Heart failure is due to either natural occurrences (88%) or outside factors (12%

Question

Heart failure is due to either natural occurrences (88%) or outside factors (12%). Outside factors are related to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection. Suppose that 19 patients will visit an emergency room with heart failure. Assume that causes of heart failure between individuals are independent. Round your answers to three decimal places (e.g. 98.765). (a) What is the probability that 3 individuals have conditions caused by outside factors? Probability = (b) What is the probability that 3 or more individuals have conditions caused by outside factors? Probability = (c) What is the mean and standard deviation of the number of individuals with conditions caused by outside factors? Mean = Standard deviation =

Explanation / Answer

Solution

Let X = number of patients out of 19 patients visiting the emergency room with heart failure due to outside factors. Then, X ~ B(19, 0.12)

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then probability mass function (pmf) of X is given by

p(x) = P(X = x) = (nCx)(px)(1 - p)n – x, x = 0, 1, 2, ……. , n ………………………………..(1)

Mean (average) of X = E(X) = np…………………………………………………………..(2)

Variance of X = V(X) = np(1 – p)…………………………………………………………..(3)

Standatd Deviation of X = SD(X) = { np(1 – p)} ………………………………………...(4)

Part (a)

Probability that 3 individuals have conditions caused by outside factors

= P(X = 3)

= 0.216566

= 0.217 ANSWER

Part (b)

Probability that 3 or more individuals have conditions caused by outside factors

= P(X 3)

= 1- P(X 2)

= 1 – 0.596763

= 0.403237

= 0.403 ANSWER

Part (c)

Mean of the number of individuals with conditions caused by outside factors

= E(X)

= [Vide (2) of Back-up Theory], np

= 19 x 0.12

= 2.28 ANSWER 1

Standard deviation of the number of individuals with conditions caused by outside factors

= SD(X)

= [Vide (4) of Back-up Theory], sqrt{np(1 - p)}

= sqrt(19 x 0.12 x 0.88)

= 1.416 ANSWER 2

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