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 (a) less than 490. 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 less than 490. This involves calculating the cumulative probability for a normal distribution.
2. Step 2: Standardize the score. To find the probability, we first convert the raw score (490) into a z-score using the formula: z = (X - μ) / σ. Here, X is the raw score, μ is the mean, and σ is the standard deviation. Substitute the given values into the formula.
3. Step 3: Use the z-score to find the cumulative probability. Once the z-score is calculated, use a z-table, statistical software, or a calculator to find the cumulative probability corresponding to the z-score. This cumulative probability represents the probability that a score is less than 490.
4. Step 4: Interpret the result. Compare the calculated probability to determine if the event is unusual. An event is typically considered unusual if its probability is less than 0.05 (5%).
5. Step 5: Repeat for other parts if needed. If there are additional parts to the problem (e.g., parts (b) and (c)), follow the same process: standardize the score, find the cumulative probability, and interpret the result. Clearly explain why any event is considered unusual based on the probability threshold.

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 the Z-score for a score of 490 will help determine how unusual this score is compared to the average, allowing us to find the corresponding probability.

Probability Calculation

Probability calculation in statistics involves determining the likelihood of a particular outcome occurring within a defined set of data. For normally distributed data, this often involves using Z-scores and standard normal distribution tables or technology to find the area under the curve. In this case, we will calculate the probability of a medical student scoring less than 490 using the mean and standard deviation provided.