91Ó°ÊÓ

Quantitative SAT scores are approximately Normally distributed with a mean of 500 and a standard deviation of 100 . On the horizontal axis of the graph, indicate the SAT scores that correspond with the provided \(z\) -scores. (See the labeling in Exercise 6.14.) Answer the questions using only your knowledge of the Empirical Rule and symmetry. a. Roughly what percentage of students earn quantitative SAT scores greater than \(500 ?\) i. almost all iii. \(50 \%\) v. about \(0 \%\) ii. \(75 \%\) iv. \(25 \%\) b. Roughly what percentage of students earn quantitative SAT scores between 400 and 600 ? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) c. Roughly what percentage of students earn quantitative SAT scores greater than 800 ? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) d. Roughly what percentage of students earn quantitative SAT scores less than \(200 ?\) i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) e. Roughly what percentage of students earn quantitative SAT scores between 300 and 700 ? i. almost all iii. \(68 \%\) v. \(2.5 \%\) ii. \(95 \%\) iv. \(34 \%\) f. Roughly what percentage of students earn quantitative SAT scores between 700 and 800 ? i. almost all iii. \(68 \%\) v. \(2.5 \%\) ii. \(95 \%\) iv. \(34 \%\)

Step by step solution

01

Question (a)

Understand that a score of 500 is exactly the mean, and thus it stands to reason that half of the scores would be above and half would be below it. Therefore, about 50% of students earn scores greater than 500.

02

Question (b)

Recognize that 400 and 600 are exactly one standard deviation below and above the mean, respectively. According to the empirical rule, about 68% of the scores land within one standard deviation from the mean. Therefore, roughly 68% of students earn scores between 400 and 600.

03

Question (c)

Observe that 800 is actually three standard deviations above the mean. Recalling the empirical rule, one can deduce that almost all (about 99.7%) scores should be below the 800 mark. Therefore, the percentage of students with scores above 800 is roughly about 0%.

04

Question (d)

Understand that 200 is 3 standard deviations below the mean. According to the empirical rule, about 99.7% of the data falls within 3 standard deviations from the mean. Therefore, the percentage of students with scores less than 200 is, similarly, roughly about 0%.

05

Question (e)

Notice that 300 is two standard deviations below the mean and 700 is two standard deviations above the mean. According to the empirical rule, about 95% of the data falls within 2 standard deviations from the mean. Therefore, roughly 95% of students earn scores between 300 and 700.

06

Question (f)

Remember that 700 is two standard deviations from the mean, and 800 is three. According to the empirical rule, about 95% of the data falls within two standard deviations and about 99.7% falls within three. By subtracting these two, one can find that about 4.7% of the data is between 700 and 800. Therefore, about 4.7% of students earn scores between 700 and 800.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a symmetrical, bell-shaped distribution curve where the bulk of the data points lie near the mean, which is also the highest point on the curve. It represents how a set of data is dispersed, showing that data near the mean are more frequent in occurrence compared to data far from the mean.

In the case of SAT quantitative scores, the normal distribution implies that most students score around the average mark, with fewer students achieving either very high or very low scores. This distribution can be graphically depicted, with the mean at the center of the curve and the rest of the scores spreading out evenly to either side. When we apply the empirical rule to this distribution, it tells us that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean of the data set, whereas a high standard deviation indicates that the data points are spread out over a wider range of values.

For example, in the context of SAT scores, a standard deviation of 100 points tells us that the scores of most students are within a 100-point range above or below the mean score. It's a way to describe how spread out the scores are, and it allows us to predict the range in which a certain percentage of students' scores will fall. In the case of SAT scores with a normal distribution, if we know the standard deviation and the mean, we can use the empirical rule to estimate the distribution of students' scores.

SAT Quantitative Scores

SAT Quantitative scores are a measure of a student’s mathematical abilities, gauged by standardized testing used for college admissions in the United States. These scores typically follow a normal distribution, as evidenced in many years of SAT testing. The SAT has an average or mean score set at a benchmark to help colleges interpret the abilities of applicants.

The mean score is generally set to be 500 with a standard deviation of 100. Students, parents, and educators can use this information to compare individual performance against national averages. For instance, a student scoring 600 on the SAT Quantitative section is performing above average compared to their peers. This knowledge, combined with the empirical rule, can aid students in understanding where they stand percentile-wise, which is crucial for competitive college admissions.