Question

17. (15pts] Consider a surgical procedure to rebuild damaged cartilage in the knee. According to previous surgeries, the procedure has an 85% of succeeding with no further complications. A doctor performs this procedure 20 times. (a) The doctor would like to use the normal approximation for binomial. Find the mean and standard deviation for X, the number of patients with further complications. (b) The doctor is interested in calculating the probability of seeing at most one patient with a future complication. State this using probability notation. (c) Calculate this probability using the standard deviation above. (d) After performing the above calculations, is the normal approximation for binomial appropriate here? (e) Calculate the probability of the above calculation using the binomial distribution.

Answer 1

Answer:

We need to use bernoulli and binomial distribution theory to solve the problems.

A.

Mean of X bar = 0.85

Standard deviation of X bar = 0.0798

B.

X : number of successful surgeries out of 20

P(X >=19)

C.

0.1057

D.

The normal approximation would not be much appropriate as by central limit theorem, the sample size should be large,( atleast 30 recommended)

Z = (X - E(X) /$\sqrt$Var(x) follows N(0,1)

E.

0.1756

failure in each success procedure a ) m Bernoulli (p=0.85) Elxi) = 085 var(xi) Pli-P) = 011275 Mean of X :- El 2x) 1 EECXD } nP P = 0,85 standard deviation of X:- 1 Vuar (2x) I e var(xi) ha V V 1 np (1-P) 2

V P(1-P) n 0.0798 the probability of seeing cne b) The doctor is interested in calculating atmost 1 persen with future complication this as, doctor interested calculating probability of atleast 19 persen successful surgery. same Is in the had P ( x 719) [X: Number of successful surgeries P [ X 719 x - E(X) Varcx > 19-E(x) Vuar (x) Binomial (n=20, p=0.85) E(X) = np var(x) = np(1-P) =P[NCO,1) 19 (20)(0,85) V2020.85 (0.15) P (NCO, 1) z 1:25] 0.1057

52 e) P ( x 2,19) =P(x=19) + P(x=20) 0,1756 [ ने since, X nou PUX320 C& (p) (1-02

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