Consider a binomial experiment with 15 trials and probability 0.45 of success on

geduiwelh 2021-09-21 Answered
Consider a binomial experiment with 15 trials and probability 0.45 of success on a single trial.
(a) Use the binomial distribution to find the probability of exactly 10 successes. (Round your answer to three decimal places.)
(b) Use the normal distribution to approximate the probability of exactly 10 successes. (Round your answer to three decimal places.)
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Expert Answer

coffentw
Answered 2021-09-22 Author has 103 answers

Step 1
Solution: It is given here that a random variable say x follows the binomial distribution with parameters n=15 and p=0.45
The binomial probability function is:
P(X=x)=n!(nx)!x!px(1p)nx;x=0,1,2,..,n
Step 2
(a) Use the binomial distribution to find the probability of exactly 10 successes.
Answer: It is required to find:
P(x=10)
Using the binomial distribution function:
P(x=10)=15!(1510)!10!0.4510(10.45)1510
=3003×0.000340506×0.050328438
=0.051
Therefore, the probability of exactly 10 successes is 0.051
Step 3
(b) Use the normal distribution to approximate the probability of exactly 10 successes.
Answer:
The mean and standard deviation of the random variable x is:
μ=np=15×0.45=6.75
σ=np(1p)=15×0.45(10.45)=1.92678
It is required to find:
P(x=10)
Using the continuity correction factor, the above probability can be written as:
P(x=10)=P(100.5<x<10+0.5)
=P(9.5<x<10.5)
Using the z-score formula:
P(9.5<x<10.5)=P(9.56.751.92678<xμσ<10.56.751.92678
=P(1.4272<z<1.9462)
=P(z<1.9462)P(z<1.4272)
Now using the excel functions:
P(9.5<x<10.5)=P(z<1.9462)P(z<1.4272)=0.97420.9232=0.051
The excel functions are:
=NORMSDIST(1.4272)=0.9232
=NORMSDIST(1.9462)=0.9742
Therefore, Using the normal distribution to approximate the probability of exactly 10 successes is 0.051

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