Question

Critical Reading on the SAT Exam In Table P.16 with Exercise \(\mathrm{P} .157\), we see that scores on the Critical Reading portion of the SAT (Scholastic Aptitude Test) exam are normally distributed with mean 495 and standard deviation \(116 .\) Use the normal distribution to answer the following questions: (a) What is the estimated percentile for a student who scores 700 on Critical Reading? (b) What is the approximate score for a student who is at the 30 th percentile for Critical Reading?

Short answer

The estimated percentile for a student scoring 700 on Critical Reading is the value corresponding to the calculated Z-score in the Z-table. The expected score for a student lying in the 30th percentile of Critical Reading can be derived using the score formula and the Z-score corresponding to 30th percentile from the Z-table.

Step-by-step solution

Estimating the percentile for a student scoring 700

Firstly, calculate the Z-score, which is the number of standard deviations a particular score is away from the mean. The formula is \(Z = (X - μ) / σ\), where X represents the score, μ the mean, and σ the standard deviation. Substitute X with 700, μ with 495, and σ with 116 to get the Z score. Look up this Z-score in a standard Normal distribution table to find the percentile.

Determining the score for the 30th percentile

To find out the score at the 30th percentile, look up '0.30' in the Z-table to locate the corresponding Z-score. Then, use the formula \(X = μ + Z * σ\) to determine the score, X. Replace μ with 495, Z with the derived Z-score, and σ with 116.

Key concepts

Understanding Normal Distribution

Normal distribution, often depicted as a bell curve, is a fundamental concept in statistics that describes how data points are distributed around a mean (average). In many natural and social phenomena, the majority of observations cluster around the central peak, while fewer observations fall towards the tails at either end.

In the context of the SAT Critical Reading scores, the mean score is stated to be 495, and scores are normally distributed around this value. The standard deviation, another important term in this context, is 116. This value indicates how spread out the scores are from the mean. The larger the standard deviation, the more spread out the scores are.

A crucial property of the normal distribution is that it is symmetrical, meaning the proportions under the curve to the left and right of the mean are equal. Moreover, certain percentages of the data fall within certain ranges. For example, approximately 68% of data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations in a normal distribution.

Calculating Z-Scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is exactly the average score. A positive Z-score indicates the number of standard deviations the data point is above the mean, while a negative Z-score represents the data point being below the mean.

To calculate a Z-score, you use the formula \(Z = (X - μ) / σ\), where \(X\) is the value in question, \(μ\) is the mean of the data, and \(σ\) is the standard deviation. In our SAT example, to find the estimated percentile for a student who scores 700, we subtract the mean, 495, from the score and then divide by the standard deviation, 116. This calculation gives us the Z-score that we can then use to determine the student's position within the normal distribution compared to peers.

Understanding Percentile Calculation

The percentile rank of a score is the percentage of scores in its frequency distribution that are equal to or lower than it. It gives you an idea of how a score compares to a larger sample. For example, a student in the 70th percentile scored higher than 70% of other students. To determine percentile, you typically refer to a Z-table which provides the percentage of values to the left of the given Z-score in a standard normal distribution.

If you know the Z-score, as you would after calculating it for the SAT Critical Reading score, you can find the approximate percentile by consulting a Z-score table or using percentile calculators available online.

To find the score associated with a specific percentile, you look up the percentile in the Z-table to find the corresponding Z-score, then rearrange the Z-score formula to solve for \(X\), the score. If we know a student is at the 30th percentile, we locate the Z-score for 0.30 in the Z-table and use the formula \(X = μ + Z * σ\) to find the SAT score. By substituting the values for \(μ\) and \(σ\), we can calculate the exact SAT score corresponding to the 30th percentile.