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 \(x=\mathrm{de}-\) foliation 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

Without directly testing, it seems that the quadratic model does specify a useful relationship between y and x. This is because the given \(R^2\) value is high (.902), which generally suggests a strong relationship. To confirm this at a .01 level of significance, the exact p-value would need to be calculated using an F-test, which is not possible with the information provided in the exercise.

Step-by-step solution

Formulate Null and Alternative Hypotheses

The null hypothesis (\(H_0\)) is that there is no relationship between \(y\) and \(x\), implying that the slope coefficients are equivalent to zero. In mathematical terms: \(H_0: \beta_1 = \beta_2 = 0\). The alternative hypothesis (\(H_1\)) is that at least one of the slope coefficients is not zero, suggesting there is a relationship between \(y\) and \(x\). In mathematical terms: \(H_1: \beta_1 \neq 0\) or \(\beta_2 \neq 0\)

Use F-statistic to Conduct Hypothesis Test

The F-statistic is typically used to test these hypotheses in multiple regression. However, the problem does not provide sufficient information to compute the F-statistic. In this case, we can look at the given \(R^2\) value. The F-statistic is a function of the \(R^2\) value, so its significance must be derived from the \(R^2\) value.

Interpret Results

The \(R^2\) value for our model is .902, which indicates that 90.2% of the variation in \(y\) can be explained by our model. Generally, a high \(R^2\) suggests a strong relationship between the dependent and independent variables. Although we cannot test the hypotheses directly without the F-statistic, the high \(R^2\) value suggests that our model is statistically significant and can provide useful information about the relationship between \(y\) and \(x\). If we knew the degrees of freedom, we could perform a test to obtain a p value which would let us know if this model is statistically significant at the 0.01 level.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make a decision about the nature of data and to test if certain assumptions are likely true. It is commonly used in various scientific fields to assess theories and models. In the context of a quadratic regression model, hypothesis testing helps determine whether the model significantly predicts the dependent variable based on the independent variables. The process begins with the researcher proposing two potential scenarios: a null hypothesis, which represents skepticism or a status quo belief that there is no effect or relationship, and an alternative hypothesis, which represents a belief that there is some significant effect or relationship. The subsequent step involves collecting data and calculating a test statistic that compares the observed data to what would be expected under the null hypothesis. If the test statistic exceeds a critical value determined by the desired level of significance (e.g., 0.01), the null hypothesis can be rejected, favoring the alternative hypothesis.

F-statistic

The F-statistic is a ratio that compares model variance with residual variance, effectively evaluating whether the group means in a dataset are significantly different from each other. In regression analysis, the F-statistic is used to test the overall significance of the model, answering the question: 'Does this regression model provide better predictions of the outcome than a model with no predictors?' The high value of an F-statistic generally indicates that at least one predictor variable in the regression model has a significant linear relationship with the dependent variable. To interpret the F-statistic, one needs to look at the F-distribution with the correct degrees of freedom and a chosen level of significance to determine whether the computed F-value is large enough to reject the null hypothesis that the model with no predictors is just as good as the current model.

Coefficient of Determination

The coefficient of determination, denoted by the symbol \(R^2\), is a key statistic in regression analysis that measures the proportion of variance in the dependent variable that is predictable from the independent variable(s). It is the square of the correlation coefficient and provides an indication of the goodness-of-fit of the model. A coefficient near 1 implies that the regression model does an excellent job in predicting the outcome, while a coefficient near 0 indicates that the model fails to accurately predict the dependent variable's variance. In the given exercise, the reported \(R^2\) value of .902 suggests that about 90.2% of the variability in bean yield can be accounted for by looking at the level of defoliation, implying a strong predictive power of the quadratic model.

Null and Alternative Hypotheses

In statistical testing, the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\) or \(H_a\)) are the two exclusive statements among which we choose based on the evidence provided by the sample. The null hypothesis represents a theory that has been put forward, either because it is believed to be true or because it is to be used as a basis for argument, but has not been proved. It posits no effect or no difference, and in the case of our quadratic regression model, the null hypothesis is that the coefficients for the independent variables, defoliation level and its square, are zero, which means they do not affect the yield. On the other hand, the alternative hypothesis posits that there is an effect or a difference; in this scenario, it means that at least one of the coefficients is different from zero and therefore the defoliation level does have an impact on the yield. The results of the hypothesis test will inform us whether we can reject the null hypothesis and accept the alternative hypothesis, indicating that our regression model specifies a useful relationship between yield and defoliation level.