91Ó°ÊÓ

\(X\) is a random variable that denotes the number of times the coin lands on heads.

Number of times the coin is flipped, \(n=100\)

Probability that it comes up heads, \(p=20%=0.2\)

Mean, \(\mu=np=100*0.2=20\)

Standard deviation, \(\sigma=\sqrt{100*0.2(1-0.2)}=4\)

For \(z=\pm1, x_{1}=\mu-z\sigma\) and \(x_{2}=\mu+z\sigma\)

\(\Rightarrow x_{1}=20-1*4=16\) and \(x_{2}=20+1*4=24\)

Thus, \(68%\) of the values of \(X\) lie between \(16\) and \(24\).

Mean, \(\mu=np=100*0.2=20\)

Standard deviation, \(\sigma=\sqrt{100*0.2(1-0.2)}=4\)

For \(z=\pm2, x_{1}=\mu-z\sigma\) and \(x_{2}=\mu+z\sigma\)

\(\Rightarrow x_{1}=20-2*4=12\) and \(x_{2}=20+2*4=28\)

Thus, \(95%\) of the values of \(X\) lie between \(12\) and \(28\).

Mean, \(\mu=np=100*0.2=20\)

Standard deviation, \(\sigma=\sqrt{100*0.2(1-0.2)}=4\)

For \(z=\pm3, x_{1}=\mu-z\sigma\) and \(x_{2}=\mu+z\sigma\)

\(\Rightarrow x_{1}=20-3*4=8\) and \(x_{2}=20+3*4=32\)

Thus, \(99.7%\) of the values of \(X\) lie between \(8\) and \(32\).