5. The four conditions required for using a Binomial distribution are (a) A fixe

Question

5. The four conditions required for using a Binomial distribution are (a) A fixed number of trials (n). Trials can be the repeat of a process (like flipping a coin), the repeat of an experiment (like giving a person a drug to treat a disease), or the repeat of a sampling process (like randomly sampling n people from a large population) (b) On each trial, there are two possible outcomes, one of which we call a "success" (c) On each trial, Pr(success) is the same (d) The outcomes of each trial are independent For each of the following decide if the random variable defined (X) is a Binomial variable or not. If it is not a Binomial variable, say which of the four conditions are not met. If it is a Binomial, give the probability function, Pr(X = r) = substituting in the values for n and given by the question. (a) A box contains 100 batteries. Ten of the batteries are dead. Five batteries are ran- domly selected (simultaneously sampling 5 is the same as sampling one-by-one without replacement). Define X as the number of dead batteries in the sample (b) In a large population 5% of the people are unemployed. We randomly select 10 people Define X as the number of employed people in the sample. When the question says "large population" this means that there is very little difference between sampling with and without replacement. So even though sampling 10 people is the same as sampling one-by-one without replacement, we can treat the situation as if it were sampling with replacement (c) In a large population of married couples (husbands and wives) 5% all people are unem- ployed. We randomly select 5 couples (husbands and wives). Define X as the number of unemployed people in the 10 people sampled (d) Roll a six-sided die repeatedly until it comes up 5. Defined X as the total number of rolls

Explanation / Answer

A) X is not a Binomial random variable since probability of success in each trial doesn't remain same and the trials are not independent. Because in one by one WOR sampling, if one dead battery is obtained, the probability of success reduces for the next one and so on.

B) Since the experiment can be considered a WR sampling experiment, X is a Binomial variable with P (success) = 0.05 and n = 10.

C) Same as part (b). X is a Binomial variable.

D) X is not Binomial since the trials are not independent.

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