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Question
Statistics
Posted 4 months ago

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=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=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=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, nn. 2. Each trial has only two possible outcomes: success or failure. 3. The probability of success, pp, 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=Q= the number of red pens obtained. - Fixed number of trials: Yes, n=5n=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=S= the number of red pens obtained. - Fixed number of trials: Yes, n=5n=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=R= the number of red pens obtained. - Fixed number of trials: Yes, n=5n=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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