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

SchachtN

SchachtN

Answered question

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 μ2.5σ  μ+2.5σ 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 n=10 and p=0.2.

Answer & Explanation

smallq9

smallq9

Skilled2021-09-16Added 106 answers

a) Calculation: The mean of the binomial probability distribution is:
μ=np
Here, n= number of trails
p= probability of success in a single trail
Substitute, n=10 and p=0.2 in the above formula, thus,
μ=10×0.2
=2
Therefore, the mean of the binomial probability distribution is 2.
The standard deviation of the binomial probability distribution is:
σ=npq
Substitute, n=10,p=0.2 and q=0.8 in the above formula,
σ=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 μ and σμ+2.5σ
μ+2.5σ=2+(2.5×1.26)
=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=number of correct answer choice for a questiontotal number of choices
=15
=0.2
Thus, n=10 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 n=10 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.

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