8.1 Binomial Distributions

Transcription

1 8.1 Binomial Distributions

2 The Binomial Setting The 4 Conditions of a Binomial Setting: 1.Each observation falls into 1 of 2 categories ( success or fail ) 2

3 2.There is a fixed # n of observations. 3.All observations are independent. 4.The probabilities of the two outcomes remain the same after each tria *(they may not be equal to each other but they are constant) 3

4 Binomial random variable (X) -# of successes in data that is produced in a binomial setting. 4

5 Binomial Distribution -the distribution of the count X of successes in the binomial setting -n is the # of observations 5

6 Binomial Distribution -p is the probability of a success on any one observation -As an abbreviation, say that X is B(n,p) 6

7 Example 5-20 According to an Allstate Survey, 56% of Baby Boomers have car loans and are making payments on these loans (USA TODAY, October 28, 2002). Assume that this result holds true for the current population of all Baby Boomers. Let x denote the number in a random sample of three Baby Boomers who are making payments on their car loans. Write the probability distribution of x and draw a bar graph for this probability distribution. 7

8 Example *For 8.3, we can still use B distr if the pop is much larger than the SRS 8

9 Binomial Probability Distribution Function -assigns a probability to each value of X -binompdf(n,p,x) under 2 nd Distr 0 9

10 Figure 5.5 Bar graph of the probability distribution of x. P(x) x 10

11 Cumulative Distribution Function -calculates the sum of the probabilities for 0,1,2, up to the value X -it calculates the prob of obtaining at most X successes in n trials 11

12 8.1 (cont.) Binomial Formulas

13 Example Each child born to a certain set of parents has prob of having blood type O. If these parents have 5 children, find P(X2). 13

14 Binomial Coefficient -the # of ways of arranging r successes among n observations n n! r r!( n r )! 14

15 Alternative Formula -Just use n C r button on calculator!! -Ex C 2 15

16 Binomial Formula -for a binomial setting, the prob. Of exactly r successes in n trials is: P( X ) C p r (1 p) n r n r 16

17 Example The # X of switches that fail inspection has approximately B(10,.1). Find P that no more than 1 switch will fail. 17

18 8.1 Mean & Std Dev of Binom Distrs.

19 Mean & Std Dev σ np( 1 p) 19

20 Mean & Std Dev -these short formulas are good only for binomial distributions -they cannot be used for other discrete random variables 20

21 Example If a basketball player makes 75% of her free throws, the mean # made in 12 tries is 75% of

22 Using the Table of Binomial Probabilities Example 5-21 According to a 2001 study of college students by Harvard University s School of Public health, 19.3% of those included in the study abstained from drinking (USA TODAY, April 3, 2002). Suppose that of all current college students in the United States, 20% abstain from drinking. A random sample of six college students is selected. 22

23 Example 5-21 Using Table IV of Appendix C, answer the following. a) Find the probability that exactly three college students in this sample abstain from drinking. b) Find the probability that at most two college students in this sample abstain from drinking. c) Find the probability that at least three college students in this sample abstain from drinking. d) Find the probability that one to three college students in this sample abstain from drinking. e) Let x be the number of college students in this sample who abstain from drinking. Write the probability distribution of x and draw a bar graph for this probability distribution. 23

24 Solution 5-21 a) P (x 3).0819 b) P (at most 2) P (0 or 1 or 2) P (x 0) + P (x 1) + P (x 2) c) P (at least 3) P(3 or 4 or 5 or 6) P (x 3) + P (x 4) + P (x 5) + P (x 6) d) P (1 to 3) P (x 1) + P (x 2) + P (x 3)

25 Table 5.13 Probability Distribution of x for n 6 and p.20 x P(x)

26 Figure 5.6 Bar graph for the probability distribution of x. P(x) x 26

27 Probability of Success and the Shape of the Binomial Distribution 1.The binomial probability x P(x) distribution Distribution of x for 0 is.0625 n 4 and p.50 symmetric 1if p Table 5.14 Probability 27

28 Figure 5.7 Bar graph from the probability distribution of Table P(x) x

29 and the Shape of the Binomial Distribution cont. 2.The binomial probability Table 5.15 Probability distribution Distribution of x for x is P(x n 4 and p.30 skewed to the 0 right.24 ) if p is less than

30 Figure 5.8 Bar graph for the probability distribution of Table P(x) x

31 and the Shape of the Binomial Distribution cont. The binomial 3. probability distribution x is P( Table 5.16 Probability 0.0 skewed to the x) Distribution of x left if for n 4 and p is greater 1 01 p.80 than

32 Figure 5.9 Bar graph for the probability distribution of Table P(x) x 32

33 Standard Deviation of the Binomial Distribution The mean and standard deviation μ np and of σ a npq binomial distribution are 33

