The National Institute of Mental Health reports that there is a 30% chance of an

Question

The National Institute of Mental Health reports that there is a 30% chance of an adult American suffering from a psychiatric disorder. Three randomly selected adult Americans are examined for a psychiatric disorders.

a) If you let a success correspond to an adult American having a psychiatric disorder, what is the success probability, p? (Note: The use if the word success in Bernoulli trials need not reflect its usually positive connotation.)

P(Success = having the psychiatric disorder) = _____________________

b) Construct a probability distribution table for three people examined using Binomial distribution. Let X be the number of adults who are suffering from the psychiatric disorder.

X

0

1

2

3

Total

Prob.

c) Determine the mean and variance for the distribution.     

X

0

1

2

3

Total

Prob.

Explanation / Answer

a) p = 0.3

b) n = 3

P(X = x) = 3Cx * 0.3x * (1 - 0.3)3-x

P(X = 0) = 3C0 * 0.30 * 0.73 = 0.343

P(X = 1) = 3C1 * 0.31 * 0.72 = 0.441

P(X = 2) = 3C2 * 0.32 * 0.71 = 0.189

P(X = 3) = 3C3 * 0.33 * 0.70 = 0.027

Total = 1

c) Mean = n * p = 3 * 0.3 = 0.9

Standard deviation = sqrt(n * p * (1 - p)) = sqrt(3 * 0.3 * 0.7) = 0.7937

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