Question

Suppose that combined verbal and math SAT scores follow a normal distribution with mean 896 and standard deviation \(174 .\) Suppose further that Peter finds out that he scored in the top \(3 \%\) of SAT scores. Determine how high Peter's score must have been.

Short answer

Peter's SAT score must be at least 1223.

Step-by-step solution

Understand the Problem

We are given that the SAT scores follow a normal distribution with a mean \(\mu = 896\) and standard deviation \(\sigma = 174\). We need to determine the cutoff SAT score that falls in the top \(3\%\) of this distribution.

Identify the Required Probability

Since Peter's score is in the top \(3\%\), we must find the score above which \(3\%\) of the data lies. This translates to finding a score above which \(97\%\) of the scores are lower, which corresponds to the \(97\text{th}\) percentile.

Use the Z-Score Formula

The Z-score formula is \(Z = \frac{X - \mu}{\sigma}\). We need to determine the Z-score corresponding to the \(97\text{th}\) percentile in a standard normal distribution.

Find the Z-Score

Using a standard normal distribution table or a calculator, we find the Z-score that corresponds to the \(97\text{th}\) percentile, which is approximately \(Z = 1.88\).

Calculate the SAT Score

We use the Z-score formula to solve for \(X\): \(X = Z \times \sigma + \mu\).Substitute the known values: \(X = 1.88 \times 174 + 896 \approx 1223.12\).Therefore, the SAT score must be at least \(1223\) (rounding up to the nearest whole number).

Key concepts

Understanding SAT Scores

SAT scores are a significant factor for students applying to colleges, as many institutions consider them in the admissions process. The SAT exam is standardized to ensure that students from different backgrounds are evaluated on the same scale. SAT scores for verbal and math generally follow a normal distribution, which means that most students score around the mean, with fewer students scoring much higher or lower.
The mean SAT score in this case is 896, with a standard deviation of 174. This distribution implies that most students have scores close to 896, but some will score significantly higher or lower. This concept of normal distribution helps in determining how an individual score compares to the scores of all test-takers.

Explaining Z-Score

A Z-score is a statistical measure that tells us how many standard deviations a data point is from the mean. In simpler terms, it quantifies the position of a score within a distribution. The formula for the Z-score is given by:

• \[ Z = \frac{X - \mu}{\sigma} \]

Where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
To find Peter's Z-score, we identify the percentile he belongs to. Since his score is in the top 3%, he is in the 97th percentile—meaning 97% of students scored below him. To find the exact Z-score for this percentile, we can use a Z-table or calculator, which shows that a Z-score of 1.88 corresponds to the 97th percentile.
Calculating the SAT score using this Z-score involves manipulating the formula to solve for \(X\):

• \[ X = Z \times \sigma + \mu = 1.88 \times 174 + 896 \approx 1223.12 \]

This estimation tells us that scoring at least 1223 places Peter in the top 3% of test-takers.

Comprehending Percentile Rank

Percentile rank is a way of comparing a score with the rest of the scores in a distribution. It tells us the percentage of scores that are below a particular score. In Peter's case, being in the 97th percentile means his SAT score is higher than 97% of test-takers. This is a useful way to interpret scores, as it provides context about where a score lies within the broader distribution.
Percentile ranks are particularly valuable in educational settings, as they allow students, teachers, and institutions to evaluate individual performances relative to peers. Understanding the percentile rank helps determine if students perform above, below, or at the average in a given group.
Essentially, the higher the percentile rank, the better the performance relative to others. A top percentile score, such as Peter's, indicates a standout performance and can be a favorable indicator when seeking admission to competitive programs.