Question

write the binomial probability and the normal probability for the shaded region of the graph. find the value of each probability and compare the results. write the normal probability for the shaded region of the graph and find its value. select the correct choice below and fill in the answer box within your choice. (round to four decimal places as needed.) a. ( p(5 leq x leq 7)=p(4.5 leq x leq 7.5)= ) b. ( p(4<x<8)=p(4.5 leq x leq 8.5)= )

Explanation:

Step1: Calculate the mean and standard deviation

For a binomial distribution \(n = 16\), \(p=0.5\).
The mean \(\mu=np = 16\times0.5=8\).
The standard deviation \(\sigma=\sqrt{np(1 - p)}=\sqrt{16\times0.5\times(1 - 0.5)}=\sqrt{4}=2\).

Step2: Standardize the values for option A

For \(x_1 = 4.5\), \(z_1=\frac{4.5 - 8}{2}=\frac{-3.5}{2}=-1.75\).
For \(x_2 = 7.5\), \(z_2=\frac{7.5 - 8}{2}=\frac{-0.5}{2}=-0.25\).
Using the standard normal table \(P(-1.75<Z<-0.25)=\varPhi(-0.25)-\varPhi(-1.75)\).
\(\varPhi(-0.25) = 0.4013\), \(\varPhi(-1.75)=0.0401\).
\(P(-1.75<Z<-0.25)=0.4013 - 0.0401=0.3612\).

Step3: Standardize the values for option B

For \(x_1 = 4.5\), \(z_1=\frac{4.5 - 8}{2}=-1.75\).
For \(x_2 = 8.5\), \(z_2=\frac{8.5 - 8}{2}=0.25\).
Using the standard normal table \(P(-1.75<Z<0.25)=\varPhi(0.25)-\varPhi(-1.75)\).
\(\varPhi(0.25) = 0.5987\), \(\varPhi(-1.75)=0.0401\).
\(P(-1.75<Z<0.25)=0.5987 - 0.0401 = 0.5586\).

From the graph, the shaded region is \(P(5\leq x\leq7)\) which corresponds to \(P(4.5\leq x\leq7.5)\) after continuity correction.

Answer:

A. \(P(5\leq x\leq7)=P(4.5\leq x\leq7.5)=0.3612\)