1. REVIEW Normal Distribution Normal Distribution

2. Characterizing a Normal Distribution To completely characterize a normal distribution, we need to know only 2 things: –The mean ---  –The standard deviation --- 

3. HAND CALCULATIONS FOR THE NORMAL DISTRIBUTION Probability tables have been created for the normal distribution expressed in terms of z, where z = the number of standard deviations x is from its mean, , i.e.

4. TYPES OF NORMAL TABLES Two types of normal tables –Tables giving probabilities from z = 0 to a positive value of z –Cumulative normal tables giving probabilities from z = - ∞ to any value of z Keller’s Book Table 3 (Page B-8 & B-9) Excel uses this approach

5. 0 z Z  Normal Curve with X and Z Scales Probabilities from 0 to z  a X Some tables give probability of falling between 0 and a positive z value µ a X

6. 0 z Z  Curve with X and Z Scales Cumulative Probabilities from - ∞ to z  a X µ a X A cumulative normal table gives the probability of falling between - ∞ and any z value

7. Calculating Normal Probabilities Using Excel Function Calculating Normal Probabilities Using Excel Function LEFT TAIL PROBABILITIES Calculating Normal Probabilities Using Excel Function P(X<a) or P(Z<z) = area between -∞ and a (or z) –Probability to the leftEXCEL: =NORMDIST(a,µ,σ,TRUE) =NORMDIST(a,µ,σ,TRUE) or =NORMSDIST(z)

8. Calculating Normal Probabilities Using Cumulative Probabilities Calculating Normal Probabilities Using Cumulative Probabilities RIGHT TAIL PROBABILITIES Calculating Normal Probabilities Using Cumulative Probabilities P(X>a) or P(Z>z) = area between a (or z) and +∞ Probability to the right 1 - (Cumulative normal table value)EXCEL: =1-NORMDIST(a,µ,σ,TRUE) =1-NORMDIST(a,µ,σ,TRUE) or =1-NORMSDIST(z)

9. Calculating Normal Probabilities Using Cumulative Probabilities Calculating Normal Probabilities Using Cumulative Probabilities IN BETWEEN PROBABILITIES Calculating Normal Probabilities Using Cumulative Probabilities P(a<X<b) or P(z a <Z<z b ) –Probability between a and b on the X scale or between z a and z b on the Z scale –(Cumulative normal table value for z b ) - (Cumulative normal table value for z a )EXCEL: =NORMDIST(b,µ,σ,TRUE) - NORMDIST(a,µ,σ,TRUE) =NORMSDIST(z b ) - NORMSDIST(z a ) =NORMDIST(b,µ,σ,TRUE) - NORMDIST(a,µ,σ,TRUE) or =NORMSDIST(z b ) - NORMSDIST(z a )

10. Calculating x Values from Cumulative Normal Probabilities Calculating x Values from Cumulative Normal Probabilities Determining the x value such that the probability of getting a value less than x is p leftFind the cumulative normal probability, p, (approximately) in the table (to the left of x) and note the corresponding z value x = µ + zσEXCEL: = NORMINV(p,µ,σ) = NORMINV(p,µ,σ) or = µ + NORMSINV(p)*σ

11. EXERCISE Given random variable X has normal distribution with  = 320 and  = 20 Find P(X=315) Find P(X<335) Find P(X>325) Find P(303 <X<348) For P(x<X) = 0.75 find X For P(x>X) = 0.85 find X –The average flight time is 320 minutes –The standard deviation is 20 minutes

12. P(X = 315 ) P(X = 315 ) = 0 –Since X is a continuous random variable

13. P(X < 335 ).4332 σ = 20 320 X 335 0 Z 0.75 EXCEL =NORMDIST(335,320,20,TRUE) OR =NORMSDIST(.75) P(X<335) =.7734

14. P(X > 325 ).4332 σ = 20 320 X 0 Z 0.25 EXCEL =1-NORMDIST(325,320,20,TRUE) OR =1-NORMSDIST(.25) 325 P(X<325) =.5987 1 -.5987 =.4013

15. P(303  X  348 ).4332 σ = 20 320 X 303 348 -0.85 0 Z 1.40 EXCEL =NORMDIST(348,320,20,TRUE)-NORMDIST(303,320,20,TRUE) OR =NORMSDIST(1.40)-NORMSDIST(-0.85).9192 -.1977 =.7215 P(X≤348) =.9192 P(X≤303) =.1977

16. P(x≤X) = 0.75 find X.4332 σ = 20 320 X 0 Z EXCEL =NORMINV(.75,320,20) OR =320 + NORMSINV(.75)*20 x.7500 is to the left of x 0.67 NORMSINV(.75) = 0.67 x = 320 +.67(20) 333.4

17. P(x  X) = 0.85 find X.4332 σ = 20 320 X 0 Z EXCEL =NORMINV(.15,320,20) OR =320 + NORMSINV(.15)*20 x.8500 is to the right of x Thus, 1-.8500 =.1500 is to the left of x -1.04 NORMSINV(.15) = -1.04 x = 320 + (-1.04)(20) 299.2

18. REVIEW Normal distribution is completely characterized by µ and σ Calculation of: –“ ” probabilities, and “in between” probabilities using: Cumulative probability table NORMDIST and NORMSDIST functions –“x values” and “z values” corresponding to a cumulative probability using: NORMINV and NORMSINV functions