We believe that 77% of the population of all Business Statistics I students consider statistics to be an exciting subject. Suppose we randomly and independently selected 38 students from the population. If the true percentage is really 77%, find the probability of observing 37 or more students who consider statistics to be an exciting subject.
To find the probability of observing 37 or more students who consider statistics to be an exciting subject, we can use the binomial distribution.
The binomial distribution is used when there are two possible outcomes (success or failure) for each trial, and the trials are independent.
In this case, the success would be a student considering statistics to be an exciting subject, and the failure would be a student not considering statistics to be an exciting subject.
The formula for the probability of observing exactly k successes in n trials is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of observing exactly k successes
- n is the total number of trials
- k is the number of successes
- p is the probability of success for each trial
- (n C k) represents the number of ways to choose k successes from n trials (binomial coefficient)
In this case, we want to find the probability of observing 37 or more students who consider statistics to be an exciting subject. So, we need to calculate the sum of probabilities for k = 37, 38.
P(X ≥ 37) = P(X = 37) + P(X = 38)
Using the binomial distribution formula, we can substitute the values into the equation and calculate the probability.