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. (round to four decimal places as needed.)
○a. ( p(4<x<8)=p(4.5)+p(5.5)+p(6.5)+p(7.5)= )
○b. ( p(4 leq x leq 8)=p(4)+p(5)+p(6)+p(7)+p(8)= )
○c. ( p(5 leq x leq 7)=p(5)+p(6)+p(7)= )
○d. ( p(5<x<7)=p(6)= )

Explanation:

Step1: Binomial Probability Formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\), \(n = 16\), \(p=0.5\).
For \(P(5\leq x\leq7)\):

• When \(k = 5\):

\(C(16,5)=\frac{16!}{5!(16 - 5)!}=\frac{16\times15\times14\times13\times12}{5\times4\times3\times2\times1}=4368\)
\(P(X = 5)=C(16,5)\times(0.5)^{5}\times(0.5)^{16 - 5}=4368\times(0.5)^{16}\)

• When \(k = 6\):

\(C(16,6)=\frac{16!}{6!(16 - 6)!}=\frac{16\times15\times14\times13\times12\times11}{6\times5\times4\times3\times2\times1}=8008\)
\(P(X = 6)=C(16,6)\times(0.5)^{6}\times(0.5)^{16 - 6}=8008\times(0.5)^{16}\)

• When \(k = 7\):

\(C(16,7)=\frac{16!}{7!(16 - 7)!}=\frac{16\times15\times14\times13\times12\times11\times10}{7\times6\times5\times4\times3\times2\times1}=11440\)
\(P(X = 7)=C(16,7)\times(0.5)^{7}\times(0.5)^{16 - 7}=11440\times(0.5)^{16}\)

\(P(5\leq x\leq7)=(4368 + 8008+11440)\times(0.5)^{16}\)
\((4368 + 8008+11440)=23816\)
\(P(5\leq x\leq7)=23816\times\frac{1}{65536}\approx0.3634\)

Step2: Normal Approximation to Binomial

The mean of the binomial distribution \(\mu=np=16\times0.5 = 8\), the standard deviation \(\sigma=\sqrt{np(1 - p)}=\sqrt{16\times0.5\times(1 - 0.5)}=\sqrt{4}=2\)
For the normal approximation of \(P(5\leq x\leq7)\), using the continuity correction.
\(P(5\leq x\leq7)\) for binomial is approximated by \(P(4.5<X<7.5)\) for normal.
\(z_1=\frac{4.5 - 8}{2}=\frac{- 3.5}{2}=-1.75\), \(z_2=\frac{7.5 - 8}{2}=\frac{-0.5}{2}=-0.25\)
\(P(-1.75<Z<-0.25)=\Phi(-0.25)-\Phi(-1.75)\)
From the standard - normal table: \(\Phi(-0.25)=0.4013\), \(\Phi(-1.75)=0.0401\)
\(P(-1.75<Z<-0.25)=0.4013 - 0.0401=0.3612\)

Answer:

The binomial probability \(P(5\leq x\leq7)\approx0.3634\) and the normal probability (with continuity correction) \(P(4.5 < X < 7.5)\approx0.3612\). The two probabilities are relatively close.