Question

Consider a binomial experiment with n = 20 and p = 0.80. (Round your answers to four decimal places.)

(a) Compute f(12).
f(12) =
(b) Compute (16).
f(16) =
(c) Compute P(x ≥ 16).
P(x ≥ 16) =
(d) Compute P(x ≤ 15).
P(x ≤ 15) =
(e) Compute E(x).
E(x) =
(f) Compute Var(x) and σ.
Var(x) =
a =

Answer 1

The question involves calculating various probabilities, expected value, and variance for a binomial distribution with 20 trials and a success probability of 0.80. Using the binomial probability formula and known statistical properties, one can find these values.

Step-by-step explanation:

The student's question deals with a binomial experiment where n = 20 and p = 0.80. The various parts of the question involve computing specific probability outcomes, the expected value, and variance for this binomial distribution.

To answer part (a), we compute the probability of getting exactly 12 successes (f(12)) using the binomial probability formula: P(x) = (n choose x) * p^x * (1-p)^(n-x). For part (b), we find the probability of getting exactly 16 successes (f(16)), following the same formula.

For part (c), to compute the probability of getting at least 16 successes (P(x ≥ 16)), we need to consider the probabilities of getting 16, 17, 18, 19, and 20 successes and sum them up. Part (d) involves finding the cumulative probability of getting 15 or fewer successes (P(x ≤ 15)).

Finally, we calculate the expected value (E(x)) and variance and standard deviation (Var(x) and σ) using the formulas E(x) = n*p and Var(x) = n*p*(1-p), respectively, with the standard deviation being the square root of the variance.