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. X can take any value between 0 and 60, inclusive.
b. X can only take values from 0 to 4, inclusive.
c. X can take values from 5 to 9, inclusive.
d. X can take values from 10 to 60, inclusive.

Answer 1

The expected number of seniors who participated in after-school sports for all four years is 4.8, rounded up usually to 5. It would be surprising if none participated, which can be numerically justified by calculating the binomial probability of having zero students participating. To compare the likelihood between four or five participating seniors, one can calculate and compare their respective binomial probabilities.

Step-by-step explanation:

We are looking at a probability question involving a binomial distribution. The given scenario informs us that approximately 8% of students at a local high school participate in after-school sports all four years of high school. A random sample of 60 seniors is selected, and we want to know the number of students, X, who have participated in after-school sports all four years.

To calculate the expected number of seniors who participated in after-school sports for all four years, we use the expected value formula for a binomial distribution.

• Expected number = n * p

where n = total number of trials (students), and p = probability of success (participating in sports all four years).

In this case:

• n = 60
• p = 0.08

So the expected number of seniors = 60 * 0.08 = 4.8.

As the expected number of students X cannot be fractional, we would predict that 5 seniors participated in after-school sports all four years. However, we could actually observe any whole number from 0 to 60.

If no seniors participated, it would be unexpected, but not impossible. To numerically justify whether this would be surprising, we'd calculate the probability of observing zero successes using the binomial probability formula.

• Probability(X = 0) = (60 choose 0) * (0.08)^0 * (1-0.08)^(60-0)

This value would be quite low, indicating that such an outcome would indeed be surprising.

To determine if it is more likely that four or five seniors participated, we can calculate the binomial probabilities for each.

• Probability(X = 4) = (60 choose 4) * (0.08)^4 * (0.92)^(56)
• Probability(X = 5) = (60 choose 5) * (0.08)^5 * (0.92)^(55)

We would compare these probabilities to see which is higher, giving us the more likely number of students participating in after-school sports for all four years.