Binomial Random Variables
When solving statistics problems, you must know the ways to find binomial probabilities. In these practice questions, pay special attention to the normal approximation. Solve the following problems about the basics of binomial random variables.
You roll a six-faced die ten times and record which face comes up each time (X). Why is X not a binomial random variable?
Answer: because each trial has more than two possible outcomes
In this case, you have six possible outcomes on each trial, but a binomial trial may have only two possible outcomes: success or failure. Here, X represents the outcome of one roll of the die (1, 2, 3, 4, 5, or 6), not the total number of rolls with a certain outcome (such as the total number of 6’s rolled).
You interview a number of employees selected at random and ask them whether they’ve graduated from high school. You continue the interviews until you have 30 employees who say they graduated from high school. If X is the number of people you had to ask until you got 30 “yes” responses, why isn’t X a binomial random variable?
Answer: because the number of trials isn’t fixed
For a binomial experiment, the number of trials must be specified in advance. In this case, although you know in the end that you need 30 employees who say they graduated from high school, you don’t know how many employees you’ll have to ask before finding 30 who graduated from high school. Because the total number of trials, n, is unknown, X isn’t binomial.
You recruit 30 pairs of siblings and test each individual to see whether he or she is carrying a particular genetic mutation, and then you add up the total number of people (not pairs) who have the mutation. Why is this not a binomial experiment?
Answer: because the trials aren’t independent
If one sibling has the mutation, a higher chance exists that the other one will, too, so the results for each person aren’t independent. Instead of recruiting 30 pairs of siblings for this test, you should recruit 60 people at random.
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