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.
a.
b. p(5 < x < 9)=p(5.5≤x≤9.5)=
c. p(6 < x < 8)=p(6.5 < x < 7.5)=
d. p(6 ≤ x ≤ 8)=p(5.5 ≤ x ≤ 8.5)=0.2694
compare the results. choose the correct answer below.
the results are about the same.
the results are totally different.

Explanation:

Step1: Identify binomial - normal conversion

For binomial distribution \(X\sim B(n,p)\) with \(n = 16\) and \(p=0.6\), when using normal - approximation to the binomial, we have \(\mu=np\) and \(\sigma=\sqrt{np(1 - p)}\). Here, \(\mu = 16\times0.6=9.6\) and \(\sigma=\sqrt{16\times0.6\times(1 - 0.6)}=\sqrt{16\times0.6\times0.4}=\sqrt{3.84}\approx1.96\). For the binomial probability \(P(6\leq x\leq8)\), when using normal approximation, we apply the continuity correction. The binomial \(P(6\leq x\leq8)\) is approximated by the normal \(P(5.5\leq x\leq8.5)\).

Step2: Standardize the values

We standardize \(x_1 = 5.5\) and \(x_2 = 8.5\) using \(z=\frac{x-\mu}{\sigma}\). For \(x = 5.5\), \(z_1=\frac{5.5 - 9.6}{1.96}=\frac{-4.1}{1.96}\approx - 2.09\). For \(x = 8.5\), \(z_2=\frac{8.5 - 9.6}{1.96}=\frac{-1.1}{1.96}\approx - 0.56\).

Step3: Find the normal probabilities

We know that \(P(5.5\leq x\leq8.5)=P(-2.09\leq z\leq - 0.56)=\Phi(-0.56)-\Phi(-2.09)\), where \(\Phi(z)\) is the cumulative - distribution function of the standard normal distribution. From the standard normal table, \(\Phi(-0.56)=0.2877\) and \(\Phi(-2.09)=0.0183\). So \(P(-2.09\leq z\leq - 0.56)=0.2877 - 0.0183=0.2694\).
For the binomial probability \(P(6\leq x\leq8)=\sum_{k = 6}^{8}C(16,k)\times0.6^{k}\times(1 - 0.6)^{16 - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\).
\(C(16,6)=\frac{16!}{6!(16 - 6)!}=\frac{16!}{6!10!}=\frac{16\times15\times14\times13\times12\times11}{6\times5\times4\times3\times2\times1}=8008\)
\(P(X = 6)=C(16,6)\times0.6^{6}\times0.4^{10}=8008\times0.046656\times0.0001048576\approx0.039\)
\(C(16,7)=\frac{16!}{7!(16 - 7)!}=\frac{16!}{7!9!}=\frac{16\times15\times14\times13\times12\times11\times10}{7\times6\times5\times4\times3\times2\times1}=11440\)
\(P(X = 7)=C(16,7)\times0.6^{7}\times0.4^{9}=11440\times0.0279936\times0.000262144\approx0.084\)
\(C(16,8)=\frac{16!}{8!(16 - 8)!}=\frac{16!}{8!8!}=12870\)
\(P(X = 8)=C(16,8)\times0.6^{8}\times0.4^{8}=12870\times0.01679616\times0.00065536\approx0.138\)
\(P(6\leq x\leq8)=0.039 + 0.084+0.138 = 0.261\)

Answer:

The binomial probability \(P(6\leq x\leq8)\approx0.261\) and the normal - approximated probability \(P(5.5\leq x\leq8.5)=0.2694\). The results are about the same. So the correct multiple - choice answers are:
D. \(P(6\leq x\leq8)=P(5.5\leq x\leq8.5)=0.2694\)
The results are about the same.