For a binomial distribution, the variance is npq and the standard deviation is the square root of npq. Here, n=2600, p=0.78, and q=0.22. Calculating npq gives 2600 * 0.78 * 0.22 = 446.16. The standard deviation is the square root of this product, which is approximately 21.12. Note: The provided answer 446.16 is the variance, not the standard deviation.
372
In a binomial distribution, how does the probability of success behave across trials?
A key requirement for a binomial distribution is that the probability of success, often denoted as p, must remain identical for every trial conducted. If the probability of success changes from one trial to the next, the experiment does not follow a binomial distribution.
373
Given a mean of 5 and a standard deviation of 2.5, what can be inferred about the nature of the binomial distribution?
A binomial distribution is defined by parameters n and p. The mean is np and variance is npq. If mean=5 and SD=2.5, then variance=6.25. Since variance (npq) must be less than mean (np) for a binomial distribution, a variance of 6.25 and mean of 5 is mathematically impossible for a binomial model, rendering the premise 'Wrong'.
374
What is the standard normal transformation formula used to approximate a Poisson distribution with parameter λ?
For a Poisson distribution with mean λ, the variance is also λ. As λ becomes large, the Poisson distribution can be approximated by a normal distribution. The standard normal variable Z is obtained by subtracting the mean (λ) and dividing by the standard deviation (√λ), resulting in the formula (x - λ) / √λ.
375
How many possible outcomes are associated with each individual trial in a Binomial distribution?
The Binomial distribution is defined by a sequence of independent Bernoulli trials. Each trial must result in exactly one of two mutually exclusive outcomes, typically categorized as 'success' or 'failure'. This binary structure is a foundational requirement for the distribution's probability mass function.
376
Which probability distribution assumes that successive trials are conducted with replacement?
The binomial distribution is based on a sequence of independent Bernoulli trials. Sampling with replacement ensures that the probability of success remains constant across all trials, which is a fundamental requirement for the binomial model. In contrast, sampling without replacement typically leads to the hypergeometric distribution.
377
Under what specific condition does a binomial distribution exhibit perfect symmetry?
A binomial distribution is perfectly symmetrical when the probability of success (p) equals the probability of failure (q), meaning p = q = 0.5. Since q = 1 - p, the condition p = q is equivalent to p = 0.5. The option 'q' represents the condition where the distribution is balanced around the mean.
378
What is the valid range of possible outcomes (random variable X) for a binomial distribution?
In a binomial distribution, the random variable X represents the number of successes in n trials. Since it is possible to have zero successes and it is impossible to have more than n successes, the range of X is the set of integers from 0 to n inclusive.
379
If the mean of a binomial distribution is 4.8, what is the variance of this distribution?
The mean of a binomial distribution is np = 4.8. The variance is npq = 4.8q. Since q = 1 - p, the variance must be less than the mean. Given the options, 2.3 is the only value less than 4.8 that could logically result from a valid probability p. Note that the exact variance depends on n, but 2.3 is the provided answer.
380
What is the relationship between the mean and variance of a binomial distribution?
For a binomial distribution, the mean is np and the variance is npq. Since q = 1-p and 0 < p < 1, it follows that 0 < q < 1. Thus, npq < np, meaning the variance is always less than the mean. The provided answer B is factually incorrect as the mean is greater than the variance.
