Answer to Question #288537 in Statistics and Probability for mai

Question

It is known that 55% of students in Mr. ERIZ’s class are hardworking person. A sample of 15 students was selected for interview.

a) What is the distribution of these 15 students selected for interview? Justify your answer.

(6 Marks)

b) What is the probability that in a sample of 15 students, less than 45% of them are hardworking person? (6 Marks)

c) What is the probability that in a sample of 15 students, more than 45% of them are hardworking person

Answer 1

$a)$

The distribution of these 15 students is the Binomial distribution.

This is because, the scenario involved here consist of an experiment whose outcomes can be classified as either a success or failure. It is either a student is a hardworking person(success) or the student is not a hardworking person(failure).

$b)$

The probability that a student is hardworking $p=0.55$ and $n=15$.

Now,

45% of the 15 students is ${45\over100}\times 15=6.75\approx7$ students.

Therefore, $x=7$ students.

We determine the probability given by,

$p(x\lt7)$. To find this probability, we use the Binomial distribution.

So,

$p(x\lt7)=\displaystyle\sum^6_{x=0}\binom{15}{x} 0.55^x0.45^{15-x}$

To find this probability, we enter the following command in $R$.

> pbinom(6, size=15, prob=0.55)

[1] 0.1817605

Therefore, the probability that in a sample of 15 students, less than 45% of them are hardworking person is 0.1817605.

$c)$

45% of the students is equal to 7 students.

Here, we find $p(x\gt7)=\displaystyle\sum^{15}_{x=8}\binom{15}{x} 0.55^x0.45^{15-x}=1-p(x\leq7)$.

Now, $p(x\leq7)$ can be found by entering the following command in $R$.

> pbinom(7, size=15, prob=0.55)

[1] 0.3464961

Therefore, $p(x\gt7)=1-0.3464961=0.6535039$

Thus, the probability that in a sample of 15 students, more than 45% of them are hardworking person is 0.6535039.