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$\mu -2.5\sigma \text{}\to \text{}\mu +2.5\sigma$ 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$n=10\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}p=0.2$ .

Given: The number of successes lying outside the range

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

Skilled2021-09-16Added 106 answers

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,$n=10\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}p=0.2$ in the above formula, thus,

$\mu =10\times 0.2$

$=2$

Therefore, the mean of the binomial probability distribution is 2.

The standard deviation of the binomial probability distribution is:

$\sigma =\sqrt{npq}$

Substitute,$n=10,p=0.2\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}q=0.8$ in the above formula,

$\sigma =\sqrt{10\times 0.2\times 0.8}$

$=1.26$

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 $\sigma \in \mu +2.5\sigma$

$\mu +2.5\sigma =2+(2.5\times 1.26)$

$=5.15$

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:

$p=\frac{\text{number of correct answer choice for a question}}{\text{total number of choices}}$

$=\frac{1}{5}$

$=0.2$

Thus,$n=10\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}p=0.2$

According to the explanation in part (a), it is unusual to get more than 5 successes for a binomial experiment with$n=10\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}p=0.2$ .

So, it is unusual to answer more than half of the questions (more than 5 questions) correct by randomly quessing it.

Here,

Substitute,

Therefore, the mean of the binomial probability distribution is 2.

The standard deviation of the binomial probability distribution is:

Substitute,

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

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,

The probability of success for a single trail can be calculated as:

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.