Sta220 - Statistics Mr. Smith Room 310 Class #12

Section 4.3

4.3- The Binomial Random Variable Many experiments result in responses for which there exist two possible alternatives, such as Yes- No, Pass-Fail, Defective-Nondefective or Male- Female. These experiments are equivalent to tossing a coin a fixed number of times and observing the number of times that one of the two possible outcomes occurs. Random variables that possess these characteristics are called Binomial random variables.

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Definition

Example 4.10 The Heart Association claims that only 10% of U.S. adults over 30 years of age meet the minimum requirements established by President’s Council of Fitness, Sports and Nutrition. Suppose four adults are randomly selected and each is given the fitness test.

a)Find the probability that none of the four adults passes the test. b)Find the probability that three of the four adults pass the test. c)Let x represent the number of four adults who pass the fitness test. Explain why x is a binomial random variable. d)Use the answers to parts a and b to derive a formula for p(x), the probability distribution of the binomial random variable x.*

a) 1) First step is define the experiment. We are observing the fitness test results of each of the four adults: Pass(S) or Fail(F)

2) Next we need to list the sample points associated with the experiment. Example: FSSS represents the sample point denoting that adult 1 fails, while adult 2, 3, and 4 pass the test. There are 16 sample points.

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Table 4.2

3) Now we have to assign probabilities to the sample points. Since sample points can be viewed as the intersection of four adults’ test results, and assuming that the results are independent, we can obtain the probability of each sample point by the multiplicative rule.

c. I.We can characterize this experiment as a consisting on four identical trials: the four test results II.There are two possible outcomes to each trail, S of F. III.The probability of passing, p =.1, is the same for each trial. IV.Assuming that each adult’s test result is independent of all others, so that the four trials are independent.

d. *Since time is limited in this class, we will not derive a formula for p(x) in class. However, I have attached it to the end of the PowerPoint.

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Procedure

Example 4.11 Refer to Example use the formula for a binomial random variable to find the probability distribution of x, where x is the number of adults who pass the fitness test. Graph the distribution.

For this application, we have n = 4 trails. Since a success S is defined as an adult who passes the test, p = P(S) =.1 andq = 1- p =.9 So we have the following: n = 4 p =.1 q =.9

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Table 4.3

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 4.8 Probability distribution for physical fitness example: graphical form

Example 4.12

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Definition

Using Binomial Tables Calculating binomial probabilities becomes tedious when n is large. For some values of n and p, the binomial probabilities have been tabulated to Table II of Appendix A. The entries in the table represent cumulative binomial probabilities.

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 4.9 Binomial probability distribution for n=10 and p=.10, with highlighted

Example 4.13 Suppose a poll of 20 voters taken in a large city. The purpose is to determine x, the number who favor a certain candidate for mayor. Suppose that 60% of all the city’s voters favor the candidate.

First, n = 20 p =.60 q =.40

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 4.10 The binomial probability distribution for x in Example 4.13; n=20 and p=.6