Question

The Scholastic Aptitude Test (SAT) was taken by 1,698,521 college-bound students in the class of \(2015 .^{24}\) The test has three parts: Critical Reading, Mathematics, and Writing. Scores on all three parts range from 200 to 800 . The means and standard deviations for the three tests are shown in Table P.16. Assuming that the Critical Reading scores follow a normal distribution, draw a sketch of the normal distribution and label at least 3 points on the horizontal axis. $$ \begin{array}{lcc} & \text { Mean } & \text { St. Dev. } \\ \hline \text { Critical Reading } & 495 & 116 \\ \text { Mathematics } & 511 & 120 \\ \text { Writing } & 484 & 115 \end{array} $$

Short answer

The sketch of the normal distribution for the Critical Reading scores should be a 'bell curve' peaked at 495 (the mean). The curve should depict values around 495 being more frequent in occurrence than data far from it. The horizontal axis would be labeled with three points, the mean (495) and the points one standard deviation away from the mean on either side (379 and 611).

Step-by-step solution

Understanding Normal Distribution

Firstly, it should be understood that a normal distribution is a statistical function that describes possible values and likelihoods that a random variable can assume within a given set of data. It is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.\nHere, it's given that the Critical Reading scores follows a normal distribution, with Mean (µ) = 495 and Standard Deviation (σ) = 116.

Labeling the Mean

The mean is the peak of the normal distribution curve, which is at the center of the horizontal axis. So, we label the center point of the horizontal axis as 495.

Labeling the Standard Deviations

The properties of a normal distribution explain that approximately 68% of the data falls within one standard deviation of the mean. This corresponds to the points on the horizontal axis that are one standard deviation away from the mean. Therefore, we mark additional points on each side of the mean, 116 units away (because the standard deviation is 116). These points will represent µ - σ and µ + σ, or 495 - 116 = 379 and 495 + 116 = 611 respectively.

Sketching the Distribution

Now, we need to sketch the curve that represents the probability density of the Critical Reading scores. It's a 'bell curve' which peaks at the mean and gradually tails off towards both ends of the graph.

Key concepts

SAT Scores

The Scholastic Aptitude Test (SAT) is a standardized test widely used for college admissions in the United States. It assesses a student's readiness for college and provides colleges with a common data point that can be used to compare all applicants. The SAT is divided into three sections: Critical Reading, Mathematics, and Writing, each scored on a scale from 200 to 800.

Students' performance on the SAT is often analyzed with various statistical methods to draw conclusions about their abilities. It's vital for students to understand their SAT scores within the context of normal distribution, which helps in interpreting how their scores compare to those of the general population of test-takers. This understanding also aids educational institutions in making informed decisions during the admissions process.

Statistical Function

A statistical function is a mathematical function that describes the relationship between different statistical variables. In the context of SAT scores and normal distribution, we often refer to specific statistical functions that help us visualize and understand the distribution of scores.

The most common statistical function for this purpose is the 'probability density function', which for a normal distribution is shaped like a bell curve. It is centered around the mean and spreads out with decreasing likelihood of scores as one moves away from the mean. The properties of this function are crucial for interpreting the distribution of SAT scores, as they illustrate the average performance as well as the variability within the scores.

Probability Density

Probability density explains how the proportions of values or scores are distributed over possible outcomes. For SAT scores following a normal distribution, the probability density function (PDF) demonstrates where scores are more concentrated - which is around the mean.

Typically, the PDF for a normal distribution is known informally as a 'bell curve' due to its shape. This curve shows that outcomes close to the mean are more likely than those further away, symbolizing that most students score near the average. Key characteristics of the normal distribution help further understand the probability density, such as the '68-95-99.7 rule', which states that about 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This intuitive sense of 'spread' helps contextualize an individual's SAT score among the entire population of test-takers.