1 / 25

Computing Probabilities From the Standard Normal Distribution

Computing Probabilities From the Standard Normal Distribution. Table B.1 p. 687. Table B.1 A Closer Look. The Normal Distribution: why use a table?. From x or z to P To determine a percentage or probability for a normally distributed variable.

ross-osborn
Download Presentation

Computing Probabilities From the Standard Normal Distribution

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. Computing Probabilities From the Standard Normal Distribution Anthony J Greene

  2. Table B.1p. 687

  3. Table B.1A Closer Look Anthony J Greene

  4. The Normal Distribution: why use a table? Anthony J Greene

  5. From x or z to PTo determine a percentage orprobability for a normally distributed variable Step 1 Sketch the normal curve associated with the variable Step 2 Shade the region of interest and mark the delimiting x-values Step 3 Compute the z-scores for the delimiting x-values found in Step 2 Step 4 Use Table B.1 to obtain the area under the standard normal curve delimited by the z-scores found in Step 3Use Geometry and remember that the total area under the curve is always 1.00. Anthony J Greene

  6. From x or z to PFinding percentages for a normally distributed variable from areas under the standard normal curve Anthony J Greene

  7. Finding percentages for a normally distributed variable from areas under the standard normal curve • ,  are given. • a and b are any two values of the variable x. • Compute z-scores for a and b. • Consult table B-1 • Use geometry to find desired area. Anthony J Greene

  8. Given that a quiz has a mean score of 14 and an s.d. of 3, what proportion of the class will score between 9 & 16? •  = 14 and  = 3. • a = 9 and b = 16. • za = -5/3 = -1.67, zb = 2/5 = 0.4. • In table B.1, we see that the area to the left of a is 0.0475 and that the area to the right of b is 0.3446. • The area between a and b is therefore 1 – (0.0475 + 0.3446) = 0.6079 or 60.79% Anthony J Greene

  9. Finding the area under the standard normal curve to the left of z = 1.23 Anthony J Greene

  10. What if you start with x instead of z? What is the probability of selecting a random student who scored above 650 on the SAT? z = 1.50: Use Column C; P = 0.0668 Anthony J Greene

  11. Finding the area under the standard normal curve to the right of z = 0.76 The easiest way would be to use Column C, but lets use Column B instead Anthony J Greene

  12. Finding the area under the standard normal curve that lies between z = –0.68 and z = 1.82 P = 1 – 0.0344 – 0.2483 = 0.7173 One Strategy: Start with the area to the left of 1.82, then subtract the area to the right of -0.68. Second Strategy: Start with 1.00 and subtract off the two tails Anthony J Greene

  13. Determination of the percentage of people having IQs between 115 and 140 Anthony J Greene

  14. From x or z to PReview of Table B.1 thus far • Using Table B.1 to find the area under the standard normal curve that lies • to the left of a specified z-score, • to the right of a specified z-score, • between two specified z-scores Then if x is asked for, convert from z to x Anthony J Greene

  15. From P to z or xNow the other way aroundTo determine the observations corresponding to a specified percentage or probability for a normally distributed variable Step 1 Sketch the normal curve associated the the variable Step 2 Shade the region of interest (given as a probability or area Step 3 Use Table B.1 to obtain the z-scores delimiting the region in Step 2 Step 4 Obtain the x-values having the z-scores found in Step 3 Anthony J Greene

  16. From P to z or xFinding z- or x-scores corresponding to a given region. Finding the z-score having area 0.04 to its left x = σ× z + μ If μ is 242 σ is 100, thenx = 100 × -1.75 + 242 x = 67 Use Column C: The z corresponding to 0.04 in the left tail is -1.75 Anthony J Greene

  17. The zNotation The symbol zα is used to denote the z-score having area α (alpha) to its right under the standard normal curve. We read “zα” as “z sub α” or more simply as “zα.” Anthony J Greene

  18. The z notation : P(X>x) = α P(X>x)= α This is the z-score that demarks an area under the curve with P(X>x)= α Anthony J Greene

  19. The z notation : P(X<x) = α P(X<x)= α Z This is the z-score that demarks an area under the curve with P(X<x)= α Anthony J Greene

  20. The z notation : P(|X|>|x|) = α P(|X|>|x|)= α α/2 α/2 1- α This is the z-score that demarks an area under the curve with P(|X|>|x|)= α Anthony J Greene

  21. Finding z0.025 Use Column C: The z corresponding to 0.025 in the right tail is 1.96 Anthony J Greene

  22. Finding z0.05 Use Column C: The z corresponding to 0.05 in the right tail is 1.64 Anthony J Greene

  23. Finding the two z-scores dividing the area under the standard normal curve into a middle 0.95 area and two outside 0.025 areas Use Column C: The z corresponding to 0.025 in both tails is ±1.96 Anthony J Greene

  24. Finding the 90th percentile for IQs z0.10 = 1.28 z = (x-μ)/σ 1.28 = (x – 100)/16 120.48 = x Anthony J Greene

  25. What you should be able to do • Start with z-or x-scores and compute regions • Start with regions and compute z- or x-scores Anthony J Greene

More Related