Question

Properties of the normal curve The National Assessment of Educational Progress (NAEP) conducts a nationwide assessment of students' proficiency in nine subjects: mathematics, reading, writing, science, the arts, civics, economics, geography, and U.S. history. The main NAEP assessments are conducted annually on samples of students from grades 4, 8, and 12. mean of 278 with a standard deviation of 37. Assume that these scores are normally In 2003, the mathematics scores for male students had distributed with the given mean and standard deviation. Which of the following are properties of the normal curve representing the distribution of the scores? Check all that apply.

A. A quarter of the total area under the curve is above 278.
B. The curve symmetrical about vertical line through 352. The curve is bell shaped, with a single peak over 278.
C. One-tenth of the total area under the curve is below 278. The curve bell shaped, with a single peak over 204. The two ends of the curve approach the horizontal axis but never touch it. above the mean. The value 167 is Therefore, below the mean, and the value 389 is of the male students have scores between 167 and 389.

Answer 1

The normal distribution curve is symmetrical about the mean, bell-shaped, and its tails approach but never touch the horizontal axis. For the NAEP math scores, the curve is centered at the mean of 278 with a standard deviation of 37. Correct understanding requires recognizing these properties rather than focusing on specific areas without reference to standard deviations.

Step-by-step explanation:

Properties of the Normal Distribution Curve

The normal curve is widely used in various fields such as psychology, business, economics, and of course, mathematics. Properties of the normal distribution curve that are relevant to interpreting the NAEP mathematics scores for male students in 2003 include:

• The curve is symmetrical about a vertical line through the mean (μ).
• The curve is bell-shaped, with a single peak over the mean.
• Approximately 68% of the data falls within one standard deviation of the mean, approximately 95% within two standard deviations, and 99.7% within three.
• The tails of the normal distribution curve approach the horizontal axis asymptotically, meaning they get closer and closer to the axis but never touch it.

In the case of the NAEP mathematics scores, the mean (μ) is 278 and the standard deviation (o) is 37. Therefore:

• A quarter of the total area under the curve is not specifically above 278, since 278 is the mean and the distribution is symmetrical.
• The curve is not symmetrical about a vertical line through 352, as it is always symmetrical around the mean, which is 278.
• One-tenth of the total area under the curve being below 278 is not a standard property; specific proportions relate to standard deviations.