Question

Which statement is not true for a binomial distribution with n = 15 and p = 1/20 ? a) The standard deviation is 0.8441 b) The number of trials is equal to 15 c) The probability that x equals 1 is 0.3658 d) The highest probability occurs when x equals 0.7500 e) The mean equals 0.7500 f) None of the above

Answer 1

Answer:

d) The highest probability occurs when x equals 0.7500

Step-by-step explanation:

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability, given by the following formula:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

In this problem, we have that:

$n = 15, p = \frac{1}{20}$

a) The standard deviation is 0.8441

$\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{15*0.05*0.95} = 0.8441$

This is correct

b) The number of trials is equal to 15

n is the number of trials and $n = 15$. So this option is correct.

c) The probability that x equals 1 is 0.3658

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 1) = C_{15,1}.(0.05)^{1}.(0.95)^{14} = 0.3658$

This option is correct.

d) The highest probability occurs when x equals 0.7500

False. The number of sucesses is a discrete number, that is, 0, 1, 2,...,15. P(X = 0.75), for example, does not exist.

e) The mean equals 0.7500

$E(X) = np = 15*0.05 = 0.75$

This option is correct.

f) None of the above

d is false