Which of the following graphs best represents the probability mass function of a binomial distribution with $n=20$ and $p=0.25$?

• A. A histogram with bars peaking at $15$ and decreasing towards both $0$ and $20$, showing a left-skewed distribution.
• B. A histogram with bars peaking at $10$ and symmetric about $10$, showing a bell-shaped distribution.
• C. A histogram with all bars of equal height from $0$ to $20$, showing a uniform distribution.
• D. A histogram with bars starting at $0$, peaking around $5$, and then decreasing towards $20$, showing a right-skewed distribution.

Answer: (D)

Verified step by step guidance

1. Recall that the binomial distribution models the number of successes in \( n \) independent Bernoulli trials, each with success probability \( p \). Here, \( n = 20 \) and \( p = 0.25 \).
2. Calculate the mean (expected value) of the binomial distribution using the formula: \[ \mu = n \times p \] This gives the center or peak location of the distribution.
3. Understand the shape of the binomial distribution based on \( p \): - If \( p = 0.5 \), the distribution is symmetric and bell-shaped. - If \( p < 0.5 \), the distribution is right-skewed (peak near 0). - If \( p > 0.5 \), the distribution is left-skewed (peak near \( n \)).
4. Since \( p = 0.25 < 0.5 \), expect the distribution to be right-skewed, with the highest probability mass near \( k = 5 \) (the mean), and probabilities decreasing as \( k \) approaches 20.
5. Compare the given histogram descriptions to this expected shape: the correct graph should show bars starting at 0, peaking around 5, and then decreasing towards 20, indicating a right-skewed distribution.