Question

Which of the following graphs represent a binomial distribution with n=20 and p=0.25

Answer 1

Option B represents a binomial distribution with n=20 and p=0.25.

Step-by-step explanation:

To determine the graph representing a binomial distribution with n=20 and p=0.25, we can calculate the mean (μ) and standard deviation (σ) using the formulas:

μ = np

σ = sqrt(np(1 - p))

For n=20 and p=0.25:

μ = 20 * 0.25 = 5

σ = sqrt(20 * 0.25 * 0.75) ≈ 2.7386

The mean (μ) gives the center of the distribution, which is 5, and the standard deviation (σ) measures the spread, approximately 2.7386 units from the mean. In a binomial distribution, the shape is influenced by these values.

Option B's graph shows a center around 5 and the spread seems appropriate for a standard deviation of approximately 2.7386. It aligns with the expected characteristics of a binomial distribution with n=20 and p=0.25. Options A, C, and D don't match these calculated values for the mean and standard deviation.

Therefore, based on the calculated mean and standard deviation, option B is the correct representation of a binomial distribution with n=20 and p=0.25.

Answer 2

A graph representing a binomial distribution with n=20 and p=0.25 would show a discrete distribution, where the mean is 5 and the standard deviation is roughly 1.9365. It would typically be skewed to the right because p is less than 0.5. A Poisson distribution is not suitable for this scenario since p is not less than 0.05.

Step-by-step explanation:

The question is asking which graphs would represent a binomial distribution with parameters n = 20 and p = 0.25. A binomial distribution is used to model the number of successes in a fixed number, n, of independent trials, with each trial having a success probability p. Considering the parameters given (n=20 and p=0.25), in order to graph the distribution accurately, one would expect a discrete graph where the x-axis represents the number of successes and the y-axis represents the probability of those successes.

For a binomial distribution with n=20 and p=0.25, we calculate the mean (μ) as μ = np = 20*0.25 = 5 and the standard deviation (o) as o = √(npq) = √(20*0.25*0.75). A correct graph should reflect the skewness and probabilities consistent with these calculations. The distribution's shape is typically skewed to the right when p < 0.5, which would be the case here.

Furthermore, for approximating a binomial distribution with a normal distribution, both np and nq should be greater than five, which is satisfied here since np = 5 and nq = 15. However, because p is not less than 0.05, the Poisson distribution would not be a suitable approximation in this case.