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 (b) between 490 and 510. 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 score lies between 490 and 510.
2. Step 2: Standardize the scores to convert them into z-scores. The z-score formula is given by: $z=\frac{x-\mu }{\sigma }$, where x is the raw score, μ is the mean, and σ is the standard deviation. Compute the z-scores for x = 490 and x = 510.
3. Step 3: Use the z-scores to find the cumulative probabilities. For each z-score, refer to a standard normal distribution table or use technology (e.g., a calculator or statistical software) to find the cumulative probability up to each z-score.
4. Step 4: Subtract the cumulative probability for the lower z-score (corresponding to x = 490) from the cumulative probability for the higher z-score (corresponding to x = 510). This difference gives the probability that a score lies between 490 and 510.
5. Step 5: Interpret the result. If the probability is very small (e.g., less than 0.05), it may indicate an unusual event. Compare the calculated probability to determine if the event is unusual and explain your reasoning based on the context of the problem.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In the context of the MCAT scores, the normal distribution allows us to understand how scores are spread around the average score of 500.9.

Z-Scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the score and then dividing by the standard deviation. For the MCAT scores, calculating Z-scores for the scores of 490 and 510 will help determine their positions relative to the mean, allowing us to find the corresponding probabilities using the standard normal distribution table.

Probability Calculation

Probability calculation in the context of normal distributions often involves finding the area under the curve between two Z-scores. This area represents the probability of a score falling within that range. For the MCAT scores between 490 and 510, we would calculate the Z-scores for both values and then use the standard normal distribution to find the probability that a randomly selected student scores within this range.