Concept explainers
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.
a.
Find the value of
Answer to Problem 33E
The value of
Explanation of Solution
Calculation:
The binomial experiment with
The probability of obtaining x successes in n independent trails of a binomial experiment is,
Where, p is the probability of success.
Substitute values
Thus, the probability of 12 successes is 0.1144.
b.
Find the value of,
Answer to Problem 33E
The value of,
Explanation of Solution
Calculation:
Consider
Thus, the probability of 16 successes is 0.1304.
c.
Find the value of,
Answer to Problem 33E
The value of
Explanation of Solution
Calculation:
Here,
The probability
The probability value of
The probability value of
The probability value of
The probability value of
The required probability becomes,
Thus, the probability of at least 16 successes is 0.2374.
d.
Find the value of,
Answer to Problem 33E
The value of
Explanation of Solution
Calculation:
The required probability of at most 15 successes can be obtained as the compliment of the probability of at least 15 successes. That is,
Substituting the value from part (c),
Thus, the probability of at most 15 successes is 0.7626.
e.
Compute the expected value.
Answer to Problem 33E
The expected value of the binomial random variable is 14.
Explanation of Solution
Calculation:
The expected value of a binomial random variable is given by,
Substituting the values
The expected value of the binomial random variable is 14.
f.
Compute the variance and standard deviation.
Answer to Problem 33E
The variance of the binomial random variable is 4.2.
The standard deviation of the binomial random variable is 2.0494.
Explanation of Solution
Calculation:
The variance of a binomial random variable is given by,
Substituting the values
The variance of the binomial random variable is 4.2.
The standard deviation of the random variable x is obtained by taking the square root of variance.
Thus, the standard deviation of the binomial random variable is given by,
The standard deviation of the binomial random variable is 2.0494.
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