Statistik Chapter 5

CHAPTER 5 SPECIAL DISTRIBUTION CHAPTER OUTLINE 5.1 Discrete distribution 5.1.1 Binomial distribution 5.1.2 Poisson distribution 5.2 Continuous distribution 5.2.1 Normal distribution 5.3 Introduction to t-distribution OBJECTIVES Student should be able to: Determine the distribution and also differentiate the distribution form. Solve any problem related to Binomial, Poisson and Normal distribution. 5.1DISCRETE DISTRIBUTION Consist of the values a random variable can assume and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation. 5.1.1BINOMIAL DISTRIBUTION Binomial Distribution - the outcomes of a binomial experiment and the corresponding probabilities of these outcomes. It is applied to find the probability that an outcome will occur x times in n performances of experiment. For example: The probability of a defective laptop manufactured at a firm is 0.05 in a random sample of ten. The probability of 8 packages will not arrive at its destination. To apply the binomial probability distribution, the random variable x must be a discrete dichotomous random variable. Each repetition of the experiment must result in one of two possible outcomes. Conditional of a Binomial Experiment Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. These outcomes can be considered as either success or failure. There must be a fixed number of trials. The outcomes of each trial must be independent of each other. In other words, the outcome of one trial does not affect the outcome of another trial. The probability of success is denoted by p and that of failure by q, and p + q =1. The probabilities p and q remain constant for each trial. Note:The success does not mean that the corresponding outcome is considered favorable or desirable and vice versaThe outcome to which the question refers is called a success; the outcome to which it does not refer is called a failure. A.Calculating Binomial Probabilities by Using Binomial Formula For a binomial experiment, the probability of exactly x successes in n trials is given by the binomial formula: P(x) = nCx px qn-x Where; n = the total number of trials p = probability of success q = 1 - p = probability of failure x = number of successes in n trials n - x = number of failures in n trials Example 1 Compute the probabilities of X successes, using the binomial formula. a. n = 6, X = 3, p = 0.03 b. n = 4, X = 2, p = 0.18 Solution: a. P(x = 3) = 6C3(0.03)3(1-0.03)6-3 = 6C3(0.03)3(0.97)3 = 0.0005 b. P(x = 2) = 4C2(0.18)2(1-0.18)4-2 = 4C2(0.18)2(0.82)2 = 0.1307 Example 2 A survey found that one out five Malaysian says he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. Solution: In this case, n = 10, x = 3, p = 1/5 and q = 4/5 P(x = 3) = 10C3(1/5)3(4/5)7 = 0.2013 Exercise 1 1.A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find these probabilities: a)Exactly 3 will fail b)Less than 2 will fail c)None will fail 2. A survey from Teenage Research Unlimited found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs. R. H Bruskin Associates Market Research found that 40% of Americans do not think that having a college education is important to succeed in the business world. If a random sample of five American is selected, find these probabilities. a)Exactly two people will agree with that statement. b)At most three people will agree with that statement c)At least two people will agree with that statement d)Fewer than three people will agree with that statement. It was found that 60% of American victims of health care fraud are senior citizens. If 10 victims are randomly selected, find the probability that exactly 3 are senior citizens. B.Using Table of Binomial Probabilities The probabilities for a binomial experiment can also be read from the table of binomial probabilities. For any number of trials n: The binomial probability distribution is symmetric if p = 0.5. The binomial probability distribution is skewed to the right if p is less than 0.5. The binomial probability distribution is skewed to the left if p is greater than 0.5. 0172085 From the table also, it is easier to calculate various from of binomial distribution such as: EquallyP(X = x) = P(X ≤ x) - P(X ≤ x - 1)At mostP(X ≤ x) = P(X ≤ x) (directly from the table)Less thanP(X < x) = P(X ≤ x - 1) At leastP(X ≥ x) = 1 – P(X ≤ x - 1)Greater thanP(X > x) = 1 – P(X ≤ x)From x1 to x2P(x1 ≤ X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 - 1)Between x1 and x2P(x1,[object Object]

Statistik Chapter 5

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Statistik Chapter 5