of x 0 such that P(X < x 0 ) = p is given by: =NORMINV(p, μ, σ) The value of z such that P(Z<z) = p is given by:=NORMSINV(p)

=NORMINV( ,320,20) =NORMINV(.1250,320,20)=NORMINV(.1000,320,20) =NORMSINV(.1000)=NORMSINV(.1250) =NORMSINV( )

What Would the Mean Have to Be So That 80% of the Flights Take Less Than 330 Minutes? Since x = μ + zσ, then μ = x - z σ  = 20 μ X μ X Z Look up.8000 in the middle of the z-table.84 μ = (20) =

REVIEW Normal Distribution Importance and Properties Converting X to Z Use of Tables to Calculate Probabilities Use of Excel to Calculate Probabilities