Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.


MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (c) more than 515. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)

Verified step by step guidance

1. Step 1: Understand the problem. The MCAT scores are normally distributed with a mean (μ) of 500.9 and a standard deviation (σ) of 10.6. We are tasked with finding the probability that a randomly selected student has a score greater than 515. This involves calculating the area under the normal curve to the right of 515.
2. Step 2: Standardize the score. To find the probability, we first convert the raw score (X = 515) into a z-score using the formula: z = (X - μ) / σ. Substituting the given values, z = (515 - 500.9) / 10.6.
3. Step 3: Use the z-score to find the probability. Once the z-score is calculated, use a z-table or statistical software to find the cumulative probability corresponding to this z-score. The cumulative probability gives the area to the left of the z-score under the standard normal curve.
4. Step 4: Calculate the probability for 'more than 515'. Since we are interested in the probability of scoring more than 515, subtract the cumulative probability (area to the left of the z-score) from 1. This gives the area to the right of the z-score.
5. Step 5: Interpret the result and identify unusual events. Compare the calculated probability to a threshold (e.g., 0.05) to determine if the event is unusual. An event is typically considered unusual if its probability is less than 0.05. Provide reasoning based on the calculated probability and the context of the problem.

Key Concepts

Normal Distribution

Normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the MCAT scores follow a normal distribution, meaning that most scores cluster around the mean (500.9), with fewer scores appearing as you move away from the mean in either direction. Understanding this distribution is crucial for calculating probabilities related to specific score thresholds.

Z-Scores

A Z-score represents the number of standard deviations a data point is from the mean of a distribution. It is calculated using the formula Z = (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation. For the MCAT scores, calculating the Z-score for a score of 515 allows us to determine how unusual this score is compared to the average, facilitating the probability calculation.

Probability Calculation

Probability calculation in the context of normal distributions often involves using Z-scores to find the area under the curve corresponding to a specific score. This area represents the probability of a randomly selected score falling within a certain range. In this case, we would use the Z-score for 515 to find the probability that a medical student scores above this threshold, which can be done using statistical tables or technology.