CHAPTER 5 THE BINOMIAL PROBABILITY DISTRIBUTION AND RELATED TOPICS BINOMIAL PROBABILITY DISTRIBUTIONS (SECTIONS 5.2, 5.3 OF UNDERSTANDABLE STATISTICS) The binomial probability distribution is a discrete probability distribution controlled by the number of trials, n, and the probability of success on a single trial, p. MINITAB has three main commands for studying probability distributions. The PDF (probability density function) gives the probability of a specified value for a discrete distribution. The CDF (cumulative distribution function) for a value X gives the probability a random variable with distribution specified in a subcommand is less than or equal to X. The INVCDF gives the inverse of the CDF. In other words, for a probability P, INVCDF returns the value X such that P CDF(X). In this case of a binomial distribution. INVCDF often gives the two values of X for which P lies between the respective CDF(X). The three commands PDF, CDF, and INVCDF apply to many probability distributions. To apply them to a binomial distribution, we need to use the menu selections. Example Calc Probability distributions Binomial Dialog Box Responses Select Probability for PDF; Cumulative probability for CDF; Inverse cumulative probability for INVCDF Number of trials: use the value of n in a binomial experiment, Probability of success: use the value of p, the probability of success on a single trial, Input column: Put the values of r, the number of successes in a binomial experiment in a column such as C1. Select an optional storage column. Note: MINITAB uses X instead of r to count the number of successes, Input constant: Instead of putting values of r in a column, you can type a specific value of r in the dialog box. A surgeon performs a difficult spinal column operation. The probability of success of the operation is p = 0.73. Ten such operations are scheduled. Find the probability of success for 0 through 10 successes out of these ten operations. First enter the possible values of r, 0 through 10, in C1 and name the column r. We will put the probabilities in C2, so name the column P(r). 218
Part III: MINITAB Guide 219 Fill in the dialog box as shown below. Then use the Data Display data command.
220 Technology Guide Understandable Statistics, 8th Edition Next use the CDF command to find the probability of 5 or fewer successes. In this case use the option for an input constant of 5. Leave Optional storage blank. The output will be P(r 5). Note that MINITAB uses X in place of r. The results follow. Finally use INVCDF to determine how many operations should be performed in order for the probability of that many or fewer successes to be 0.5. We select Inverse cumulative probability. Use.5 as the input constant. The results follow.
Part III: MINITAB Guide 221 LAB ACTIVITIES FOR BINOMIAL PROBABILITY DISTRIBUTIONS 1. You toss a coin 8 times. Call heads success. If the coin is fair, the probability of success P is 0.5. What is the probability of getting exactly 5 heads out of 8 tosses? of exactly 20 heads out of 100 tosses? 2. A bank examiner s record shows that the probability of an error in a statement for a checking account at Trust Us Bank is 0.03. The bank statements are sent monthly. What is the probability that exactly two of the next 12 monthly statements for our account will be in error? Now use the CDF option to find the probability that at least two of the next 12 statements contain errors. Use this result with subtraction to find the probability that more than two of the next 12 statements contain errors. You can use the Calculator key to do the required subtraction. 3. Some tables for the binomial distribution give values only up to 0.5 for the probability of success p. There is symmetry to the values for p greater than 0.5 with those values of p less than 0.5. (a) Consider the binomial distribution with n = 10 and p =.75. Since there are 0 10 successes possible, put 0 10 in C1. Use PDF option with C1 and store the distribution probabilities in C2. Name C2 = P =.75. We will print the results in part (c). (b) Now consider the binomial distribution with n = 10 and p =.25. Use PDF option with C1 and store the distribution probabilities in C3. Name C3 = P =.25. (c) Now display C1 C2 C3 and see if you can discover the symmetries of C2 with C3. How does P(K = 4 successes with p =.75) compare to P(K = 6 successes with p =.25)? (d) Now consider a binomial distribution with n = 20 and p =.035. Use PDF on the number 5 to get P(K = 5 successes out of 20 trials with p =.35). Predict how this result will compare to the probability P(K = 15 successes out of 20 trials with p =.65). Check your prediction by using the PDF on 15 with the binomial distribution n = 20, p =.65. The INVCDF command for a binomial distribution can be used in the solution of Quota problems as described in Section 5.3 of Understandable Statistics. 4. Consider a binomial distribution with n = 15 and p = 0.64. Use the INVCDF to find the smallest number of successes K for which P(X K) = 0.98. What is the smallest number of successes K for which P(X K) = 0.09? COMMAND SUMMARY To Find Probabilities PDF E [E] calculates probabilities for the specified values of a discrete distribution and calculates the probability density function for a continuous distribution. CDF E [E] gives the cumulative distribution. For any value X, CDF X gives the probability that a random variable with the specified distribution has a value less than or equal to X. INVCDF E [E] gives the inverse of the CDF. Each of these commands apply the following distributions (as well as some others). If no subcommand is used, the default distribution is the standard normal. BINOMIAL n = K p = K POISSON µ = K (note that for the Poisson distribution µ = λ) INTEGER a = K b = K DISCRETE values in C, probabilities in C NORMAL [µ = K [σ = K]] UNIFORM [a = k b = K]
222 Technology Guide Understandable Statistics, 8th Edition T F CHISQUARE d.f. = K d.f. numerator = K d.f. denominator = K d.f. = K WINDOWS menu selection: Calc Probability Distribution Select distribution In the dialog box, select Probability for PFD; Cumulative probability for CDF; Inverse cumulative for INV; Enter the required information such as E, n, p, or µ, d.f. and so forth.