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 40 students from the population. If the true percentage is really 77%, find the probability of observing 39 or more students who consider statistics to be an exciting subject
0.000344
To find the probability of observing 39 or more students who consider statistics to be an exciting subject, we can use the binomial distribution.
The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. In this case, the trials are the 40 randomly selected students, and the probability of success is 77%.
The probability of observing exactly k successes in n trials, where the probability of success is p, can be calculated using the formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where (n choose k) is the number of combinations of n items taken k at a time, given by:
(n choose k) = n! / (k! * (n - k)!)
In this case, we want to find the probability of observing 39 or more students who consider statistics to be an exciting subject. We can calculate this as the sum of probabilities of observing exactly 39, 40, and any number greater than 40 successes:
P(X >= 39) = P(X = 39) + P(X = 40) + P(X = 41) + ... + P(X = 40)
Plugging in the values into the formula:
P(X >= 39) = [(40 choose 39) * (0.77)^39 * (1 - 0.77)^(40 - 39)] + [(40 choose 40) * (0.77)^40 * (1 - 0.77)^(40 - 40)]
To calculate this, we need to evaluate the binomial coefficients [(40 choose 39) and (40 choose 40)] and simplify the expression:
[(40 choose 39) * (0.77)^39 * (1 - 0.77)^(40 - 39)] = [40 * (0.77)^39 * 0.23]
[(40 choose 40) * (0.77)^40 * (1 - 0.77)^(40 - 40)] = [(0.77)^40]
Then we can sum up these terms:
P(X >= 39) = [40 * (0.77)^39 * 0.23] + [(0.77)^40]
By calculating this expression, you will find the probability of observing 39 or more students who consider statistics to be an exciting subject.