r the following data on x = rainfall volume (m3) and y = runoff volume (m³) for a particular location. 7 12 14 17 23 30 40 49 55 67 72 83 96 112 127 y 4 10 13 15 15 25 27 46 38 46 53 70 82 99 103 accompanying Minitab output to decide whether there is a useful linear relationship between rainfall and runoff. The regression equation is runoff = -2.00 + 0.841 rainfall Predictor Coef Stdev t-ratio Constant -2.000 2.194 -0.91 0.379 rainfall 0.84079 0.03369 24.96 0.000 = = 4.823 R-sg = 98.08 R-sq (adj) = 97.8% e appropriate null and alternative hypotheses. B 0 B, = 0 B, > 0 B, = 0 e the test statistic value and find the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.) =24.96 =0.000 e conclusion in the problem context. (Use a = 0.05.) ct H. There is a useful linear relationship between runoff and rainfall at the 0.05 level. ct H. There is not a useful linear relationship between runoff and rainfall at the 0.05 level. co reject H.. There is not a useful linear relationship between runoff and rainfall at the 0.05 level. co reject H.. There is a useful linear relationship between runoff and rainfall at the 0.05 level. e a 95% confidence interval for the true average change in runoff volume associated with a 1 m3 increase in rainfall volume. (Round your answers to three decimal places.)
r the following data on x = rainfall volume (m3) and y = runoff volume (m³) for a particular location. 7 12 14 17 23 30 40 49 55 67 72 83 96 112 127 y 4 10 13 15 15 25 27 46 38 46 53 70 82 99 103 accompanying Minitab output to decide whether there is a useful linear relationship between rainfall and runoff. The regression equation is runoff = -2.00 + 0.841 rainfall Predictor Coef Stdev t-ratio Constant -2.000 2.194 -0.91 0.379 rainfall 0.84079 0.03369 24.96 0.000 = = 4.823 R-sg = 98.08 R-sq (adj) = 97.8% e appropriate null and alternative hypotheses. B 0 B, = 0 B, > 0 B, = 0 e the test statistic value and find the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.) =24.96 =0.000 e conclusion in the problem context. (Use a = 0.05.) ct H. There is a useful linear relationship between runoff and rainfall at the 0.05 level. ct H. There is not a useful linear relationship between runoff and rainfall at the 0.05 level. co reject H.. There is not a useful linear relationship between runoff and rainfall at the 0.05 level. co reject H.. There is a useful linear relationship between runoff and rainfall at the 0.05 level. e a 95% confidence interval for the true average change in runoff volume associated with a 1 m3 increase in rainfall volume. (Round your answers to three decimal places.)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:### Linear Regression Analysis of Rainfall and Runoff Data
The dataset provided contains the following variables:
- \( x \): Rainfall volume in cubic meters (\( m^3 \))
- \( y \): Runoff volume in cubic meters (\( m^3 \))
**Data Points:**
- Rainfall (\( x \)) and Runoff (\( y \)) values for specific locations are given in pairs. For example:
- Rainfall: 7, 12, 14, 17, 23, ... 96, 127
- Runoff: 4, 10, 13, 15, 25, ... 93, 103
**Regression Analysis Output:**
1. **Regression Equation:**
- Runoff = \(-2.00 + 0.841 \times \text{rainfall}\)
2. **Statistical Details:**
- **Predictor Variables:**
- **Constant (Intercept):**
- Coefficient = \(-2.000\)
- Standard Deviation (Stdev) = 2.194
- T-ratio = \(-0.91\)
- P-value = 0.379
- **Rainfall:**
- Coefficient = 0.84709
- Standard Deviation (Stdev) = 0.03369
- T-ratio = 24.96
- P-value = 0.000
- **Model Statistics:**
- Standard Error of the Estimate (s) = 4.823
- Coefficient of Determination (\( R^2 \)) = 98.0%
- Adjusted \( R^2 \) = 97.8%
**Hypothesis Testing:**
- **Null Hypothesis (\( H_0 \)):** \(\beta_1 = 0\) (No linear relationship)
- **Alternative Hypothesis (\( H_a \)):** \(\beta_1 \neq 0\) (There is a linear relationship)
The hypothesis test selected corresponds to \(\beta_1 \neq 0\), indicating a relationship between the variables.
**Test Statistic and P-value:**
- Test Statistic (\( t \)) = 24.96
- P-value = 0.000
**Conclusion:**
- Reject \( H_0 \).
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