Chapter 18, Problem 24E

To determine

To find out what is the probability that they find enough subjects for their study.

Answer to Problem 24E

  $0.09608$ .

Explanation of Solution

It is given in the question that $4\%$ of children have a gene that may be linked to juvenile diabetes. And the researchers hoping to track $20$ of these children for several years test $732$ new-borns for the presence of this gene. Thus, let us first verify the assumptions and conditions for the analysis as:

Random condition: It is satisfied as we assume that the children were randomly selected.

  $10\%$ condition: The sample size less than $10\%$ of the population size.

Success/failure condition: as $np=29.28,n(1-p)=702.72$ and both are greater than ten then this is also true.

Thus, all the conditions and assumptions are met. Thus, now find the mean and the standard deviation for the following as:

  $\begin{array}{l}\mu =p=0.04\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.04(0.96)}{732}}\\ =0.072\end{array}$

Thus, let us find the z -score to standardized as:

  $\begin{array}{l}z=\frac{\frac{20}{732}-0.04}{0.072}\\ =-1.761\end{array}$

So, the probability that the shipment will be accepted anyway is calculated as:

  $\begin{array}{l}P(z<-1.761)=normalcdf(-E99,-1.761,0,1)\\ =0.09608\end{array}$

Therefore, the probability that they find enough subjects for their study is $0.09608$ .