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 who 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? Justify your answer numerically.

f. Based upon 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. The random variable X is the number who participated in after-school sports all four years of high school.

b. The values that X may take on are $X=0,1,2,.......,60.$

c. The distribution of $X~B(60,0.08)$

d. $4.8$is the number of seniors are expected to have participated in after-school sports all four years of high school.

e. Yes, it would be surprising if none of the seniors participated in after-school sports all four years of high school.

f. More likely four of the seniors participated in after-school sports all four years of high school.

Step-by-step solution

Content Introduction

The binomial distribution determines the probability of looking at a specific quantity of a hit results in a specific quantity of trials.

Part (a) Step 1: Explanation

We are given,

$8\%$of students at a local high school participate in after-school sports all four years of high school and a group of $60$seniors is randomly chosen.

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case the random variable X is the number who participated in after-school sports all four years of high school.

Part (b) Step 1: Explanation

Make the list of values that you want to use X may take on.

As we can see there is an upper bound for the situation at hand, $60$, then X is given by:

$X=0,1,2,.........,60.$

Part (c) Step 1: Explanation

The random variable is distributed by the data provided X will keep the track of online offerings.

According to the given information, $8\%$of students at a local high school participate in after-school sports all four years of high school and a group of $60$seniors is randomly chosen.

The probability distribution of binomial distribution has two parameters role="math" localid="1649086363932" $$\begin{gathered} n=60 \\ numberoftrials \\ p=0.08 \\ probabilityofsuccess \end{gathered}$$

The binomial distribution is of the form: $X~B(n,p)$

Therefore,

$X~B(60,0.08)$

Part (d) Step 1: Explanation

The expected binomial distribution is calculated as:

$\mu =np$

$\mu$is the number of seniors are expected to have participated in after-school sports all four years of high school,

$n=60$

$p=0.08$

Therefore, the number of seniors are expected to have participated in after-school sports all four years of high school is:

$$\begin{gathered} \mu =np \\ \mu =60\times 0.08 \\ \mu =4.8 \end{gathered}$$

Part (e) Step 1: Explanation

As we have found the number of seniors that have participated in after-school sports all four years of high school is $4.8$

Therefore, the probability of the seniors who have not participated at all in after-school sports all four years of high school is $0.$

$P\left(0\right)=0.0067$

Part (f) Step 1: Explanation

Using the $TI-83$calculator the probability of four or that five of the seniors participated in after-school sports all four years of high school is

$P\left(4\right)=0.1873$and $P\left(5\right)=0.1824$

We can conclude that,

More likely four of the seniors participated in after-school sports all four years of high school.