a) Whether it is unusual to have more than five successes or not. Given: The...

a) Calculation: The mean of the binomial probability distribution is:
${\displaystyle \mu =np}$
Here, ${\displaystyle n=}$ number of trails
${\displaystyle p=}$ probability of success in a single trail
Substitute, ${\displaystyle n=10\text{and}p=0.2}$ in the above formula, thus,
${\displaystyle \mu =10\times 0.2}$
${\displaystyle =2}$
Therefore, the mean of the binomial probability distribution is 2.
The standard deviation of the binomial probability distribution is:
${\displaystyle \sigma =\sqrt{npq}}$
Substitute, ${\displaystyle n=10,p=0.2\text{and}q=0.8}$ in the above formula,
${\displaystyle \sigma =\sqrt{10\times 0.2\times 0.8}}$
${\displaystyle =1.26}$
Therefore, the standard deviation of the binomial probability distribution is 1.26.
Now, the range for considering the number of successes to be unusual is:
Substituting the values of ${\displaystyle \mu }$ and ${\displaystyle \sigma \in \mu +2.5\sigma }$
${\displaystyle \mu +2.5\sigma =2+(2.5\times 1.26)}$
${\displaystyle =5.15}$
Thus, number of successes more than 5.25 will be considered unusual. Hence, it is unusual to have more than five successes, that is, 6 or more successes.
b) Calculation:
According to the provided information, the number of questions is 10. So, ${\displaystyle n=10}$.
The probability of success for a single trail can be calculated as:
${\displaystyle p=\frac{\text{number of correct answer choice for a question}}{\text{total number of choices}}}$
${\displaystyle =\frac{1}{5}}$
${\displaystyle =0.2}$
Thus, ${\displaystyle n=10\text{and}p=0.2}$
According to the explanation in part (a), it is unusual to get more than 5 successes for a binomial experiment with ${\displaystyle n=10\text{and}p=0.2}$.
So, it is unusual to answer more than half of the questions (more than 5 questions) correct by randomly quessing it.