Certainly! Below is a transcription suitable for an educational website:---**Title: Analyzing the Linear Relationship Between Rainfall and Runoff****Introduction:**This example involves analyzing the linear relationship between rainfall volume ($x$) and runoff volume ($y$) for a particular location. We use the accompanying Minitab output to make statistical inferences about this relationship.**Data:**- Independent variable ($x$): Rainfall volume (in cm)- Dependent variable ($y$): Runoff volume (in m$^3$)Given data pairs:- $x$ values: 4, 12, 13, 35, 57, 35, 48, 75, 82, 96, 112, 127- $y$ values: 14, 13, 15, 27, 45, 46, 57, 83, 99, 100, 111, 127**Statistical Output from Minitab:**The regression equation is computed as:\[ \hat{y} = 0.971 + 0.602x \]- **Predictor**: - Constant: 0.971 - Coefficient for rainfall: 0.602- **Standard Error (SE) for Coefficient**: 0.021- **T-statistic** for coefficient: 28.71- **P-value**: 0.000**Statistical Measures:**- Standard deviation of residuals ($s$): 5.021- Coefficient of determination ($R^2$): 97.6%- Adjusted $R^2$: 97.4%**Hypothesis Testing:****Null and Alternative Hypotheses:**- $H_0: \beta = 0$- $H_a: \beta \neq 0$**Conclusion Steps:**1. **Compute the test statistic value** and find the **P-value**.2. **Conclusion at $\alpha = 0.05$:** - Reject $H_0$: There is a significant linear relationship between rainfall and runoff at the 0.05 level. - Fail to reject $H_0$: There is not a significant linear relationship between rainfall and runoff at the 0.05 **Instruction for Educational Website: Confidence Interval Calculation****Task:**Calculate 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.**Answer Box:**( _______ , _______ ) m³**Additional Information:**You may need to use the appropriate table in the Appendix of Tables to answer this question.**Support:**If you need help, click the "Read It" button.

The question is about regression.Given :No. of observations ( n ) = 15Coefficient of rainfall = 0.824t - ration for rainfall ( t ) = 22.77p value for rainfall = 0.000s = 5.211To find :1 ) Test whether there is relationship between rainfall and runoff volume.2 ) Construct confidence interval for slope.1 )Null and alternative hypothesis :Answer : Option 4Test statistic :t = 22.77 p value = 0.000Decision rule :i ) Reject H0 if ii ) Fail to reject H0 if Given : Here, Hence, reject H0Conclusion :There is useful linear relationship between rainfall and runoff volume.Answer : Option 12 )The formula is given bytcritical value : Given : Confidence level ( C ) = 0.95and,Using excel command, So,and,Plugging all the values in the above formulaeHence, the confidence interval is ( 0.746, 0.902 )1 )t = 22.77p value = 0.000Conclusion :Reject H0 . There is useful linear relationship between rainfall and runoff volume.Answer : Option 12 )The confidence interval is ( 0.746, 0.902 ) m3