Question
A human resources manager recorded the experience and hourly wage for a sample of 10 technology workers. 6-85 HW eTool Homework Help ✎
Experience (years)
Hourly Wage ($)
x=1,2,3,4,5,6,7,8,9,10
Y=12,13.25,14,16,17,18,19.50,21,22,23
Sketch a scatterplot showing the association between the wage and the years of experience. Describe the association.
Sketch the residual plot. Is a linear model appropriate?
What is the correlation coefficient? What does it tell you?
Experience (years)
Hourly Wage ($)
x=1,2,3,4,5,6,7,8,9,10
Y=12,13.25,14,16,17,18,19.50,21,22,23
Sketch a scatterplot showing the association between the wage and the years of experience. Describe the association.
Sketch the residual plot. Is a linear model appropriate?
What is the correlation coefficient? What does it tell you?
Answers
GPT-4o mini
To analyze the relationship between years of experience and hourly wage using the given data:
Experience (x, years):
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Hourly Wage (y, $):
- 12, 13.25, 14, 16, 17, 18, 19.50, 21, 22, 23
### Step 1: Create a Scatterplot
You would plot the data points (x, y) where:
- (1, 12)
- (2, 13.25)
- (3, 14)
- (4, 16)
- (5, 17)
- (6, 18)
- (7, 19.50)
- (8, 21)
- (9, 22)
- (10, 23)
On the X-axis (horizontal), plot the years of experience; on the Y-axis (vertical), plot the hourly wage.
### Description of the Association
- Based on the plotted points, you would typically observe a positive association indicating that as years of experience increase, the hourly wage tends to increase. This suggests a linear relationship between the two variables.
### Step 2: Calculate the Residuals
To analyze the appropriateness of a linear model, calculate the predicted values of the hourly wage using a linear regression equation. The general form of the linear relationship can be expressed as:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the intercept.
1. **Perform Linear Regression**: If you were to conduct a linear regression analysis, you'd obtain slope (m) and intercept (b) values.
2. **Calculate Residuals**: The residual for each observation is calculated as:
\[ \text{Residual} = \text{Observed value} - \text{Predicted value} \]
3. **Create a Residual Plot**: Plot the residuals on the Y-axis and the years of experience on the X-axis.
### Is a Linear Model Appropriate?
- If the residuals do not show a clear pattern (such as no trend or if they are scattered randomly around zero), it suggests that a linear model is appropriate. However, if you notice a pattern or a curve in the residuals, this indicates that a linear model may not be suitable.
### Step 3: Calculate the Correlation Coefficient (r)
The correlation coefficient measures the strength and direction of the linear relationship between the two variables. It can be calculated using:
\[ r = \frac{n(\sum (xy)) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
Where:
- \( n \) = number of data points
- \( \sum xy \) = sum of the product of each pair of x and y
- \( \sum x \), \( \sum y \) = sums of the respective variables
- \( \sum x^2 \) and \( \sum y^2 \) = sum of squares of the respective variables
### Interpretation of the Correlation Coefficient
- The value of \( r \) ranges between -1 and 1.
- An \( r \) close to 1 indicates a strong positive linear relationship.
- An \( r \) close to -1 indicates a strong negative linear relationship.
- An \( r \) close to 0 indicates no linear relationship.
When you calculate the correlation coefficient based on the provided data, you can determine how strongly the years of experience predict hourly wage.
### Conclusion
To summarize:
- You would observe a positive association in the scatterplot.
- Check the residuals for non-linearity to validate the linear model.
- Calculate and interpret the correlation coefficient to understand the strength and direction of the relationship.
Please note that for actual calculations of regression and correlation, you would typically use software or a calculator for accurate results.
Experience (x, years):
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Hourly Wage (y, $):
- 12, 13.25, 14, 16, 17, 18, 19.50, 21, 22, 23
### Step 1: Create a Scatterplot
You would plot the data points (x, y) where:
- (1, 12)
- (2, 13.25)
- (3, 14)
- (4, 16)
- (5, 17)
- (6, 18)
- (7, 19.50)
- (8, 21)
- (9, 22)
- (10, 23)
On the X-axis (horizontal), plot the years of experience; on the Y-axis (vertical), plot the hourly wage.
### Description of the Association
- Based on the plotted points, you would typically observe a positive association indicating that as years of experience increase, the hourly wage tends to increase. This suggests a linear relationship between the two variables.
### Step 2: Calculate the Residuals
To analyze the appropriateness of a linear model, calculate the predicted values of the hourly wage using a linear regression equation. The general form of the linear relationship can be expressed as:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the intercept.
1. **Perform Linear Regression**: If you were to conduct a linear regression analysis, you'd obtain slope (m) and intercept (b) values.
2. **Calculate Residuals**: The residual for each observation is calculated as:
\[ \text{Residual} = \text{Observed value} - \text{Predicted value} \]
3. **Create a Residual Plot**: Plot the residuals on the Y-axis and the years of experience on the X-axis.
### Is a Linear Model Appropriate?
- If the residuals do not show a clear pattern (such as no trend or if they are scattered randomly around zero), it suggests that a linear model is appropriate. However, if you notice a pattern or a curve in the residuals, this indicates that a linear model may not be suitable.
### Step 3: Calculate the Correlation Coefficient (r)
The correlation coefficient measures the strength and direction of the linear relationship between the two variables. It can be calculated using:
\[ r = \frac{n(\sum (xy)) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
Where:
- \( n \) = number of data points
- \( \sum xy \) = sum of the product of each pair of x and y
- \( \sum x \), \( \sum y \) = sums of the respective variables
- \( \sum x^2 \) and \( \sum y^2 \) = sum of squares of the respective variables
### Interpretation of the Correlation Coefficient
- The value of \( r \) ranges between -1 and 1.
- An \( r \) close to 1 indicates a strong positive linear relationship.
- An \( r \) close to -1 indicates a strong negative linear relationship.
- An \( r \) close to 0 indicates no linear relationship.
When you calculate the correlation coefficient based on the provided data, you can determine how strongly the years of experience predict hourly wage.
### Conclusion
To summarize:
- You would observe a positive association in the scatterplot.
- Check the residuals for non-linearity to validate the linear model.
- Calculate and interpret the correlation coefficient to understand the strength and direction of the relationship.
Please note that for actual calculations of regression and correlation, you would typically use software or a calculator for accurate results.