1. 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Random Variables  Random variable a variable (typically represented by x) that takes a numerical value by chance.  For each outcome of a procedure, x takes a certain value, but for different outcomes that value may be different.

2. 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Examples:  Number of boys in a randomly selected family with three children. Possible values: x=0,1,2,3  The weight of a randomly selected person from a population. Possible values: positive numbers, x>0

3. 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Discrete and Continuous Random Variables  Discrete random variable either a finite number of values or countable number of values (resulting from a counting process)  Continuous random variable infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions

4. 4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Probability Distributions  Probability distribution a description that gives the probability for each value of the random variable; often expressed in the format of a table, graph, or formula

5. 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. xP(x) 01/8 13/8 2 31/8 Tables Values: Probabilities:

6. 6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Graphs The probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities.

7. 7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Requirements for Probability Distribution P ( x ) = 1 where x assumes all possible values.  0  P ( x )  1 for every individual value of x.

8. 8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean, Variance and Standard Deviation of a Probability Distribution µ =  [ x P(x) ] Mean  2 =  [ (x – µ) 2 P(x )] Variance  2 =  [ x 2 P ( x ) ] – µ 2 Variance (shortcut )  =  [ x 2 P ( x ) ] – µ 2 Standard Deviation

9. 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Roundoff Rule for µ,  and  2 Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ,  and  2 to one decimal place.

10. 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (1)Enter the values and their probabilities as separate columns Using StatCrunch

11. 11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (2)Stat → Calculators → Custom Using StatCrunch

12. 12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (2)Use var1 for “Values in” and var2 for “Weights in” Using StatCrunch

13. 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (4)The distribution, mean, and standard deviation will be displayed Using StatCrunch

14. 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = μ + 2σ Minimum usual value = μ – 2σ

15. 15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Identifying Unusual Results By Probabilities Using Probabilities to Determine When Results Are Unusual:  Unusually high: a particular value x is unusually high if P(x or more) ≤ 0.05.  Unusually low: a particular value x is unusually low if P(x or fewer) ≤ 0.05.

16. 16 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Binomial Probability Distribution A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure). 4. The probability of a success remains the same in all trials.

17. 17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notation for Binomial Probability Distributions S and F (success and failure) denote the two possible categories of all outcomes; p and q denote the probabilities of S and F, respectively: P(S) = p(p = probability of success) P(F) = 1 – p = q(q = probability of failure)

18. 18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notation (continued) n denotes the fixed number of trials. x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. p denotes the probability of success in one of the n trials. q denotes the probability of failure in one of the n trials. P(x) denotes the probability of getting exactly x successes among the n trials.

19. 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Methods for Finding Probabilities We will now discuss two methods for finding the probabilities corresponding to the random variable x in a binomial distribution.

20. 20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Method 1: Using the Binomial Probability Formula P(x) = p x q n-x ( n – x )! x ! n !n ! for x = 0, 1, 2,..., n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p)

21. 21 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Rationale for the Binomial Probability Formula P(x) = p x q n-x n !n ! ( n – x )! x ! The number of outcomes with exactly x successes among n trials

22. 22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Binomial Probability Formula P(x) = p x q n-x n !n ! ( n – x )! x ! Number of outcomes with exactly x successes among n trials The probability of x successes among n trials for any one particular order

23. 23 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (1)Stat → Calculators → Binomial Using StatCrunch

24. 24 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (2)Enter “n” (the sampe size) Using StatCrunch

25. 25 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (3)Enter “p” (the probability of success) Using StatCrunch

26. 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (4)Enter “x” (the number of successes) Using StatCrunch

27. 27 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (4)For P(x) use “=“ (probability at x) For ∑P(x) use “<=“ (summed probability) Using StatCrunch

28. 28 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. An “unfair coin” has a 0.55 probability of getting heads and is tossed 10 times Example What is the probability of getting exactly 5 heads? What is the probability of getting at least 4 heads?

29. 29 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Probability of heads: p = 0.55 Number of tosses: n = 10 “Exactly 5 heads” → P(5) “at least 4 heads” → ∑P(4) P(5) = 0.234 ∑P(4) = 0.262

30. 30 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. An “unfair coin” has a 0.55 probability of getting heads and is tossed 10 times Example What is the probability of getting exactly 5 heads? What is the probability of getting at least 4 heads? p = 0.55n = 10 P(5) = 0.234 ∑P(4) = 0.262

31. 31 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Binomial Distribution: Formulas Std. Dev.  = n p q Mean µ = n p Variance  2  = n p q Where n = number of fixed trials p = probability of success in one of the n trials q = probability of failure in one of the n trials

32. 32 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Interpretation of Results Maximum usual values = µ + 2  Minimum usual values = µ – 2  It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these limits: