Final answer:
D. The tails of a normal distribution touch the x-axis at the 3 SD from the mean.
The incorrect statement about the normal distribution is D. The tails of a normal distribution approach but never touch the x-axis, even at or beyond 3 standard deviations from the mean.
Step-by-step explanation:
The question pertains to the characteristics of a normal distribution. Let's address the points mentioned:
- A. It is true that the mean of a normal distribution can be any positive or negative number; it determines the location of the center of the distribution.
- B. The variance (and thereby the standard deviation) can indeed be any positive number. This affects the spread or width of the distribution but not its overall bell-shaped appearance.
- C. The shape of the normal distribution is symmetrical about the mean, which is also the median and mode in such a distribution.
- D. The statement that the tails of a normal distribution touch the x-axis at 3 standard deviations from the mean is incorrect. In fact, the tails of the normal distribution approach but never actually touch the x-axis; they extend infinitely in both directions.
Therefore, the correct answer is D. The tails of a normal distribution do not touch the x-axis at exactly 3 standard deviations from the mean; they decrease asymptotically towards the x-axis.