Question

It is believed that 44​% of children have a gene that may be linked to juvenile diabetes. Researchers at a firm would like to test new monitoring equipment for diabetes. Hoping to have 22 children with the gene for their​ study, the researchers test 735 newborns for the presence of the gene linked to diabetes. What is the probability that they find enough subjects for their​ study?

What is the​ probability? Select the correct answer below​ and, if​ necessary, fill in the answer box to complete your choice.

A. P(enough subjects)equals= nothing ​(Round to three decimal places as​ needed.)

B. The conditions for finding the probability are not satisfied

Answer 1

n=	735	p=	0.0400
here mean of distribution=μ=np=		29.4
and standard deviation σ=sqrt(np(1-p))=		5.3126
for normal distribution z score =(X-μ)/σx

P(enough subjects) =P(X>22) =P(Z>(22-29.4)/5.3126)=P(Z>-1.39)=0.918

(please try 0.932 if continuity correction is required)

Answer 2

To find the probability that the researchers find enough subjects for their study (at least 22 children with the gene), we can use the binomial probability formula.

The binomial probability formula is given by:

P(X ≥ k) = 1 - P(X < k)

where: P(X ≥ k) is the probability of getting at least k successes (in this case, k children with the gene), P(X < k) is the cumulative probability of getting less than k successes.

In this case, the probability of a newborn having the gene linked to diabetes is 44%, which can be represented as a success in a binomial experiment.

Now, we can calculate the probability of getting less than 22 successes (children with the gene) out of 735 trials (newborns tested) and then subtract it from 1 to get the probability of getting at least 22 successes.

Using the binomial probability formula:

P(X < 22) = Σ [n choose x] * p^x * (1 - p)^(n - x)

where: n = 735 (number of trials) x = 0, 1, 2, ..., 21 (number of successes) p = 0.44 (probability of success)

Now, calculate the cumulative probability:

P(X < 22) = Σ [735 choose x] * (0.44)^x * (1 - 0.44)^(735 - x) for x = 0 to 21

The probability of getting at least 22 successes is:

P(X ≥ 22) = 1 - P(X < 22)

Use the above formula to find the answer. The correct option is either A or B, depending on the calculated probability.