34 Example 5-22 In a Martiz poll of adult drivers conducted in July 2002, 45% said that they often or sometimes eat or drink while driving (USA TODAY, October 23, 2002). Assume that this result is true for the current population of all adult drivers. A sample of 40 adult drivers is selected. Let x be the number of drivers in this sample who often or sometimes eat or drink while driving. Find the mean and standard deviation of the probability distribution of x. 34

35 Solution 5-22 n 40 p.45, and q.55 μ np 40(.45) 18 σ npq (40)(.45)(.55)

36 HYPERGEOMETRIC PROBABILITY DISTRIBUTION Let N total number of elements in the population r number of successes in the population N r number of failures in the population n number of trials (sample size) x number of successes in n trials n x number of failures in n trials 36

37 HYPERGEOMETRIC PROBABILITY DISTRIBUTION The probability of x successes in n trials is C given C by r x N r n x P( x) N C n 37

38 Example 5-23 Brown Manufacturing makes auto parts tha are sold to auto dealers. Last week the company shipped 25 auto parts to a dealer Later on, it found out that five of those parts were defective. By the time the company manager contacted the dealer, four auto parts from that shipment have already been sold. What is the probability that three of those four parts were good parts and one was defective? 38

39 Solution 5-23 P( x 3) r C x N r N C C n n x (1140)(5) 12, C3 5C C ! 5! 3!(20 3)! 1!(5 1)! 25! 4!(25 4)! Thus, the probability that three of the four parts sold are good and one is defective is

40 Example 5-24 Dawn Corporation has 12 employees who hold managerial positions. Of them, seven are female and five are male. The company is planning to send 3 of these 12 managers to a conference. If 3 managers are randomly selected out of 12, a) Find the probability that all 3 of them are female b) Find the probability that at most 1 of them is a female 40

41 Solution 5-24 (a) P( x 3) r C x N r N C C n n x C C (35)(1) C Thus, the probability that all three of managers selected re female is

42 42 Solution ) ( 0) ( 1) ( (7)(10) 1) ( (1)(10) 0) ( x P x P x P C C C C C C x P C C C C C C x P n N x n r N x r n N x n r N x r (b)

43 THE POISSON PROBABILITY DISTRIBUTION Using the Table of Poisson probabilities Mean and Standard 43

44 PROBABILITY DISTRIBUTION cont. Conditions to Apply the Poisson Probability Distribution 44

45 Examples 1.The number of accidents that occur on a given highway during a one-week period. 45

46 PROBABILITY DISTRIBUTION cont. Poisson Probability Distribution Formula According to the Poisson probability distribution, the probability of x occurrences x λ in an interval ( is λ e P x ) x! where λ is the mean number of occurrences i that interval and the value of e is approximately

47 Example 5-25 On average, a household receives 9.5 telemarketing phone calls per week. Using the Poisson distribution 47

48 Solution 5-25 P( x 6) λ x e x! λ (9.5) e 6! (735, )( )

49 Example 5-26 A washing machine in a laundromat breaks down an average of three times per month. Using the Poisson 49

50 50 Solution (3)( ) 1 (1)( ) 1! (3) 0! (3) 1) ( 0) ( ) ( (9)( ) 2! (3) 2) ( ) ( e e x P x P b e x P a

51 Example 5-27 Cynthia s Mail Order Company provides free examination of its products for seven days. If not completely satisfied, a customer can return the product within that period and get a full refund. According to past records of the company, an average of 2 of every 10 products sold by this company are returned for a refund. Using the Poisson probability distribution formula, find the probability that exactly 6 of the 40 products sold by this company on a given day will be returned for a refund. 51

52 Solution 5-27 λ 8 x 6 P( x 6) x λ e x! λ 6 (8) e 6! 8 (262,144)( )

53 Using the Table of Poisson Probabilities Example 5-28 On average, two new accounts are opened per day at an Imperial Saving 53

54 Table 5.17 Portion of Table of Poisson Probabilities for λ 2.0 λ x λ 2.0 x 6 P (x 6) 54

55 Solution 5-28 a) P (x 6).0120 b) P (at most 3) P (x 0) + P (x 1) + P (x 2) + P (x 3)

56 Deviation of the Poisson Probability Distribution μ λ σ 2 λ σ λ 56

57 Example 5-29 An auto salesperson sells an average of.9 car per day. Let x be the number of cars sold by this salesperson on any given day. Using the Poisson probability distribution table, a) Write the probability distribution of x. b) Draw a graph of the probability distribution. c) Find the mean, variance, and standard deviation. 57

58 Table 5.18 Probability Distribution of x for λ.9 Solution 5-29 a x P.40 (x)

59 Figure 5.10 Bar graph for the probability distribution of Table Solution 5-29 b P(x) x 59

60 Solution 5-29 Solution 5-29 c μ λ.9 car σ 2 λ.9 σ λ car 60