By: M.A.Yulianto. *)

Sometimes populations composed of two classes (two*outcomes*)*,* for example men and women, married and unmarried. For a population that consists of two classes, it will be valid provided that if the proportion of cases in one class is in the proportion of P, then the other class must (1-P) or usually written by Q. when in a single experiment (*single trial*) the possibility of the occurrence of an event is equal to P then it is likely not the occurrence of such events is Q or (1-P). If the experiment is done over and over again as many as n times, done is *independent*, we want to know is how likely the event will occur exactly X times as much as n times of experiment done?. To answer the question, do the calculation using the formula:

Where:

P is the proportion of case that is expected to occur within one of the categories.

Q is the proportion of case expected in other categories

Example:

A survey of HRD society stated that 68% of employees agree that his superior has the right to monitor the phone being used (“snapshot,” usatoday.com, April 18, 2006). If 20 employees are selected by random and they asked whether their boss have the right to monitor the usage of their telephone, how is probability that:

a) 8 employees will agree.

b) 18 or less than employees would agree

solution:

n = 20 P = 0.68 Q = 0.32

**BINOMIAL TEST**

Binomial distribution is used to determine the odds of the outcome might be if a sample is taken from a population of binomial. If the hypothesis is Ho: P = Po, this test States that whether proportion (frequency) of the two categories of the samples are taken from a population with the hypothetical value of Po and (1-Po). For small samples (n ≤ 35), a critical value can be found from a binomial table (can be seen in table D *nonparametric statistics* book written by Sidney Siegel or Siegel-Castelan). For one-way test if the value on the table is smaller than the value of α, then the decision is rejecting Ho. For two-way test if the value in the table is smaller than the value of α/2, then the decision is rejecting Ho. For great samples (n > 35), critical values can be approached with the standard normal distribution Z (Z constitute approaches a normal distribution with the mean equal to 0 and the standard deviation equal to 1).

Where (x + 0.5) is used if x < (n p), and (x – 0.5) is used if x > (n p). If the value of Z > Z_{α/2} or Z <-Z_{α/2} then the decision is rejecting Ho (for two-way test). For one-way test, if the value of Z > Z_{α} or Z <-Z_{α} then the decision is rejecting Ho.

Example: (small sample)

From the 15 cars stopped at the rest area in *jagorawi* toll, 10 driver order chicken soup. What is probability that the drivers who ordered chicken soup are greater than not ordered chicken soup with α = 5%, or in other words we can say that whether the proportion of samples originated from a population that has a chance of ordering chicken soup is greater than not ordering?.

Ordered Not ordered Total

Frequency 5 10 15

Ho: P = Q = ½ (there is no difference between the frequency of driver ordering chicken soup with not ordering).

H1: P > Q (the frequency of driver ordering chicken soup is greater than not ordering).

From binomial table (table D), On n = 15 and X = 5 we gained the value = 0.151. Since 0.151 > α then the decision is to accept Ho. It means that the frequency of driver ordering chicken soup is the same as the frequency of driver not ordering chicken soup.

Example: (large sample)

If the frequencies are added

Ordered Not ordered Total

Frequency 11 25 36

H_{0}: P = Q = ½ (there is no difference between the frequency of driver ordering chicken soup with not ordering).

H_{1}: P > Q (the frequency of driver not ordering chicken soup is greater than ordering).

Z_{0.05} = 1.645

The decision reject Ho because Z > Z_{0.05} , which means that the frequency of driver not ordering chicken soup is greater than the frequency of driver ordering chicken soup.

See you in other writing sessions, have enjoying statistics.

If you have questions can be send to e-mail address: yuliantoyorki@yahoo.com

*) The author is a lecturer in Institute of Statistics, Jakarta.

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