Step 1

Given:

\(\displaystyle{p}={0.70}\)

\(\displaystyle{n}={20}\)

Formula binomial probability:

\[f(k)=(\begin{matrix} n \\ k \end{matrix})\times p^{k}\times(1-p)^{n-k}\]

a) \[f(12)=(\begin{matrix} 20 \\ 12 \end{matrix})\times0.70^{12}\times(1-0.70)^{20-12}=0.114397\]

b \[f(16)=(\begin{matrix} 20 \\ 16 \end{matrix})\times0.70^{16}\times(1-0.70)^{20-16}=0.130421\]

c) Add the corresponding probabilities:

\(\displaystyle{P}{\left({X}\geq{16}\right)}={f{{\left({16}\right)}}}+{f{{\left({17}\right)}}}+{f{{\left({18}\right)}}}+{f{{\left({19}\right)}}}+{f{{\left({20}\right)}}}={0.2375}\)

d) Use the complement rule for probabilities:

\(\displaystyle{P}{\left({X}\le{15}\right)}={1}-{P}{\left({X}\geq{16}\right)}={1}-{0.2375}={0.7625}\)

Step 2

e) The mean of binomial distribution is the product of the sample size n and the probability p:

\(\displaystyle\mu={n}{p}={20}\times{0.20}={14}\)

f) The standard deviation of a binomial distribution is the square root of the product of the sample size n and the probabilities p and q. The variance is the square of the standard deviation.

\(\displaystyle\sigma^{{{2}}}={n}{p}{q}={n}{p}{\left({1}-{p}\right)}={20}{\left({0.70}\right)}{\left({1}-{0.70}\right)}={4.2}\)

\(\displaystyle\sigma=\sqrt{{{n}{p}{q}}}=\sqrt{{{n}{p}{\left({1}-{p}\right)}}}=\sqrt{{{20}{\left({0.70}\right)}{\left({1}-{0.70}\right)}}}\approx{2.0494}\)

Given:

\(\displaystyle{p}={0.70}\)

\(\displaystyle{n}={20}\)

Formula binomial probability:

\[f(k)=(\begin{matrix} n \\ k \end{matrix})\times p^{k}\times(1-p)^{n-k}\]

a) \[f(12)=(\begin{matrix} 20 \\ 12 \end{matrix})\times0.70^{12}\times(1-0.70)^{20-12}=0.114397\]

b \[f(16)=(\begin{matrix} 20 \\ 16 \end{matrix})\times0.70^{16}\times(1-0.70)^{20-16}=0.130421\]

c) Add the corresponding probabilities:

\(\displaystyle{P}{\left({X}\geq{16}\right)}={f{{\left({16}\right)}}}+{f{{\left({17}\right)}}}+{f{{\left({18}\right)}}}+{f{{\left({19}\right)}}}+{f{{\left({20}\right)}}}={0.2375}\)

d) Use the complement rule for probabilities:

\(\displaystyle{P}{\left({X}\le{15}\right)}={1}-{P}{\left({X}\geq{16}\right)}={1}-{0.2375}={0.7625}\)

Step 2

e) The mean of binomial distribution is the product of the sample size n and the probability p:

\(\displaystyle\mu={n}{p}={20}\times{0.20}={14}\)

f) The standard deviation of a binomial distribution is the square root of the product of the sample size n and the probabilities p and q. The variance is the square of the standard deviation.

\(\displaystyle\sigma^{{{2}}}={n}{p}{q}={n}{p}{\left({1}-{p}\right)}={20}{\left({0.70}\right)}{\left({1}-{0.70}\right)}={4.2}\)

\(\displaystyle\sigma=\sqrt{{{n}{p}{q}}}=\sqrt{{{n}{p}{\left({1}-{p}\right)}}}=\sqrt{{{20}{\left({0.70}\right)}{\left({1}-{0.70}\right)}}}\approx{2.0494}\)