NORMAL PROBABILITY DISTRIBUTIONS - The Most Important Probability Distribution in Statistics Lecture 10

1. NORMAL PROBABILITY DISTRIBUTIONS - The Most Important Probability Distribution in Statistics Lecture 10 William F. Hunt, Jr. Statistics 361 Sec. 1.4

2. A random variable X with mean m and standard deviation s is normally distributed if its probability density function is given by Normal Distribution

3. The Shape of the Normal Distribution

4. Normal Probability Distributions The expected value (also called the mean) ??can be any number The standard deviation ? can be any nonnegative number There are infinitely many normal distributions

12. Property and Notation Property: normal density curves are symmetric around the population mean ?, so the population mean = population median = population mode = ?? Notation: X ~ N(????? is written to denote that the random variable X has a normal distribution with mean ? and standard deviation ?.

13. Standardizing Suppose X~N(????? Form a new random variable by subtracting the mean ? from X and dividing by the standard deviation ?: (X?????? This process is called standardizing the random variable X.

14. Standardizing (cont.) (X??????? is also a normal random variable; we will denote it by Z: Z = (X?????? ???has mean 0 and standard deviation 1:E(Z) = ? = 0; SD(Z) = ? = 1. ??????????? The probability distribution of Z is called the standard normal distribution.

15. Standardizing (cont.) If X has mean ? and stand. dev. ?, standardizing a particular value of x tells how many standard deviations x is above or below the mean ?. Exam 1: ?=80, ?=10; exam 1 score: 92 Exam 2: ?=80, ?=8; exam 2 score: 90 Which score is better?

17. Pdf of a standard normal rv A normal random variable x has the following pdf:

18. Z = standard normal random variable ? = 0 and ? = 1 Standard Normal Distribution

19. Important Properties of Z #1. The standard normal curve is symmetric around the mean 0 #2. The total area under the curve is 1; so (from #1) the area to the left of 0 is 1/2, and the area to the right of 0 is 1/2

20. Finding Normal Percentiles by Hand (cont.) Table Z is the standard Normal table. We have to convert our data to z-scores before using the table. The figure shows us how to find the area to the left when we have a z-score of 1.80:

21. Areas Under the Z Curve: Using the Table

23. Example – continued X~N(60, 8)

24. Class Problem What proportion of values selected from the standard normal distribution will satisfy each of the following conditions? Be at most 1.78 Exceed .55 Exceed -.80 Be between .21 and 1.21 Be either at most -2.00 or at least 2.00

25. Examples P(0 ? z ? 1.27) =

26.

27. Examples P(-2.24 ? z ? 0) =

28. P(z ? -1.85) = .0322

29. Examples (cont.) P(-1.18? z? 2.73) = A - A1 = .9968 - .1190 = .8778

30. P(-1 = Z = 1) = .8413 - .1587 =.6826

31. Class Problem Suppose that values are successively chosen from the standard normal distribution. How large must value a value be to be among the largest 15% of all values selected ? How small must a value be to be among the smallest 25% of all values selected?

32. Class Problem SolutionLinear Interpolation

34. Examples (cont.) viii)

35. Examples (cont.) ix)

36. X~N(275, 43) find k so that P(x<k)=.9846

38. Example Regulate blue dye for mixing paint; machine can be set to discharge an average of ? ml./can of paint. Amount discharged: N(?, .4 ml). If more than 6 ml. discharged into paint can, shade of blue is unacceptable. Determine the setting ? so that only 1% of the cans of paint will be unacceptable

39. Solution