Question

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

Explanation:

Step1: Analyze Mean (Center)

The peaks of all normal distributions are at the same horizontal position (around 75), so they share the same mean.

Step2: Analyze Standard Deviation (Spread)

The distributions have different widths (spreads). A narrower peak means a smaller standard deviation, and a wider peak means a larger one. So their standard deviations differ.

Step3: Evaluate Options

• Option 1: Different means? No, same center. Eliminate.
• Option 2: Different means? No. Eliminate.
• Option 3: Same standard deviation? No, different spreads. Eliminate.
• Option 4: Same mean (same center) and different standard deviation (different spreads). Correct.

Answer:

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