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. compare the results. choose the correct answer below. the results are about the same. the results are totally different.

Explanation:

Step1: Calculate binomial probability

For a binomial distribution \(X\sim B(n = 16,p=0.5)\), the probability mass function is \(P(X = k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\), where \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\).
\(P(4\leq X\leq8)=\sum_{k = 4}^{8}\binom{16}{k}(0.5)^{k}(0.5)^{16 - k}=\sum_{k = 4}^{8}\binom{16}{k}(0.5)^{16}\)
\(\binom{16}{4}=\frac{16!}{4!(16 - 4)!}=\frac{16\times15\times14\times13}{4\times3\times2\times1}=1820\)
\(\binom{16}{5}=\frac{16!}{5!(16 - 5)!}=\frac{16\times15\times14\times13\times12}{5\times4\times3\times2\times1}=4368\)
\(\binom{16}{6}=\frac{16!}{6!(16 - 6)!}=\frac{16\times15\times14\times13\times12\times11}{6\times5\times4\times3\times2\times1}=8008\)
\(\binom{16}{7}=\frac{16!}{7!(16 - 7)!}=\frac{16\times15\times14\times13\times12\times11\times10}{7\times6\times5\times4\times3\times2\times1}=11440\)
\(\binom{16}{8}=\frac{16!}{8!(16 - 8)!}=\frac{16!}{8!8!}=12870\)
\(\sum_{k = 4}^{8}\binom{16}{k}=1820 + 4368+8008+11440+12870=38506\)
\(P(4\leq X\leq8)=\frac{38506}{65536}\approx0.5876\)

Step2: Calculate normal - approximation probability

For a binomial distribution \(X\sim B(n,p)\), the mean \(\mu=np = 16\times0.5=8\), and the standard deviation \(\sigma=\sqrt{np(1 - p)}=\sqrt{16\times0.5\times(1 - 0.5)} = 2\)
For the normal approximation \(Y\sim N(\mu = 8,\sigma = 2)\), we use the continuity correction. We want to find \(P(3.5\leq Y\leq8.5)\)
First, standardize:
\(z_1=\frac{3.5 - 8}{2}=\frac{- 4.5}{2}=-2.25\), \(z_2=\frac{8.5 - 8}{2}=\frac{0.5}{2}=0.25\)
Using the standard normal table \(P(Z\leq z_2)-P(Z\leq z_1)\)
\(P(Z\leq0.25)=0.5987\), \(P(Z\leq - 2.25)=0.0122\)
\(P(3.5\leq Y\leq8.5)=P(-2.25\leq Z\leq0.25)=0.5987-0.0122 = 0.5865\)

Answer:

The binomial probability \(P(4\leq X\leq8)\approx0.5876\), the normal - approximation probability \(P(3.5\leq X\leq8.5)\approx0.5865\). The results are about the same.