Let's analyze the uploaded image step by step to help solve the question.

The image presents a multiple-choice question on linear models, residuals, and least squares methods in statistics. Each part of the question asks whether a specific statement is true or false. I'll address each one individually:

Part (a)

To choose the best linear model, choose the linear model that passes through the most data points on the scatterplot.

1. Option A: "True. Choose the linear model that passes through the most data points on the scatterplot."
   False. A linear regression model minimizes the sum of squared residuals, not the number of points it passes through.

2. Option B: "False. The linear model line usually passes through exactly half of the data points."
   False. A regression line doesn’t guarantee that it passes through any specific number or proportion of points.

3. Option C: "False. The line usually touches none of the points. Minimize the sum of the squared errors."
   True. The regression line minimizes the sum of squared residuals and often does not pass through any data points.

4. Option D: "False. All of the data points either touch the line or fall below the line."
   False. Not all data points touch or fall below the line; the line minimizes squared residuals, meaning points can lie above or below.

Correct Answer: C

Part (b)

The residuals are the observed y-values minus the y-values predicted by the linear model.

1. Option A: "True. The residuals are the observed y-values minus the y-values predicted by the linear model."
   True. This is the correct definition of residuals.

2. Option B: "False. The residuals are the observed y-values minus the mean y-value."
   False. This describes deviations, not residuals.

3. Option C: "False. The residuals are the predicted y-values minus the y-values observed by the linear model."
   False. Residuals are calculated as (observed - predicted), not the reverse.

4. Option D: "False. The residuals are the observed x-values minus the x-values predicted by the linear model."
   False. Residuals pertain to y-values, not x-values.

Correct Answer: A

Part (c)

Least squares means that the square of the largest residual is as small as it could possibly be.

1. Option A: "True. Least squares means that the square of the largest residual is as small as it could possibly be."
   False. Least squares minimizes the sum of the squared residuals, not the square of the largest residual.

2. Option B: "False. Least squares means that the square of the median residual is minimized."
   False. Least squares does not minimize the square of the median residual.

Correct Answer: Neither A nor B is correct based on the given options.

Summary of Correct Answers

• Part (a): C
• Part (b): A
• Part (c): Both A and B are incorrect.

Let me know if you want more details or explanations! 😊

Related Questions:

1. What are residuals, and how are they used to assess a model's performance?
2. Why does the least squares method focus on minimizing the sum of squared residuals?
3. Can a regression line ever pass through all data points? If so, when?
4. How do outliers impact the sum of squared residuals in a linear regression model?
5. What alternative methods exist for fitting a line when least squares isn't appropriate?

Tip: When interpreting regression, always verify the assumptions (e.g., linearity, normality of residuals) before trusting the results!