Problem 25 - Calculus for Biology and Medicine

Question

Assume that the mathematics score \(X\) on the Scholastic Aptitude Test (SAT) is normally distributed with mean 500 and standard deviation 100 . (a) Find the probability that an individual's score exceeds 700 . (b) Find the math SAT score so that \(10 \%\) of the students who took the test have that score or greater.

Step-by-Step Solution

Answer

(a) 0.0228; (b) Approximately 628.

Step 1: Understanding the Problem

We need to calculate two probabilities involving a normally distributed SAT math score. For part (a), we need to determine the probability of a score exceeding 700. For part (b), we find the score beyond which the top 10% of students scored.

Step 1: Standardize the Score for Part (a)

To find the probability of a score greater than 700, we first standardize 700 using the formula for a z-score: \[ z = \frac{X - \mu}{\sigma} \]where \(X = 700\), \(\mu = 500\), and \(\sigma = 100\). Calculating this gives:\[ z = \frac{700 - 500}{100} = 2 \]

Step 2: Calculate Probability for Part (a)

Using the standard normal distribution table, find \(P(Z > 2)\). The value for \(P(Z < 2)\) is typically about 0.9772. Therefore, \[ P(Z > 2) = 1 - 0.9772 = 0.0228 \]Thus, the probability that an individual's score exceeds 700 is 0.0228.

Step 3: Find Z-Score for Top 10% for Part (b)

For part (b), we need to find the score corresponding to the top 10%. This means you need the z-score where the cumulative probability is 0.90 (since 10% are above it). From the z-table, \(P(Z < z) = 0.90\) corresponds to a \(z \approx 1.2816\).

Step 4: Convert Z-Score to Actual SAT Score for Part (b)

To convert this z-score to an SAT score, use the z-score formula in reverse:\[ X = \mu + z\sigma \]Substituting the values, we get:\[ X = 500 + (1.2816)(100) \approx 628.16 \]Therefore, about 10% of students scored 628 or higher.

Key Concepts

Standard Deviation

Standard deviation is a measure of how spread out numbers are in a dataset. In the context of a normal distribution, it helps describe how much scores deviate from the average, or mean, score.
When you're dealing with the standard deviation:

• **Small Standard Deviation**: Most data points are close to the mean.
• **Large Standard Deviation**: Data points are more spread out from the mean.

In our SAT example, the standard deviation is given as 100. This tells us that most students' scores are within 100 points of the mean score of 500. Understanding this concept helps when you're working with probabilities and the normal curve, as it describes the spread of data points in this curve.

Z-Score

The z-score is a way of standardizing scores on different scales to make them comparable. It measures the number of standard deviations a data point is from the mean.
For the SAT math scores:

• The z-score formula: \[ z = \frac{X - \mu}{\sigma} \]where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
• In step (a), to find the z-score for a score of 700, we used: \[ z = \frac{700 - 500}{100} = 2 \]

This z-score of 2 means the score of 700 is 2 standard deviations above the average score of 500. This helps in finding the probability associated with scores higher than 700 using standard normal distribution tables.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. In normal distribution, it's often visualized as the area under the curve to the left of a z-score.
Here's how it works using the SAT example:

• In step (a), we calculated the probability of a score exceeding 700, which involved finding \( P(Z > 2) \). This is 1 minus the cumulative probability of a z-score of 2: \[ P(Z > 2) = 1 - 0.9772 = 0.0228 \]
• For step (b), we sought the score where only 10% of students scored higher. We needed the z-score corresponding to a cumulative probability of 0.90, indicating 90% scored below it. This score translated back from the z-score was approximately 628.

Understanding cumulative probability is crucial for interpreting various probabilities in a normal distribution and can be applied to real-world situations like standardized testing outcomes as seen here.