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C is for cabbage ~ A researcher wants to know if the head weight of the cabbage is a good predictor for ascorbic acid (vitamin C) content. From a random sample of 60 cabbages, the researcher recorded the ascorbic acid content in milligrams and the weight of the cabbage head in pounds. Head weight is the X variable and Ascorbic acid content is the Y variable in this scenario. R output Coefficients: t value Pr(>Itl) < 2e-16 *** 9.75e-09 *** Estimate Std. Error (Intercept) 77.574 HeadWeight 3.096 25.052 -7.567 1.131 -6.689 --- Residual standard error: 7.668 on 58 degrees of freedom Multiple R-squared: 0.4355, Adjusted R-squared: 0.4257 F-statistic: 44.74 on 1 and 58 DF, p-value: 9.753e-09 Use the output above to match the value to its interpretation. The proportion of variability in Y that can be explained by the linear relationship to X. 1. 77.574 The standard error value 2. 3.096 you would use to construct a confidence interval for the actual 3. 25.052

C is for cabbage ~ A researcher wants to know if the head weight of the cabbage is a good predictor for ascorbic acid (vitamin C) content. From a random sample of 60 cabbages, the researcher recorded the ascorbic acid content in milligrams and the weight of the cabbage head in pounds. Head weight is the X variable and Ascorbic acid content is the Y variable in this scenario. R output Coefficients: t value Pr(>Itl) < 2e-16 *** 9.75e-09 *** Estimate Std. Error (Intercept) 77.574 HeadWeight 3.096 25.052 -7.567 1.131 -6.689 --- Residual standard error: 7.668 on 58 degrees of freedom Multiple R-squared: 0.4355, Adjusted R-squared: 0.4257 F-statistic: 44.74 on 1 and 58 DF, p-value: 9.753e-09 Use the output above to match the value to its interpretation. The proportion of variability in Y that can be explained by the linear relationship to X. 1. 77.574 The standard error value 2. 3.096 you would use to construct a confidence interval for the actual 3. 25.052

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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## Linear Regression Analysis Overview ### Concepts and Definitions **1. Total Variation (77.574)** - The total variation explained by the linear relationship to the independent variable \(X\). **2. Standard Error for Confidence Interval Construction (3.096)** - This is the standard error value used to construct a confidence interval for the actual slope, \(\beta_1\). **3. Total Error (25.052)** - Represents total error in the model or data. **4. Y-Intercept (-7.567)** - The expected value of the dependent variable \(Y\) when the independent variable \(X = 0\). **5. Mean Value (1.131)** - Expected mean value of the dependent variable \(Y\). **6. Test Statistic for Hypotheses (-6.689)** - The test statistic value for testing the following hypotheses: \[ H_0: \beta_1 = 0 \quad \text{vs} \quad H_A: \beta_1 \ne 0 \] **7. Variable Value (7.668)** - Another characteristic value potentially related to the data set. **8. Sample Size (58)** - Number of observations in the data set. **9. Standard Error of Regression Model (0.4355)** - The standard error associated with the regression model. **10. Slope (0.4257)** - Represents the amount of increase or decrease in the \(Y\) variable for an increase of 1 unit in the \(X\) variable. ### Explanation of In-Text Symbols and Notations **Symbols:** - \(\beta_1\): Represents the slope of the regression line. - \(H_0\): Null hypothesis in hypothesis testing. - \(H_A\): Alternative hypothesis in hypothesis testing. **Detailed Analysis:** - The information provided can be used to analyze how well the linear model fits the data and to understand the relationship between the dependent and independent variables through the calculated slope and intercept. - Understanding the standard error values and total variation is crucial for constructing confidence intervals and testing hypotheses related to the slope (\(\beta_1\)) of the regression line. This snapshot serves as a basic guide to understanding specific components of linear regression analysis, which are essential in various fields such as economics, social sciences, and
Transcribed Image Text:## Linear Regression Analysis Overview ### Concepts and Definitions **1. Total Variation (77.574)** - The total variation explained by the linear relationship to the independent variable \(X\). **2. Standard Error for Confidence Interval Construction (3.096)** - This is the standard error value used to construct a confidence interval for the actual slope, \(\beta_1\). **3. Total Error (25.052)** - Represents total error in the model or data. **4. Y-Intercept (-7.567)** - The expected value of the dependent variable \(Y\) when the independent variable \(X = 0\). **5. Mean Value (1.131)** - Expected mean value of the dependent variable \(Y\). **6. Test Statistic for Hypotheses (-6.689)** - The test statistic value for testing the following hypotheses: \[ H_0: \beta_1 = 0 \quad \text{vs} \quad