1. SADC Course in Statistics The binomial distribution (Session 06)

2. To put your footer here go to View > Header and Footer 2 Learning Objectives At the end of this session you will be able to: describe the binomial probability distribution including the underlying assumptions calculate binomial probabilities for simple situations apply the binomial model in appropriate practical situations

3. To put your footer here go to View > Header and Footer 3 Study of child-headed households One devastating effect of the HIV and AIDS pandemic is the emergence of child-headed households, i.e. ones where both parents have died and the children are left to fend for themselves. Suppose it is of interest to study in greater detail those households that are child-headed. Statistical techniques that may be employed require initially, a knowledge of the distributional pattern of the random variable X corresponding to the number of child-headed households.

4. To put your footer here go to View > Header and Footer 4 Interest is on the distribution of X = number of child-headed households Under certain assumptions, X has a binomial distribution. To introduce this distribution, we first deal with a simpler (but related) distribution A probability distribution for X

5. To put your footer here go to View > Header and Footer 5 The Bernoulli Distribution The simplest probability distribution is one describing the behaviour of a dichotomous (binary) random variable, i.e. one with two possible outcomes; (Success, Failure), (Yes, No), (Female, Male), etc. Outcome Values of random variable Probability Success1p failure01-p Total1

6. To put your footer here go to View > Header and Footer 6 Background In general, we have a sequence of n trials, each with just two possible outcomes. e.g. visiting n households in turn and recording whether it is child-headed. Call one outcome a success, the other a failure. Let probability (of a success) = p. The word success is a generic term used to represent the outcome of interest, e.g. if a sampled household is child-headed we call it a success because that is the outcome of interest.

7. To put your footer here go to View > Header and Footer 7 Let X be the number of successes in n trials. X is said to have a binomial distribution if: The probability of success p is the same for each trial. The trials have independent outcomes. In the context of our example, X=number of child-headed HHs from n HHs sampled. p=probability of a HH being child-headed. Under what conditions would X be binomial? Basics and terminology

8. To put your footer here go to View > Header and Footer 8 Binomial Probability Distribution The probability of finding k successes out of n trials is given by Here n! = n(n-1)(n-2)……. (3) (2) (1); 0!=1. Thus, for example, 4! = 4 x 3 x 2 x 1 = 24. Exercise: If p=0.2 and n=10, confirm that

9. To put your footer here go to View > Header and Footer 9 Binomial Probability Distribution In computing binomial probabilities, the value of p is often unknown. It is then estimated by the proportion of successes in the sample. i.e. Following graphs show binomial probabilities for n=10 and differing values of p.

10. To put your footer here go to View > Header and Footer 10 There are 11 possible outcomes. Graph shows P(X=2)=0.3, P(X=3)=0.2, P(X>6) almost=0. Example 1: Left-handedness Suppose the probability of a person being left- handed is p=0.2. Let X be number left-handed persons in a group of 10. Graph shows probability of 0, 1, 2, … left- handed persons

11. To put your footer here go to View > Header and Footer 11 The distribution is symmetrical. We find P(X=2)=P(X=8)=0.044, P(X=3)=P(X=7)=0.12, etc. Example 2: Tossing a coin A coin is tossed. The probability of getting a head is p=0.5. Let X be number heads in 10 tosses of the coin. Graph shows probability of getting 0, 1, 2, … heads.

12. To put your footer here go to View > Header and Footer 12 The distribution is now concentrated to the right. Here P(X<4) is almost zero. Example 3: Selecting a rural village Ratio of rural villages to urban villages is 4:1. Suppose 10 villages are selected at random. Let X be number of rural villages selected. Graph shows probability of getting 0, 1, 2, … rural villages.

13. To put your footer here go to View > Header and Footer 13 Properties of Binomial Distribution The mean (average) of the binomial distribution with parameters n and p = np. e.g. In a population of size 1000, suppose the probability of selecting a child-headed HH is p=0.03. Then the mean number of child-headed HHs is 1000x0.03 = 30. Recall that the mean = expected value of X. Thus Note: Since the binomial is a probability distribution,

14. To put your footer here go to View > Header and Footer 14 The standard deviation of the binomial distribution is For n=1000, p=0.2 the standard deviation is therefore =[1000*0.2*0.8] ½ = 12.65 The theoretical derivation is given below. Further Properties:

15. To put your footer here go to View > Header and Footer 15 Practical work follows to ensure learning objectives are achieved…