Questions: According to admissions data from the Massachusetts Institute of Technology (MIT), 4.1% of freshman applicants are admitted to the college. If 33,240 individuals applied to MIT last year, find the probability that at least 1,400 of them were admitted. Round your answer to 4 decimal places.

Solution Steps

To solve this problem, we can use the normal approximation to the binomial distribution. First, calculate the mean and standard deviation of the binomial distribution using the given probability of admission and the number of applicants. Then, use the normal distribution to find the probability that at least 1,400 applicants were admitted.

Given the probability of admission \( p = 0.041 \) and the number of applicants \( n = 33240 \), we can calculate the mean \( \mu \) and standard deviation \( \sigma \) of the binomial distribution as follows:

\[ \mu = n \cdot p = 33240 \cdot 0.041 = 1362.84 \]

\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{33240 \cdot 0.041 \cdot (1 - 0.041)} \approx 36.1519 \]

To find the probability that at least 1,400 applicants were admitted, we need to calculate:

\[ P(X \geq 1400) = 1 - P(X < 1400) \]

Using the normal approximation, we convert this to:

\[ P(X < 1400) = P\left(Z < \frac{1400 - \mu}{\sigma}\right) \]

Calculating the Z-score:

\[ Z = \frac{1400 - 1362.84}{36.1519} \approx 1.02 \]

Using the Z-score, we can find \( P(Z < 1.02) \) from the standard normal distribution table or using a cumulative distribution function (CDF):

\[ P(Z < 1.02) \approx 0.8460 \]

Thus, the probability that at least 1,400 applicants were admitted is:

\[ P(X \geq 1400) = 1 - 0.8460 = 0.1540 \]

Final Answer