Let's break down and solve each of the problems you've presented.

Problem 1: Mean of Data Points on a Curve

You have data points lying on a curve in a coordinate plane where each horizontal and vertical line is 1 unit apart. The options are points on the plane that might describe the mean of the data points.

Given that the possible answers are (4,4), (5,4), (4,6), and (5,5), these are candidate mean points.

Without seeing the actual curve, if you know the spread of the data points, the mean would typically be centered within the cluster of points.

For example:

• If the points are centered around the point (5,5), with an equal spread in all directions, (5,5) would be the mean.
• If the points are slightly skewed towards (4,4) or (5,4), the mean would shift accordingly.

Answer: From the provided options and the standard interpretations:

• (a) (4,4)
• (b) (5,4)
• (c) (4,6)
• (d) (5,5)

If the data points are centered and symmetrically distributed, (d) (5,5) might be the correct choice, assuming the mean lies exactly at the center of the data cluster.

Problem 2: Properties of MDA (Multiple Discriminant Analysis)

You are asked to evaluate the truth of certain statements related to MDA (Multiple Discriminant Analysis).

• Maximizing intra-class scatter would not make sense, as MDA aims to make data points within the same class as close to each other as possible.
• Maximizing inter-class scatter is correct, as MDA tries to make classes distinct from one another.
• Minimizing intra-class scatter is correct since the goal is to make data points within a class more similar.

Answer:

• Statement I is incorrect because we do not want to maximize intra-class scatter.
• Statement II and III are correct.

So the correct answer is (c) Only II and III.

Follow-Up Questions

Here are five follow-up questions related to the content:

1. What is the significance of the mean in statistical analysis?
2. How does Multiple Discriminant Analysis differ from Principal Component Analysis?
3. Can you describe a situation where (5,5) might not be the mean even if data points are clustered around that point?
4. What are the key factors that determine whether the mean of a dataset lies at a specific point?
5. How can you visually determine the mean of a cluster of points on a graph?

Tip: When calculating the mean of a dataset graphically, consider both the spread and the symmetry of the data distribution relative to the axis.