Question

which statement must be true?
each distribution has the same mean and the same standard deviation.
each distribution has the same mean and a different standard deviation.
each distribution has a different mean and the same standard deviation.
each distribution has a different mean and a different standard deviation.

Explanation:

Step1: Recall the properties of normal distribution curves

The mean of a normal distribution is the center of the curve. Here, the peaks of the curves are centered at the same point on the x - axis (around \(x = 30\) - like position, visually they are symmetric about the same vertical line). So, the means are the same.

Step2: Recall the relationship between standard deviation and the shape of the normal distribution curve

The standard deviation of a normal distribution determines its spread. A smaller standard deviation makes the curve more "peaked" (narrower), and a larger standard deviation makes the curve more "flat" (wider). Since the curves have different spreads (one is more peaked, one is moderately spread, and one is the most spread out), the standard deviations are different.

Answer:

Each distribution has the same mean and a different standard deviation.