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 solve this problem, we can use the binomial distribution formula to calculate the probability.
The binomial distribution calculates the probability of observing a certain number of "successes" in a fixed number of independent Bernoulli trials, where each trial can result in either a success or a failure.
In this scenario:
- The number of trials is 38 (since we randomly selected 38 students).
- The probability of success (p) is 0.77 (since 77% of the population considers statistics to be an exciting subject).
- The number of successes we want to calculate the probability for is 37 or more.
To find the probability of observing 37 or more students who consider statistics to be an exciting subject, we need to calculate the cumulative probability of 37, 38, which is the sum of the individual probabilities.
Now, let's calculate the probability using the binomial distribution formula. The formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- n is the number of trials (38 in this case)
- k is the number of successes we want to calculate the probability for
- (n C k) is the binomial coefficient, calculated as n! / (k! * (n - k)!)
- p is the probability of success (0.77 in this case)
- (1 - p) is the probability of failure (1 - 0.77 = 0.23 in this case)
Using this formula, we can calculate the probability for each value of k (from 37 to 38) and sum them up:
P(X >= 37) = P(X = 37) + P(X = 38)
Let's calculate the probability step by step:
First, let's calculate P(X = 37):
P(X = 37) = (38 C 37) * 0.77^37 * (1 - 0.77)^(38 - 37)
Next, let's calculate P(X = 38):
P(X = 38) = (38 C 38) * 0.77^38 * (1 - 0.77)^(38 - 38)
Finally, we can sum up these probabilities to find P(X >= 37).