SchachtN

2021-09-15

a) Whether it is unusual to have more than five successes or not.
Given: The number of successes lying outside the range are considered as unusual. The success probability in a single trail is 0.2 and the number of trials is 10.
b) Whether one would be likely to get more than half of the questions correct or not.
Given: A multiple-choice exam consisting of 10 questions with 5 possible responses for each questions. Consider the explanation in part (a), it is unusual to get more than 5 successes when .

smallq9

a) Calculation: The mean of the binomial probability distribution is:
$\mu =np$
Here, $n=$ number of trails
$p=$ probability of success in a single trail
Substitute, in the above formula, thus,
$\mu =10×0.2$
$=2$
Therefore, the mean of the binomial probability distribution is 2.
The standard deviation of the binomial probability distribution is:
$\sigma =\sqrt{npq}$
Substitute, in the above formula,
$\sigma =\sqrt{10×0.2×0.8}$
$=1.26$
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 $\mu$ and $\sigma \in \mu +2.5\sigma$
$\mu +2.5\sigma =2+\left(2.5×1.26\right)$
$=5.15$
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, $n=10$.
The probability of success for a single trail can be calculated as:
$p=\frac{\text{number of correct answer choice for a question}}{\text{total number of choices}}$
$=\frac{1}{5}$
$=0.2$
Thus,
According to the explanation in part (a), it is unusual to get more than 5 successes for a binomial experiment with .
So, it is unusual to answer more than half of the questions (more than 5 questions) correct by randomly quessing it.

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