Question

5-28. Assuming the binomial distribution applies with a sample size of n = 15, find

a. the probability of 5 or more successes if the probability of a success is 0.30

b. the probability of fewer than 4 successes if the probability of a success is 0.75

c. the expected value of the random variable if the probability of success is 0.40

d. the standard deviation of the random variable if the probability of success is 0.40

5-36. Dell Computers receives large shipments of microprocessors
from Intel Corp. It must try to ensure the
proportion of microprocessors that are defective is
small. Suppose Dell decides to test five microprocessors
out of a shipment of thousands of these microprocessors.
Suppose that if at least one of the microprocessors is
defective, the shipment is returned.
a. If Intel Corp.

Answer 1

5-28. Assuming the binomial distribution applies with a sample size of n = 15, find

a. the probability of 5 or more successes if the probability of a success is 0.30

Given X~Binomial(n=15, p=0.3)

P(X=x)=15Cx*(0.3^x)*(0.7^(15-x))

So the probability is

P(X>=5) = P(X=5)+P(X=6)+...+P(X=15)

=15C5*(0.3^5)*(0.7^(15-5))+...+15C15*(0.3^15)*(0.7^(15-15))

=0.4845089

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b. the probability of fewer than 4 successes if the probability of a success is 0.75

Given X~Binomial(n=15, p=0.75)

P(X=x)=15Cx*(0.75^x)*(0.25^(15-x))

So the probability is

P(X<4) = P(X=0)+P(X=1)+...+P(X=3) =0.000012

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c. the expected value of the random variable if the probability of success is 0.40

expected value= n*p= 15*0.4 =6

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d. the standard deviation of the random variable if the probability of success is 0.40

standard deviation =sqrt(n*p*(1-p))

=sqrt(15*0.4*0.6) =1.897367

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5-36. Dell Computers receives large shipments of microprocessors
from Intel Corp. It must try to ensure the
proportion of microprocessors that are defective is
small. Suppose Dell decides to test five microprocessors
out of a shipment of thousands of these microprocessors.
Suppose that if at least one of the microprocessors is
defective, the shipment is returned.
a. If Intel Corp.

Answer 2

Assuming the binomial distribution applies with a sample size of n = 15, find

a. the probability of 5 or more successes if the probability of a success is 0.30

n=15   p=0.30  q=1-p=1-0.30=0.7

P(X> 5)=?

P(X> 5)= 1-P(X< 4) 

P(X> 5)=1 -  {P(X=4)+P(X=3)+P (X=2)+P(X=1)+P(X=0)........(A)

P(X=x) = ⁿC_x p^x q^n-x

At x=0,

P(X=0) =0.00748

At x=1,

P(X=1) =0.03053

At x=2,

P(X=2) =0.09157

At x=3,

P(X=3) =0.1700

At x=4,

P(X=4) =0.2186

P(X> 5)=1-0.51818

P(X> 5)=0.48182