Questions: Area Under a Normal Curve (a) Estimate the probability that a randomly selected student will complete their final exam in less than 63 minutes. (b) The final exam is scheduled to allow 2 hours. Estimate the probability that a randomly selected student will need more than 2 hours to complete their exam. (c) In a section with 120 students, how many students do you expect will need more than 2 hours to complete their exams? students

Solution Steps

The problem states a standard deviation of 19 minutes. μ represents the mean. μ – σ = 63 minutes where σ is the standard deviation. Therefore, μ = 63 + σ = 63 + 19 = 82 minutes.

63 minutes corresponds to μ – σ. According to the normal distribution, the area to the left of μ – σ is 13.5% + 2.35% + 0.15% = 16%. Therefore, the probability is 0.16.

2 hours is 120 minutes. The mean is 82 minutes. This corresponds to roughly $\mu + 2\sigma$, as $\mu + 2\sigma = 82 + 2 \cdot 19 = 82 + 38 = 120$ minutes. The probability of a student taking _more_ than 120 minutes is 2.35% + 0.15% = 2.5%.

With 120 students, we expect 2.5% of them to take more than 2 hours. So, 120 * 0.025 = 3 students.

Final Answer: