A used concrete pumping truck can be purchased for $125,000. The operation costs are expected to be $65,000 the first year and increase 5% each year…

A defining characteristic of the standard normal distribution, and any normal distribution, is its perfect symmetry around the central point. Due to this symmetry, the mean, median, and mode are all equal and located at the peak of the bell-shaped curve. The statement that the mean and median are not equal is factually incorrect for a normal distribution. The other options correctly describe fundamental properties: the empirical rule (approximately 68% of values lie within one standard deviation), the total area under the probability density curve equaling 1, and the symmetrical shape.

B. about 68% of its value is within one standard deviation: This is a correct statement known as the empirical rule (or 68–95–99.7 rule) for normal distributions.
C. the total area underneath the curve is 1: This is a fundamental property of all continuous probability distributions, including the standard normal distribution.
D. the curve shape is symmetrical: The bell-shaped curve of the normal distribution is, by definition, symmetrical about its mean.

1. Hollmann, J. K. (Ed.). (2015). AACE International's Skills & Knowledge of Cost Engineering (6th ed.). AACE International. In Chapter 28, Section 28.2, "Probability Distributions," it is stated for the normal distribution that "the mean, median, and mode are all equal" and it is a "symmetrical, bell-shaped curve." The section also confirms that "68.3 percent of the area is within plus or minus one standard deviation of the mean."

2. Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). John Wiley & Sons. Chapter 4, Section 4-5, "The Normal Distribution," states, "A normal distribution is symmetric about its mean." and describes the empirical rule.

3. MIT OpenCourseWare. (2014). 18.05 Introduction to Probability and Statistics, Spring 2014. Massachusetts Institute of Technology. In the Class 11 Reading: Continuous random variables; Normal distribution, Section 4, "Normal distribution," it is noted that the probability density function (pdf) is symmetric about the mean μ. This symmetry implies that the mean and median are identical. The document also illustrates the 68-95-99.7 rule.