Elementary Statistics – Binomial and Normal-Approximation-to-the-Binomial Examples

Example 1 (Binomial distribution: Check conditions, determine mean, use Table C to determine probabilities)

You roll a fair 4-sided die 20 times and record the number of 2s that are rolled.

1. Is this a binomial setting? (Check the BINS assumptions.)

2. What is the expected number of 2s rolled?

3. What is the probability of getting exactly three 2s?

4. What is the probability of getting at least ten 2s?

Example 2 (Rule of thumb for approximate independence; Rephrasing problem in order to use Table C)

A factory produces a computer part that is very difficult to manufacture. Thus, 60% of the parts are defective. A simple random sample of 8 parts is selected from the large population (100,000) of computer parts, and the number of defective parts in the sample is recorded.

1. Is this an approximate binomial setting? What if the population size was 10, instead of 100,000?

2. What is the probability of getting at most one defective computer part?

Example 3 (Normal approximation in the binomial setting; phrased on terms of the sample number of successes)

According to the National Sleep Foundation, 23% of American adults have fallen asleep at the wheel in the last year. A researcher selects a random sample of 120 people and finds that 21 of them have fallen asleep at the wheel in the last year. Assuming that the National Sleep Foundation’s claim is correct, what is the chance that 21 or fewer people in the sample would have fallen asleep at the wheel?

Example 4 (Normal approximation in the binomial setting; phrased in terms of the sample proportion of successes)

A political candidate has the support of 46% of the population of a large city. A simple random sample of 200 people is selected. What is the approximate probability that more than 50% of the sample supports the candidate?