Question

1. (8 pts) there are five conditions listed at the beginning of section 14.5 that should be at least approximately true for linear regression. which of the conditions can be checked by using each of the following methods? in each case, list all the conditions that can be checked. they are listed below for your convenience.

conditions:

1. linear relationship condition: the form of the line condition linking ( x ) and ( y ) is correct. (a straight line for a straight - line pattern.)
2. no outlier condition: there are no extreme outliers.
3. constant variance condition: ( y ) values are spread around the line at all of ( x ) in a similar way.
4. normal distribution condition: ( y ) values and deviations from the mean value of ( y ) are a normal distribution.
5. independence condition: the observations are independent of each other.

a. drawing a histogram of the residuals.
b. drawing a scatterplot of the residuals versus the ( x ) values.
c. learning how the data were collected.
d. drawing scatterplot of the raw data, ( y ) versus ( x ).

Explanation:

• Condition 1 (Linear Relationship Condition): A scatter - plot of the raw data (option d) can be used to check if the relationship between \(x\) and \(y\) is linear (i.e., if a straight - line pattern exists).
• Condition 2 (No Outlier Condition): A scatter - plot of the raw data (option d) can also show if there are extreme outliers.
• Condition 3 (Constant Variance Condition): A scatter - plot of the residuals versus the \(x\) values (option b) is used. If the spread of the residuals around the horizontal line (residual = 0) is constant for all \(x\) values, the constant variance condition is met.
• Condition 4 (Normal Distribution Condition): A histogram of the residuals (option a) is used. If the histogram of residuals is bell - shaped (approximately normal), the normal distribution condition is met.
• Condition 5 (Independence Condition): This is often checked based on how the data were collected (option c). For example, if the data are from a simple random sample or a well - designed experiment with random assignment, the independence condition is more likely to be satisfied.

Answer:

a. Drawing a histogram of the residuals: Condition 4 (Normal Distribution Condition).
b. Drawing a scatterplot of the residuals versus the \(x\) values: Condition 3 (Constant Variance Condition).
c. Learning how the data were collected: Condition 5 (Independence Condition).
d. Drawing scatterplot of the raw data, \(y\) versus \(x\): Conditions 1 (Linear Relationship Condition) and 2 (No Outlier Condition).