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What is the number of trials of a binomial distribution having mean and SD as 3 and 1.5 respectively?
- a)2
- b)4
- c)8
- d)12
Correct answer is option 'D'. Can you explain this answer?
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What is the number of trials of a binomial distribution having mean an...
Understanding the Binomial Distribution
The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p). The mean (μ) and standard deviation (σ) of a binomial distribution can be expressed as:
- Mean (μ) = n * p
- Standard Deviation (σ) = √(n * p * (1 - p))
Given:
- Mean (μ) = 3
- Standard Deviation (σ) = 1.5
Calculating the Parameters
1. From the mean formula:
μ = n * p
Therefore, p = μ/n = 3/n
2. From the standard deviation formula:
σ = √(n * p * (1 - p))
Substituting p:
1.5 = √(n * (3/n) * (1 - 3/n))
Squaring both sides gives:
2.25 = 3 * (1 - 3/n)
3. Expanding this:
2.25 = 3 - 9/n
Rearranging yields:
9/n = 3 - 2.25 = 0.75
Therefore, n = 9/0.75 = 12
Conclusion
The number of trials (n) for the given mean and standard deviation of a binomial distribution is 12. Thus, the correct answer is option 'D'.
This calculation illustrates how the properties of the binomial distribution can be used to derive the number of trials based on mean and standard deviation.
The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p). The mean (μ) and standard deviation (σ) of a binomial distribution can be expressed as:
- Mean (μ) = n * p
- Standard Deviation (σ) = √(n * p * (1 - p))
Given:
- Mean (μ) = 3
- Standard Deviation (σ) = 1.5
Calculating the Parameters
1. From the mean formula:
μ = n * p
Therefore, p = μ/n = 3/n
2. From the standard deviation formula:
σ = √(n * p * (1 - p))
Substituting p:
1.5 = √(n * (3/n) * (1 - 3/n))
Squaring both sides gives:
2.25 = 3 * (1 - 3/n)
3. Expanding this:
2.25 = 3 - 9/n
Rearranging yields:
9/n = 3 - 2.25 = 0.75
Therefore, n = 9/0.75 = 12
Conclusion
The number of trials (n) for the given mean and standard deviation of a binomial distribution is 12. Thus, the correct answer is option 'D'.
This calculation illustrates how the properties of the binomial distribution can be used to derive the number of trials based on mean and standard deviation.
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What is the number of trials of a binomial distribution having mean and SD as 3 and 1.5 respectively?a)2b)4c)8d)12Correct answer is option 'D'. Can you explain this answer?
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