Question

Statistics

Posted 4 months ago

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Which of the following choices can be considered binomial random variables?
Choose all answers that apply:
A A container holds 20 pens, half of which are red. Select an SRS of 5 pens and let $Q=$ the number of red pens obtained.
B A container holds 1000 pens, half of which are red. Select an SRS of 5 pens and let $S=$ the number of red pens obtained.
c A container holds 100 pens, half of which are red. Select an SRS of 5 pens and let $R=$ the number of red pens obtained.
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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Identify the criteria for a binomial random variable. A binomial random variable must satisfy the following conditions:
1. There are a fixed number of trials, $n$.
2. Each trial has only two possible outcomes: success or failure.
3. The probability of success, $p$, is the same for each trial.
4. The trials are independent

step 2

Evaluate each scenario to see if it meets the criteria for a binomial random variable:
- **Choice A**: A container holds 20 pens, half of which are red. Select an SRS of 5 pens and let $Q=$ the number of red pens obtained.
- Fixed number of trials: Yes, $n=5$.
- Two possible outcomes: Yes, red or not red.
- Probability of success: No, the probability changes as pens are selected without replacement.
- Independence: No, the trials are not independent due to the changing probability.
- **Choice B**: A container holds 1000 pens, half of which are red. Select an SRS of 5 pens and let $S=$ the number of red pens obtained.
- Fixed number of trials: Yes, $n=5$.
- Two possible outcomes: Yes, red or not red.
- Probability of success: Approximately yes, since the population is large, the probability of selecting a red pen remains nearly constant.
- Independence: Approximately yes, for a large population, the trials can be considered nearly independent.
- **Choice C**: A container holds 100 pens, half of which are red. Select an SRS of 5 pens and let $R=$ the number of red pens obtained.
- Fixed number of trials: Yes, $n=5$.
- Two possible outcomes: Yes, red or not red.
- Probability of success: No, the probability changes as pens are selected without replacement.
- Independence: No, the trials are not independent due to the changing probability

Answer

B

Key Concept

Binomial Random Variable Criteria

Explanation

A binomial random variable requires a fixed number of trials, two possible outcomes, constant probability of success, and independent trials.

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