Let's analyze the provided questions and figures one by one.

1. Which has the larger standard deviation?

The standard deviation of a normal distribution reflects the spread or width of the curve. A larger standard deviation makes the curve wider, while a smaller standard deviation makes it narrower.

• Figure (a): The curve is wider, extending from approximately 6 to 18 on the x-axis.
• Figure (b): The curve is narrower, extending from approximately 2 to 8 on the x-axis.

Since figure (a) has a wider spread, it has the larger standard deviation.

Answer: Figure (a) has the larger standard deviation.

2. What is the mean of the curve in figure (a)?

The mean of a normal distribution is located at the peak (center) of the curve.

• For Figure (a), the peak is at 12.

Answer: The mean of the curve in figure (a) is 12.

3. What is the mean of the curve in figure (b)?

Similarly, the mean in figure (b) is located at the peak of the curve.

• For Figure (b), the peak is at 5.

Answer: The mean of the curve in figure (b) is 5.

Let me know if you would like more detailed explanations or have further questions.

Here are five related questions to deepen understanding:

1. How does changing the standard deviation affect the shape of a normal distribution curve?
2. What happens to a normal curve if the mean is shifted to a different value?
3. Why is the mean of a normal distribution equal to its median and mode?
4. How would you determine the variance if you know the standard deviation?
5. How is the area under a normal distribution curve related to probability?

Tip: In a normal distribution, about 68% of the data falls within one standard deviation of the mean.