We believe that 90% of the population of all calculus i students consider calculus an exciting subject. suppose we randomly and independently selected 21 students from the population. if the true percentage is really 90%, find the probability of observing 20 or more of the students who consider calculus to be an exciting subject in our sample of 21.

This is a binomial distribution problem. The formula to find the required probability is:

p (X) = [ n! / ((n - X) ! · X!) ] · (p) ˣ · (q) ⁿ⁻ˣ

where:

X = number of what you are trying to find the probability for = 20 or 21;

n = number of events randomly selected = 21;

p = probabiity of sucess = 0.9 (90%);

q = probability of failure = 0.1 (10%);

We need to find the probability of two events: finding 20 students and finding 21 students. Therefore P (X) = P (X = 20) + P (X = 21).

P (X = 20) = [ 21! / ((21 - 20) ! · 20!) ] · (0.9) ²⁰ · (0.1) ²¹⁻²⁰

= 0.2553

P (X = 21) = [ 21! / ((21 - 21) ! · 21!) ] · (0.9) ²¹ · (0.1) ²¹⁻²¹

= 0.1094

Therefore:

P (X) = 0.2553 + 0.1094 = 0.3647

We have a probability of 36.5% to find 20 or more students who consider calculus to be an exciting subject.