H_A: \beta_1 \ne 0 \] **7. Variable Value (7.668)** - Another characteristic value potentially related to the data set. **8. Sample Size (58)** - Number of observations in the data set. **9. Standard Error of Regression Model (0.4355)** - The standard error associated with the regression model. **10. Slope (0.4257)** - Represents the amount of increase or decrease in the \(Y\) variable for an increase of 1 unit in the \(X\) variable. ### Explanation of In-Text Symbols and Notations **Symbols:** - \(\beta_1\): Represents the slope of the regression line. - \(H_0\): Null hypothesis in hypothesis testing. - \(H_A\): Alternative hypothesis in hypothesis testing. **Detailed Analysis:** - The information provided can be used to analyze how well the linear model fits the data and to understand the relationship between the dependent and independent variables through the calculated slope and intercept. - Understanding the standard error values and total variation is crucial for constructing confidence intervals and testing hypotheses related to the slope (\(\beta_1\)) of the regression line. This snapshot serves as a basic guide to understanding specific components of linear regression analysis, which are essential in various fields such as economics, social sciences, and
## C is for Cabbage - Vitamin C Content Study A researcher aims to determine if the head weight of cabbage is a good predictor of its ascorbic acid (vitamin C) content. By taking a random sample of 60 cabbages, the ascorbic acid content (in milligrams) and the head weight (in pounds) were recorded. The head weight is designated as the X variable, while the ascorbic acid content is the Y variable in this scenario. Below is the statistical output from the study conducted using R. ### R Output Analysis: #### Coefficients: | Term | Estimate | Std. Error | t value | Pr(>|t|) | |-------------|----------|------------|---------|----------| | Intercept | 77.574 | 3.096 | 25.052 | < 2e-16 | | HeadWeight | -7.567 | 1.131 | -6.689 | 9.75e-09 | #### Additional Statistics: - Residual standard error: 7.668 on 58 degrees of freedom - Multiple R-squared: 0.4355 - Adjusted R-squared: 0.4257 - F-statistic: 44.74 on 1 and 58 DF, p-value: 9.753e-09 ### Interpretation: Use the following output to match values to their interpretations: 1. **The proportion of variability in Y that can be explained by the linear relationship to X.** - Value: **0.4355** 2. **The standard error value you would use to construct a confidence interval for the actual slope, β1.** - Value: **1.131** 3. **The t value for the intercept.** - Value: **3.096** ### Summary: In this linear regression model, the intercept (77.574) represents the estimated ascorbic acid content when the cabbage head weight is zero. The slope (-7.567) suggests that for each additional pound of cabbage, the ascorbic acid content decreases by approximately 7.567 milligrams, which is statistically significant (p-value < 0.001). The R-squared value of 0.4355 indicates that about 43.55% of the variability in ascorbic acid content can be explained by the head weight of
Transcribed Image Text:## C is for Cabbage - Vitamin C Content Study A researcher aims to determine if the head weight of cabbage is a good predictor of its ascorbic acid (vitamin C) content. By taking a random sample of 60 cabbages, the ascorbic acid content (in milligrams) and the head weight (in pounds) were recorded. The head weight is designated as the X variable, while the ascorbic acid content is the Y variable in this scenario. Below is the statistical output from the study conducted using R. ### R Output Analysis: #### Coefficients: | Term | Estimate | Std. Error | t value | Pr(>|t|) | |-------------|----------|------------|---------|----------| | Intercept | 77.574 | 3.096 | 25.052 | < 2e-16 | | HeadWeight | -7.567 | 1.131 | -6.689 | 9.75e-09 | #### Additional Statistics: - Residual standard error: 7.668 on 58 degrees of freedom - Multiple R-squared: 0.4355 - Adjusted R-squared: 0.4257 - F-statistic: 44.74 on 1 and 58 DF, p-value: 9.753e-09 ### Interpretation: Use the following output to match values to their interpretations: 1. **The proportion of variability in Y that can be explained by the linear relationship to X.** - Value: **0.4355** 2. **The standard error value you would use to construct a confidence interval for the actual slope, β1.** - Value: **1.131** 3. **The t value for the intercept.** - Value: **3.096** ### Summary: In this linear regression model, the intercept (77.574) represents the estimated ascorbic acid content when the cabbage head weight is zero. The slope (-7.567) suggests that for each additional pound of cabbage, the ascorbic acid content decreases by approximately 7.567 milligrams, which is statistically significant (p-value < 0.001). The R-squared value of 0.4355 indicates that about 43.55% of the variability in ascorbic acid content can be explained by the head weight of
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