The Normal Distribution

1. The Normal Distribution And the standard normal distribution (also known as the “bell curve”)

3. The Mathematics of the Normal Distribution The normal distribution is completely described by only two parameters: the mean μ and the standard deviation σ

4. Review: Binomial Distribution The Binomial Distribution is the outcome of a Bernoulli trial (only two possible values) repeated n times, with probability p of success. In the notation, x is the number of successes.

5. Binomial Example P = 0.25 (correct answer from a, b, c or d) N = 10 questions What is the probability of getting exactly 3 questions correct?

6. Binomial Dist. >>> Normal Dist. The Normal Distribution is a continuous distribution, and is a limiting case of a (discrete) binomial distribution. As the number of trials n -> ∞, the binomial distribution will approach the normal distribution. Here n = 5; experiment simulated 1,000 times. The binomial random variable X = number of success out of n = 5 trials, with probability p = 0.1 of success, repeated 1,000 times.

7. Simulation Continued (n increases) P = 0.1, n = 10 Simulated 1,000 times. P = 0.1, n = 25 Simulated 1,000 times.

8. Simulation Continued (n increases) P = 0.1, n = 25 Simulated 1,000 times. P = 0.1, n = 50 Simulated 1,000 times.

9. Other Simulations P = 0.5, n = 10 Simulated 1,000 times. P = 0.6, n = 100 Simulated 1,000 times.

10. Determinants of the place and shape of the normal distribution

11. Normal Distributions: Changing μ or σ Same mean μ Different standard deviation σ Different mean μ Same standard deviation σ

12. Normal Distributions: Changing both μ and σ Different mean μand Different standard deviation σ

14. The Sigma (standard deviation) Rule revisited 68% of the results will be between + or – 1 standard deviation (σ) away from the mean (μ). 95% of the results will be between + or – 2 standard deviation (σ) away from the mean (μ).

15. The Sigma (standard deviation) Rule revisited 99.7% of the results will be between + or – 3 standard deviation (σ) away from the mean (μ).

16. The StandardNormal Distribution

17. How to use a z-score Standard normal (Z-score) tables can tell you the probability of getting between the z-score and the mean 0. * Alternatively, the standard normal table could tell you the probability of getting a score anywhere below the z-score.

18. Sample Standard Normal Table To economize on space, the standard normal table has the first digit of the z-score down the first column, and the second digit of the z-score across the first row. Example: z = 0.92 0.3212 0.50 Pr(z ≤ 0.92) = 0.50 + 0.3212 = 0.8212

19. Standard Normal Curve: Example * Note that if we looked at P (-2 ≤ z ≤ ++2) = 0.4772 x 2 ≈ 0.95 or 95%, which is the 2-sigma rule.

20. Standard Normal Curve: Example • Note that the probability of getting a z-score less than z = -1.8 is P(z ≤ -1.8) = 1 – 0.50 – 0.4641 = 0.0359.

21. Standard Normal Curve: Example

22. Standard Normal Curve: Example

23. Standard Normal Curve: Example

24. Practice Problems (student)

26. Practice Problems (solutions)

27. Practice Problems (solutions)

28. Practice Problems (solutions)

29. Applying the Standard Normal Distribution • If the underlying distribution is approximately normal…… • … Transform the normal variable into a standard normal variable using its z-score. • Solve the problem as above.

30. Applying the Standard Normal Distribution: Further Examples

32. Practice Problems (student)

33. Practice Problems (solutions)

34. Practice Problems (solutions)

35. Online Help is Readily Available Google “z-score calculator” to convert normal variables into standard normal variables; Google “z-score table” or “standard normal table” to get the associated probability. Example (not necessarily recommended, and certainly not the only one available) Z-score calculator: http://www.danielsoper.com/statcalc3/calc.aspx?id=22 Standard normal z-score table: http://regentsprep.org/Regents/math/algtrig/ATS7/ZChart.htm which shows the area to the left of the z-score.