Question

Hi, I need help with a homework question that I am working on. The answer and how you came to the answer would be very helpful. This is the question:

Assume a binomial probability distribution with n = 40 and π = .55. Compute the following:
a. The mean and standard deviation of the random variable.
b. The probability that X is 25 or greater.
c. The probability that X is 15 or less.
d. Thr probability that X is between 15 and 25, inclusive.

Thanks!

Answer 1

Assume a binomial probability distribution with $n = 40$ and $\pi = .55$. Compute the following:

a. The mean and standard deviation of the random variable.

b. The probability that X is 25 or greater.

c. The probability that X is 15 or less.

d. The probability that X is between 15 and 25, inclusive.

$$\mathrm{Ex} = \sum_{i=0}^{40} i * C_{40}^{i} p^{i*} (1 - p)^{40 - i} =$$ $$\sum_{i=1}^{40} \frac{40! i 0.55^{i} (1 - 0.55)^{40 - i}}{i! (40 - i)!} = 22$$ $$\mathrm{Dx} = \sum_{i=0}^{40} i^{2} * C_{40}^{i} p^{i*} (1 - p)^{40 - i} - 22^{2} = 493.9 - 484 = 9.9$$

standard deviation = $d = \sqrt{9.9} = 3.14$

b. The probability that X is 25 or greater.

$$P = \sum_{i=25}^{40} C_{40}^{i} p^{i*} (1 - p)^{40 - i} = 0.214214$$

c. The probability that X is 15 or less

$$P = \sum_{i=0}^{15} C_{40}^{i} p^{i*} (1 - p)^{40 - i} = 0.0195775$$

d. The probability that X is between 15 and 25, inclusive.

$$P = \sum_{i=15}^{25} C_{40}^{i} p^{i*} (1 - p)^{40 - i} = 0.858791$$

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