Question

Approximately $8\%$of students at a local high school participate in after-school sports all four years of high school. A group of $60$seniors is randomly chosen. Of interest is the number that participated in after-school sports all four years of

high school.

a. In words, define the random variable $X$.

b. List the values that $X$may take on.

c. Give the distribution of$X.X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

d. How many seniors are expected to have participated in after-school sports all four years of high school?

e. Based on numerical values, would you be surprised if none of the seniors participated in after-school sports all

four years of high school.

f. Based on numerical values, is it more likely that four or that five of the seniors participated in after-school sports

all four years of high school? Justify your answer numerically.

Short answer

(a) $X=$the number of seniors who engaged in after-school sports over their entire high school career.

(b) Because the random variable $X$has a Binomial distribution with an upper bound of 60, the value is given by $x=0,1,2,\dots ,60$.

(c) Binomial distribution will be applied to the random variable $X$.

$X~B(60,0.08)$.

(d) As a result, $4.8$seniors are anticipated to have participated in after-school sports during their four years of high school.

(e) $P(x,y)=0.006$Because the likelihood of none of the seniors participating in after-school sports all four years of high school is quite low, it will come as a surprise if none of the seniors participate in after-school sports all four years of high school.

(f) As a result, when $P(x=4)$and $P(x=5)$are compared, it is more likely that four or five of the seniors participated in after-school sports throughout their four years of high school.

Step-by-step solution

Given (a)

Define the random variable X in words.

Explanation (a)

The random variable $X$is defined as follows:

$X=$the number of seniors who engaged in after-school sports over their entire high school career.

Step :3 explanation (b)

Make a list of the values that you want to use$\mathrm{X}$may be able to.

we see random variable $X$will be followed The value is given by, which has a binomial distribution and an upper bound of 60 is we get $x=0,1,2,\dots ,60$.

Given (c)

The distribution we have $\mathrm{X}~\mathrm{X}$.

Explanation (c)

Binomial distribution will be applied to the random variable$X.$

$X~B(60,0.08)$

Explanation (d)

The expected Binomial distribution is calculated as follows:

$\mu =np$

$\mu =60\times 0.08$

$\mu =4.8$

As a result, $4.8$seniors are anticipated to have participated in after-school sports during their four years of high school.

Explanation (e)

The required probability can be expressed as $P(x=0)$than it calculated by calculator.

As a result, $P(x=0)=0.00671$Because the likelihood of none of the seniors participating in after-school sports all four years of high school is quite low, it will come as a surprise if none of the seniors participate in after-school sports all four years of high school.

Explanation (f)

It's possible to write the requisite probability as,

$P(x=4)\text{and}P(x=5)$

Than it is calculated by using $Ti-83$using calculator.

$P(x=4)$

$.1873152554$

$P(x=5)$

$.1824287705$

As a result, when $P(x=4)$ and $P(x=5)$ are compared, it is more likely that four or five of the seniors participated in after-school sports throughout their four years of high school.