Solution:
Given:
[tex]\begin{gathered} p=0.75 \\ n=22 \\ \mu=? \\ \sigma=? \end{gathered}[/tex]Using the formula for the mean and standard deviation of binomial distribution;
[tex]\begin{gathered} \mu=np \\ \text{where;} \\ \mu\text{ is the mean} \\ n\text{ is the number of trials} \\ p\text{ is the probability of success} \\ \\ \\ \text{Also,} \\ \sigma=\sqrt[]{npq} \\ \text{where;} \\ \sigma\text{ is the standard deviation } \\ n\text{ is the number of trials} \\ p\text{ is the probability of success} \\ q\text{ is the probability of failure} \end{gathered}[/tex]Hence, the mean is;
[tex]\begin{gathered} \mu=np \\ \mu=22\times0.75 \\ \mu=16.5 \end{gathered}[/tex]Therefore, the mean is 16.5
The standard deviation is;
[tex]\begin{gathered} \sigma=\sqrt[]{npq} \\ n=22 \\ p=0.75 \\ q=1-0.75=0.25 \\ \\ \sigma=\sqrt[]{22\times0.75\times0.25} \\ \sigma=\sqrt[]{4.125} \\ \sigma=2.031 \end{gathered}[/tex]Therefore, the standard deviation is 2.031
