A geologist has collected 150 specimens of basaltic rock and 50 specimens of gra

Question

A geologist has collected 150 specimens of basaltic rock and 50 specimens of granite. The geologist instructs a laboratory assistant to randomly select 8 specimens for analysis. Let Y = number of selected specimens that are granite.

(a) What is the distribution of Y ? Identify all relevant parameters and write down the pmf.

(b) Calculate E (Y ) and SD (Y ).

(c) Calculate (using the true distribution) P (Y <= 1).

(d) Suggest a distribution that can be used to approximate the true distribution of Y . Verify any conditions necessary for the approximation to work.

(e) Regardless of your conclusion in (c), use the suggested approximation to calculate P (Y >= 4).

Explanation / Answer

a)

It is a hypergeometric distirbution.

Note that the probability of Y successes out of n trials is          
          
P(Y) = C(N-K, n-Y) C(K, Y) / C(N, n)          
          
where          
N = population size = 150 + 50 =    200      
K = number of successes in the population =    50      
n = sample size =    8      
Y = number of successes in the sample.

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b)

Here,

E(Y) = nK/N =    2 [ANSWER]

Also,

Var(Y) = [nK(N-K)(N-1)]/[(N^2)(N-1)] =    1.447236181  
Hence,

SD(Y) = sqrt(variance) =    1.203011297 [ANSWER]

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c)

Using a cumulative hypergeometric distribution table or technology, matching          
          
where          
N = population size =    200      
K = number of successes in the population =    50      
n = sample size =    8      
x = critical number of successes in the sample =    1      
          
Thus,          
P(at most   1   ) =    0.362306017 [ANSWER]

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d)

We may use the binomial distribution to approximate.

To use it, not that n/N < 0.05.

Here, n = 8, N = 200, so n/N = 0.04 < 0.05, so we can use the binomial approximation.

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E)

Here, the probability of a success = 50/200 = 0.25 = p.

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    8      
p = the probability of a success =    0.25      
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   3   ) =    0.886184692
          
Thus, the probability of at least   4   successes is  
          
P(at least   4   ) =    0.113815308 [ANSWER]

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