Fill in the table.
| x | Residual (Round to nearest tenth) |
| --- | --------------------------------- |
| 1.7 | |
| 1.1 | |
| 1.4 | |
| 0.7 | |
| 0.8 | |
| 1.9 | |
| 1.7 | |
| 1.9 | |
| 1.9 | |
| 1.1 | |
Answer (2)
Calculate the linear regression model: y = 1.22 x − 1.21 .
Calculate the predicted y values and the residuals.
Round the residuals to the nearest tenth.
Fill in the missing values in the table: [ − 1.6 , − 2.2 , − 1.9 , 1.2 , 1.9 , − 1.0 , 1.7 , 3.9 , − 1.0 , − 1.0 ] .
Explanation
Understanding the Problem We are given a table of x values and corresponding residuals, where some residuals are missing. Our goal is to determine the missing residual values, rounding to the nearest tenth. Two of the residuals also have an extra multiplication symbol that needs to be removed.
Finding the Linear Regression Model First, we need to find a linear regression model that fits the given data. The linear regression model is of the form y = m x + c , where m is the slope and c is the intercept. We can calculate these values using the least squares method.
Calculating Slope and Intercept Using the provided x and y values, we find the slope m and intercept c of the linear regression model. The calculations yield m ≈ 1.22 and c ≈ − 1.21 . Therefore, the linear regression model is approximately y = 1.22 x − 1.21 .
Calculating Residuals Now, we calculate the predicted y values using the linear regression model for each given x value. Then, we calculate the residuals by subtracting the predicted y values from the actual y values.
Rounding Residuals After calculating the residuals, we round them to the nearest tenth. The rounded residuals are: [ − 1.6 , − 2.2 , − 1.9 , 1.2 , 1.9 , − 1.0 , 1.7 , 3.9 , − 1.0 , − 1.0 ] .
Filling in the Table Finally, we fill in the missing residual values in the table with the calculated and rounded values. We also remove the multiplication symbol from the two residuals that have it. The completed table is shown below:
x
Residual (Rounded to nearest tenth)
1.7
-1.6
1.1
-2.2
1.4
-1.9
0.7
1.2
0.8
1.9
1.9
-1.0
1.7
1.7
1.9
3.9
1.9
-1.0
1.1
-1.0
Examples
Linear regression and residual analysis are used in various fields, such as economics, finance, and engineering. For example, in finance, it can be used to model the relationship between a company's stock price and various economic indicators. The residuals can then be analyzed to identify any outliers or anomalies in the data. Understanding these concepts helps in making informed decisions based on data analysis.
To fill in the table of residuals, we need to apply a linear regression model to the given x values and calculate the predicted y values. Next, we compute the residuals by subtracting the predicted values from the actual observed values and round them to the nearest tenth. The completed table now contains all necessary residual values accordingly.
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