Final answer:
The normal distribution widens and flattens when the standard deviation increases, but the peak of the curve, which is at the mean value, remains in the same location on the x-axis.
Step-by-step explanation:
When the standard deviation increases in a normal distribution, and the mean is held constant, the curve widens but the peak of the distribution remains at the same point on the x-axis. This indicates that option b is the correct answer: The curve widens but the peak of the distribution stays at the same x-axis point. When standard deviation increases, it results in a flatter or more spread out curve, illustrating that the data points are more variable and further dispersed from the mean. Since the mean is constant, the location of the peak does not shift; it remains at the mean value.
The concept of a normal distribution is a fundamental topic in statistics, where the mean, median, and mode are the same, lying at the center of the distribution. Changes in standard deviation affect the dispersion and 'fatness' of the curve without changing its center. This is important to understand when analyzing data and determining probabilities for various outcomes under a normal distribution.