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A geologist has collected 17 specimens of basaltic rock and 17 specimens of granite. The geologist instructs a laboratory assistant to randomly select 29 of the specimens for analysis.

a. What is the pmf of the number of granite specimens selected for analysis?
b. What is the probability that all specimens of one of the two types of rock are selected for analysis?
c. What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?

Answer :

Answer:

Step-by-step explanation:

Out of N = 34specimens, K = 17 are granite and 17 are basaltic.

Let X be the number of specimens that are granite in randomly selected n = 29 specimens

Tthe geometric random variable  can only take integer values  in range ; [max ( 0 , n + K - N), min (n, K )] = [ 12, 17] using the combination method;

nCr = n!/(n - r) r)! which is as shown below.

a)

X P(X)

12 0.0222

13 0.1454

14 0.3324

15 0.3324

16 0.1454

17 0.0222

b) P (all specimens of one of the two types of rock are selected for analysis

= C(17,17) x C (17,12) / C34,29) + C(17,12) x C (17,17) / C34,29) = 0.0445

 

c) Sample size, n =    29

  • No. of events of interest in population , k =   17
  • Population size  , N = 34
  • mean = E(X) = n(k/N) = 14.5
  • variance = nk(N-k)(N-n)/(N²(N-1)) = 1.0985

Therefore, the probability that the number of granite specimens selected is within 1 standard deviation of mean ;

P (µ-σ < X < µ+σ) = P(14.5-1.05 < X <14.5+1.05) = P(13.45 < X <15.55) = P(X=14) + P(X=15) = 0.3324 + 0.3324 = 0.6647

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