Which of these statements is not true about the variance in a binomial distribution B(n, p)? __________
A. For a fixed p, the variance increases as n increases.
B. For a fixed n, the variance is maximum when p = 0.5.
C. The variance depends only on n.
D. The variance is constant for a specific n and p.
E. None of these are true.
To determine which statement is not true about the variance in a binomial distribution B(n, p), let's analyze each statement:
A. For a fixed p, the variance increases as n increases.
This statement is true. In a binomial distribution, the variance is directly proportional to the number of trials (n). As the number of trials increases, so does the variance.
B. For a fixed n, the variance is maximum when p = 0.5.
This statement is true. The variance in a binomial distribution is symmetric around p = 0.5. When p is close to 0 or 1, there is less variability in the distribution, resulting in a smaller variance. However, when p is close to 0.5, the distribution is more spread out, leading to a larger variance.
C. The variance depends only on n.
This statement is true. The variance in a binomial distribution is solely determined by the number of trials (n). It does not depend on the value of p.
D. The variance is constant for a specific n and p.
This statement is not true. The variance of a binomial distribution is not constant for a specific n and p. It varies based on the number of trials and the probability of success.
E. None of these are true.
Since statement D is not true, the correct answer is D. The variance is not constant for a specific n and p.
The statement that is not true about the variance in a binomial distribution B(n, p) is C. The variance depends only on n.
The variance of a binomial distribution B(n,p) is np(1-p).
So can you figure out which ones are false?