Question

Athletes on social issues in a survey of college athletes, 84% said they are willing to speak up and be more active in social issues. you randomly select 25 college athletes. find the probability that the number who are willing to speak up and be more active in social issues is_________.

(a) at least 24,
(b) less than 23, and
(c) between 18 and 22, inclusive.

Answer 1

The probability for the number of college athletes willing to speak up on social issues can be calculated using the binomial probability formula. For at least 24 athletes, less than 23 athletes, and between 18 and 22 athletes inclusive, we apply the formula with success probability 0.84 over 25 trials.

Step-by-step explanation:

Calculating the Probability for Number of Athletes Willing to Speak on Social Issues:

To find the probability of a certain number of college athletes willing to speak up on social issues, we can use the binomial probability formula because each athlete represents a Bernoulli trial with only two outcomes (willing to speak up or not). Given that 84% of them are willing, the success probability for each athlete (p) is 0.84, and the failure probability (q) is 1 - p = 0.16. The number of trials (n) is 25.

To calculate the probability for (a) at least 24 athletes, (b) less than 23 athletes, and (c) between 18 and 22 athletes, inclusive, willing to speak up:

• At least 24: We calculate P(X = 24) + P(X = 25).
• Less than 23: We calculate the sum from P(X = 0) to P(X = 22).
• Between 18 and 22: We calculate the sum from P(X = 18) to P(X = 22).

The binomial probability formula is:

P(X = k) = C(n, k) * pk * q(n-k), where C(n, k) is the number of combinations of n items taken k at a time.