1 / 15

7.3

7.3. Applications of the Normal Distribution. Probabilities. We want to calculate probabilities and values for general normal probability distributions We can relate these problems to calculations for the standard normal in the previous section. Z-Score.

halima
Télécharger la présentation

7.3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 7.3 Applications of the Normal Distribution

  2. Probabilities • We want to calculate probabilities and values for general normal probability distributions • We can relate these problems to calculations for the standard normal in the previous section

  3. Z-Score • For a general normal random variable X with mean μ and standard deviation σ, the variable has a standard normal probability distribution • We can use this relationship to perform calculations for X

  4. Z-Score • Values of X Values of Z • If x is a value for X, then is a value for Z • This is a very useful relationship

  5. Example • For example, if • μ = 3 • σ = 2 then a value of x = 4 for X corresponds to a value of z = 0.5 for Z

  6. Relationships • Because of this relationship • Values of X Values of Z Then P(X < x) = P(Z < z) • To find P(X < x) for a general normal random variable, we could calculate P(Z < z) for the standard normal random variable

  7. Z is X • With this relationship, the following method can be used to compute areas for a general normal random variable X • Shade the desired area to be computed for X • Convert all values of X to Z-scores using • Solve the problem for the standard normal Z (Section 7.2 Stuff) • The answer will be the same for the general normal X

  8. Example • For a general random variable X with • μ = 3 • σ = 2 calculate P(X < 6) • This corresponds to so P(X < 6) = P(Z < 1.5) = 0.9332 Hint…Normalcdf(small, z , 0,1)

  9. Example • For a general random variable X with • μ = –2 • σ = 4 calculate P(X > –3) • This corresponds to so P(X > –3) = P(Z > –0.25) = 0.5987 Hint…(right hand) Normalcdf(z, big number,0,1)

  10. Between • For a general random variable X with • μ = 6 • σ = 4 calculate P(4 < X < 11) • This corresponds to so P(4 < X < 11) = P(– 0.5 < Z < 1.25) = 0.5858 • Hint…Normalcdf(smaller z,bigger z,0,1)

  11. Inverse • The inverse of the relationship is the relationship • With this, we can compute value problems for the general normal probability distribution

  12. Inverse • The following method can be used to compute values for a general normal random variable X • Shade the desired area to be computed for X • Find the Z-scores for the same probability problem • Convert all the Z-scores to X using • This is the answer for the original problem

  13. Example • For a general random variable X with • μ = 3 • σ = 2 find the value x such that P(X < x) = 0.3 Find the Z that corresponds to .3 (InvNorm,.3,0,1) = -.5244 • Since P(Z < –0.5244) = 0.3, we calculate so P(X < 1.95) = P(Z < –0.5244) = 0.3

  14. Notice the Greater than!! • For a general random variable X with • μ = –2 • σ = 4 find the value x such that P(X > x) = 0.2 Since it is a “Righthand” we must use 1 - .2 or .8 in our invNorm(.8,0,1) = 0.8416 so P(X > 1.37) = P(Z > 0.8416) = 0.2

  15. Same old Same old • Tough Stuff…work hard on this!

More Related