Chapter PV, Problem 25RE

(a)

To determine

To explain is it reasonable to use a Normal model to describe the possible distribution of the proportion of the group that has left-brain language control.

Answer to Problem 25RE

Yes, it is reasonable.

Explanation of Solution

It is given in the question that Neurological research has shown that in about $80\%$ of people language abilities reside in the brain's left side. There are a $10\%$ of display right-brain language centres and the remaining $10\%$ have two sided language control. Thus, yes, it is reasonable to use a Normal model to describe the possible distribution of the proportion of the group that has left-brain language control since there is a fixed number of sample observations and there is a proportion of the how much they are suffered overall and also these both np and nq are greater than ten.

(b)

To determine

To find out what is the probability that our group has at least $75\%$ left-brainers.

Answer to Problem 25RE

  $0.834$ .

Explanation of Solution

It is given in the question that Neurological research has shown that in about $80\%$ of people language abilities reside in the brain's left side. There are a $10\%$ of display right-brain language centres and the remaining $10\%$ have two sided language control. Thus, to find out the probability that our group has at least $75\%$ left-brainers we will first calculate standard deviation as:

  $\begin{array}{l}Sd=\sqrt{\frac{48\times 12}{60}}\\ =3.0984\end{array}$

Then we will calculate the z -score as:

  $\begin{array}{l}z=\frac{45-48}{3.0984}\\ =-0.96824\end{array}$

Thus, the probability will be as:

  $\begin{array}{l}normalcdf(0.96824,99.9)\\ =0.834\end{array}$

(c)

To determine

To explain would that probability be higher, lower or about the same and why.

Answer to Problem 25RE

It will be higher.

Explanation of Solution

It is given in the question that Neurological research has shown that in about $80\%$ of people language abilities reside in the brain's left side. There are a $10\%$ of display right-brain language centres and the remaining $10\%$ have two sided language control. Thus, the probability will be higher because bigger samples have smaller standard deviations.

(d)

To determine

To explain how large a group would almost certainly guarantee at least $75\%$ left-brainers.

Answer to Problem 25RE

  $1000$ people.

Explanation of Solution

It is given in the question that Neurological research has shown that in about $80\%$ of people language abilities reside in the brain's left side. There are a $10\%$ of display right-brain language centres and the remaining $10\%$ have two sided language control. Thus, the sample we assume to be $1000$ people that almost certainly guarantee at least $75\%$ left-brainers. So, we have,

  $\begin{array}{l}75\%np=750\\ \text{Expected}=800\\ Sd=\sqrt{\frac{800\times 200}{1000}}\\ =12.649\end{array}$

Then the z -score will be:

  $\begin{array}{l}z=\frac{750-800}{12.649}\\ =-3.953\end{array}$

Thus, the probability to have at least $75\%$ left-brainers language control people is $0.99996$ . Thus, $1000$ people group would almost certainly guarantee at least $75\%$ left-brainers.