Consider a binomial experiment with n = 20 and p = .70.
a. Compute f (12).
b. Compute f (16).
c. Compute P(x ≥ 16).
d. Compute P(x ≤ 15).
e. Compute E(x).
f. Compute Var(x) and σ.
The Correct Answer and Explanation is :
In this problem, we’re dealing with a binomial experiment where:
- ( n = 20 ) (number of trials),
- ( p = 0.70 ) (probability of success).
The binomial probability mass function (PMF) is given by:
[
f(x) = P(X = x) = \binom{n}{x} p^x (1 – p)^{n – x}
]
Where:
- ( \binom{n}{x} ) is the binomial coefficient, which can be calculated as ( \frac{n!}{x!(n – x)!} ).
- ( p^x ) is the probability of getting exactly ( x ) successes.
- ( (1 – p)^{n – x} ) is the probability of getting exactly ( n – x ) failures.
a. Compute ( f(12) )
To calculate ( f(12) ), use the binomial PMF formula:
[
f(12) = \binom{20}{12} (0.70)^{12} (0.30)^{8}
]
b. Compute ( f(16) )
Similarly, for ( f(16) ):
[
f(16) = \binom{20}{16} (0.70)^{16} (0.30)^{4}
]
c. Compute ( P(x \geq 16) )
To calculate the probability that ( x \geq 16 ), sum the probabilities from ( x = 16 ) to ( x = 20 ):
[
P(x \geq 16) = f(16) + f(17) + f(18) + f(19) + f(20)
]
d. Compute ( P(x \leq 15) )
To calculate the probability that ( x \leq 15 ), sum the probabilities from ( x = 0 ) to ( x = 15 ):
[
P(x \leq 15) = f(0) + f(1) + \cdots + f(15)
]
e. Compute ( E(x) )
The expected value ( E(x) ) for a binomial distribution is given by:
[
E(x) = n \cdot p = 20 \cdot 0.70 = 14
]
f. Compute ( \text{Var}(x) ) and ( \sigma )
The variance ( \text{Var}(x) ) for a binomial distribution is given by:
[
\text{Var}(x) = n \cdot p \cdot (1 – p) = 20 \cdot 0.70 \cdot 0.30 = 4.2
]
The standard deviation ( \sigma ) is the square root of the variance:
[
\sigma = \sqrt{4.2} \approx 2.05
]
Summary
- ( f(12) ) and ( f(16) ) are computed using the binomial PMF formula.
- ( P(x \geq 16) ) and ( P(x \leq 15) ) require summing appropriate probabilities.
- ( E(x) = 14 ) and ( \text{Var}(x) = 4.2 ), with ( \sigma \approx 2.05 ).
These calculations give insight into the expected outcome and variability of the binomial distribution.