Question

The article "Effect of Manual Defoliation on Pole Bean Yield" (Journal of Economic Entomology [1984]: \(1019-1023\) ) used a quadratic regression model to describe the relationship between \(y=\) yield \((\mathrm{kg} /\) plot \()\) and \(x=\) defoliation level (a proportion between 0 and 1\()\) The estimated regression equation based on \(n=24\) was \(\hat{y}=12.39+6.67 x_{1}-15.25 x_{2}\) where \(x_{1}=x\) and \(x_{2}=\) \(x^{2}\). The article also reported that \(R^{2}\) for this model was .902. Does the quadratic model specify a useful relationship between \(y\) and \(x ?\) Carry out the appropriate test using a .01 level of significance.

Short answer

Whether the quadratic model specifies a useful relationship between \(y\) and \(x\) depends on the comparison of the calculated F statistic with the critical F value for a given level of significance (.01 in this case). Without specific numerical values for the F statistic and F critical, a definite conclusion can't be provided here. But, follow the steps outlined above to make this comparison and draw a conclusion.

Step-by-step solution

Set up null and alternative hypothesis

In order to determine the significance of the regression, a null hypothesis \(H_{0}\) and alternative hypothesis \(H_{1}\) need to be established. The null hypothesis suggests that all of the regression coefficients are equal to zero: \(\beta_{1} = \beta_{2} = 0\). The alternative hypothesis suggests that at least one of the regression coefficients is different from zero: \(\beta_{1} \neq 0\) or \(\beta_{2} \neq 0\)

Calculate F statistics

The F statistic in this context can be calculated using the formula \(F = (R^{2} / k) / [(1 - R^{2}) / (n - k - 1)]\), where \(R^{2}\) is the coefficient of determination, provided as \(0.902\), \(k\) is the number of predictors, in this case 2 (\(x_{1}\) and \(x_{2}\)), and \(n\) is the number of observations, given as 24. Substituting these values, the F statistic is calculated as \(F = (0.902 / 2) / [(1 - 0.902) / (24 - 2 - 1)]\).

Determine the critical F value

The critical F value is the value beyond which the null hypothesis would be rejected. This value can be found in statistical tables or calculated using statistical software. It depends on the degrees of freedom for the numerator (which equals the number of predictors, \(k = 2\)) and the denominator (which equals \(n - k - 1 = 24 - 2 - 1 = 21\)), as well as the level of significance \(.01\). For this exercise, specific numerical results are not stated, so this step will just refer to checking tables or using software for this value.

Compare F statistic with F critical

If the calculated F statistic is greater than the F critical value from the F distribution table, then the null hypothesis would be rejected. Conversely, if the F statistic is less than or equal to the F critical value, fail to reject the null hypothesis. Rejecting the null would mean that the regression model is significant, and that at least one of the coefficients \(\beta_{1}\) or \(\beta_{2}\) is different from zero, thus indicating a significant relationship between \(y\) and \(x\) in the quadratic model.

Key concepts

Null Hypothesis and Alternative Hypothesis

Understanding the null hypothesis and the alternative hypothesis is fundamental when trying to determine if there is a significant effect or relationship in a study. In the context of quadratic regression models, the null hypothesis \(H_0\) typically proposes that there is no relationship between the dependent and independent variables. In other words, it asserts that the regression coefficients are equal to zero \(\beta_1 = \beta_2 = 0\). On the other hand, the alternative hypothesis \(H_1\) suggests that at least one of the regression coefficients is not zero \(\beta_1 eq 0\) or \(\beta_2 eq 0\), indicating a potential relationship worth exploring.

In our example regarding pole bean yield, the null hypothesis supposes that the level of defoliation has no effect on the yield, while the alternative hypothesis suggests there is an effect. Selecting a correct hypothesis is crucial as it serves as the basis for further statistical testing and subsequent findings.

F Statistic Calculation

When dealing with linear regression analysis, the F statistic plays a crucial role in understanding if any of the independent variables, collectively, have predictive power. The F statistic is calculated using the formula \(F = (R^{2} / k) / [(1 - R^{2}) / (n - k - 1)]\), where \(R^{2}\) is the coefficient of determination, \(k\) is the number of predictors, and \(n\) is the number of observations.

For the quadratic regression model of the pole bean yield study, we use the given \(R^{2}\) value of 0.902, with 2 predictors and 24 observations. The higher the F statistic, the more likely it is that the variations captured by the regression are not due to random chance, thus potentially rejecting the null hypothesis. It's important to clarify the calculation step-by-step during the teaching process, ensuring students know which values to substitute into the equation.

Significance Level

The significance level, often denoted by \(\alpha\), is a critical concept in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true, known as a type I error. Common levels of significance are 0.05, 0.01, and 0.10. In the pole bean yield example, we use a 0.01 significance level due to the precision required by the investigation.

This stringent threshold means we are willing to accept only a 1% chance of incorrectly rejecting a true null hypothesis. By teaching the implications of different levels of significance, we can provide students with the insights they need to set appropriate thresholds for their own analyses, depending on the context and their willingness to risk making errors.

Regression Coefficients

The regression coefficients in a quadratic regression model, such as the one used for the pole bean yield study, represent how much the dependent variable \(y\) is expected to change as the independent variable \(x\) increases by one unit, before reaching its peak or trough when considering \(x^2\). In the provided quadratic equation \(\hat{y}= 12.39 + 6.67x_{1} - 15.25x_{2}\), \(6.67\) and \( -15.25\) are the regression coefficients for \(x_{1}=x\) and \(x_{2}=x^2\), respectively.

These coefficients are essential for understanding the nature of the relationship between yield and defoliation level. A positive coefficient indicates a positive relationship, while a negative one indicates a negative relationship. It is critical to convey how these coefficients affect the interpretation of the quadratic model in practice and to ensure students grasp both the statistical and practical significance of regression coefficients.