### Understanding Regression Coefficients and Their ImplicationsWhen conducting regression analysis, determining the significance of predictor variables is crucial. If the regression coefficient $ \beta_1 $ has a nonzero value, it indicates that the predictor is effective in explaining the variability of the response variable. If $ \beta_1 = 0 $, the predictor is not significant and can be removed from the regression equation.To test this, we use the t-statistic $ t = (b_1 - 0)/s_{b_1} $, where $ b_1 $ is the estimated coefficient and $ s_{b_1} $ is the standard error. The critical t-values come from the t-distribution with degrees of freedom $ n - (k + 1) $, where $ n $ is the number of observations and $ k $ is the number of predictor variables.The example provided includes the standard error $ s_{b_1} = 0.0724477235 $ for the variable Height.### Testing Claims about Regression Coefficients1. **Testing $ \beta_1 = 0 $:** - **Null Hypothesis ($ H_0 $)**: $ \beta_1 = 0 $ - **Test Statistic**: (to be calculated) - **P-value**: (to be calculated) - If the P-value is less than a significance level (e.g., 0.05), reject $ H_0 $, implying that $ \beta_1 $ should be kept.2. **Testing $ \beta_2 = 0 $:** - **Null Hypothesis ($ H_0 $)**: $ \beta_2 = 0 $ - **Test Statistic**: (to be calculated) - **P-value**: (to be calculated) - If the P-value is less than a significance level (e.g., 0.05), reject $ H_0 $, indicating that $ \beta_2 $ should be kept.### Implications of Results on Regression Equation**Choose the correct statement:**- **A**. Only use waist since height is not useful.- **B**. Neither height nor waist should be included.- **C**. Both height and waist are useful predictors.- **D**. Only use height since waist is

If the coefficient p, has a nonzero value, then it is helpful in predicting the value of the response variable. If p, =0, it is not helpful in predicting the value of the response variable and can be eliminated from the regression equation. To test the claim that B, =0 use the test statistic t = (b₁ - 0) /s. Critical values or P-values can be found using the t distribution with n-(k+1) degrees of freedom, where kis the number of predictor (x) variables and n is the number of observations in the sample. The standard error s, is often provided by software. For example, see the accompanying technology display, which shows that b=0.072447235 (found in the column with the heading of "Std. Err." and the row corresponding to the first predictor variable of height). Use the technology display to test the claim that p, = 0. Also test the claim that p=0. What do the results imply about the regression equation? Click the icon to view the technology output. Test the claim that p, =0. the test statistic is t= and the P-value isso [ For Ho (Round to three decimal places as needed.) Test the claim that P₂ = 0. For H₂ (Round to three decimal places as needed.) What do the results imply about the regression equation? the test statistic is t= and the P-value isso H, and conclude that the regression coefficient b. = should H, and conclude that the regression coefficient b₂ =should kept kept OA. The results imply that the regression equation should only include the independent variable of waist since height will not be useful in predicting the response variable. OB. The results imply that the regression equation should not include either independent variable since both height and waist will not be useful in predicting the response variable. OC. The results imply that the regression equation should include both independent variables of height and waist as both are useful in predicting the response variable. OD. The results imply that the regression equation should only include the independent variable of height since waist will not be useful in predicting the response variable. Technology Output Parameter Intercept Height Waist Parameter estimates: Estimate✿ Std. Err.$ Alternative -143.50036 12.934811 0.77403064 0.072447235 1.0526946 0.033659265 Print OF T-Stat <0.0001 #0 150-11.094121 #0 150 10.684060 <0.0001 #0 150 31.275032 <0.0001 Done P-value$ X