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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

Expert verified
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

01

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 \(\beta_{2} \neq 0\)
02

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 \(F = (0.902 / 2) / [(1 - 0.902) / (24 - 2 - 1)]\).
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Data from a sample of \(n=150\) quail eggs were used to fit a multiple regression model relating $$ y=\text { eggshell surface area }\left(\mathrm{mm}^{2}\right) $$ \(x_{1}=\) egg weight \((\mathrm{g})\) \(x_{2}=e g g\) width \((\mathrm{mm})\) $$ x_{3}=\text { egg length }(\mathrm{mm}) $$ (“Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors," International Journal of Poultry Science [2008]: 85-88). The resulting estimated regression function was $$ \begin{array}{l} 10.561+1.535 x_{1}-0.178 x_{2}-0.045 x_{3} \\ \text { and } R^{2}=.996 \end{array} $$ a. Carry out a model utility test to determine if this multiple regression model is useful. b. A simple linear regression model was also used to describe the relationship between \(y\) and \(x_{1}\), resulting in the estimated regression function \(6.254+1.387 x_{1}\). The \(P\) -value for the associated model utility test was reported to be less than .01 , and \(r^{2}=.994 .\) Is the linear model useful? Explain. c. Based on your answers to Parts (a) and (b), which of the two models would you recommend for predicting eggshell surface area? Explain the rationale for your choice.

Obtain as much information as you can about the \(P\) -value for the \(F\) test for model utility in each of the following situations: a. \(k=2, n=21,\) calculated \(F=2.47\) b. \(k=8, n=25,\) calculated \(F=5.98\) c. \(\quad k=5, n=26,\) calculated \(F=3.00\) d. The full quadratic model based on \(x_{1}\) and \(x_{2}\) is fit, \(n=20,\) and calculated \(F=8.25 .\) e. \(k=5, n=100,\) calculated \(F=2.33\)

When coastal power stations take in large quantities of cooling water, it is inevitable that a number of fish are drawn in with the water. Various methods have been designed to screen out the fish. The article “Multiple Regression Analysis for Forecasting Critical Fish Influxes at Power Station Intakes" (Journal of Applied Ecology [1983]: 33-42) examined intake fish catch at an English power plant and several other variables thought to affect fish intake: \(\begin{aligned} y &=\text { fish intake (number of fish) } \\ x_{1} &=\text { water temperature }\left({ }^{\circ} \mathrm{C}\right) \\ x_{2} &=\text { number of pumps running } \\ x_{3} &=\text { sea state }(\text { values } 0,1,2, \text { or } 3) \\ x_{4} &=\text { speed }(\mathrm{knots}) \end{aligned}\) Part of the data given in the article were used to obtain the estimated regression equation $$ \hat{y}=92-2.18 x_{1}-19.20 x_{2}-9.38 x_{3}+2.32 x_{4} $$ (based on \(n=26\) ). SSRegr \(=1486.9\) and SSResid = 2230.2 were also calculated. a. Interpret the values of \(b_{1}\) and \(b_{4}\) b. What proportion of observed variation in fish intake can be explained by the model relationship? c. Estimate the value of \(\sigma\). d. Calculate adjusted \(R^{2} .\) How does it compare to \(R^{2}\) itself?

The article “Readability of Liquid Crystal Displays: A Response Surface" (Human Factors [1983]: \(185-190\) ) used an estimated regression equation to describe the relationship between \(y=\) error percentage for subjects reading a four-digit liquid crystal display and the independent variables \(x_{1}=\) level of backlight, \(x_{2}=\) character subtense, \(x_{3}=\) viewing angle, and \(x_{4}=\) level of ambient light. From a table given in the article, SSRegr \(=19.2,\) SSResid \(=20.0\), and \(n=30\). a. Does the estimated regression equation specify a useful relationship between \(y\) and the independent variables? Use the model utility test with a .05 significance level. b. Calculate \(R^{2}\) and \(s_{e}\) for this model. Interpret these values. c. Do you think that the estimated regression equation would provide reasonably accurate predictions of error percentage? Explain.

According to “Assessing the Validity of the Post-Materialism Index" (American Political Science Review [1999]: \(649-664\) ), one may be able to predict an individual's level of support for ecology based on demographic and ideological characteristics. The multiple regression model proposed by the authors was $$ \begin{aligned} y=& 3.60-.01 x_{1}+.01 x_{2}-.07 x_{3}+.12 x_{4}+.02 x_{5} \\ &-.04 x_{6}-.01 x_{7}-.04 x_{8}-.02 x_{9}+e \end{aligned} $$ where the variables are defined as follows: \(y=\) ecology score (higher values indicate a greater concern for ecology) \(x_{1}=\) age times 10 \(x_{2}=\) income (in thousands of dollars) \(x_{3}=\) gender \((1=\) male \(, 0=\) female \()\) \(x_{4}=\operatorname{race}(1=\) white \(, 0=\) nonwhite \()\) \(x_{5}=\) education (in years) \(x_{6}=\) ideology \((4=\) conservative, \(3=\) right of center, \(2=\) middle of the road, \(1=\) left of center, and \(0=\) liberal) \(\begin{aligned} x_{7}=& \text { social class }(4=\text { upper, } 3=\text { upper middle, }\\\ & 2=\text { middle }, 1=\text { lower middle, and } \\ &0=\text { lower }) \end{aligned}\) \(x_{8}=\) postmaterialist ( 1 if postmaterialist, 0 otherwise) \(x_{9}=\) materialist (1 if materialist, 0 otherwise) a. Suppose you knew a person with the following characteristics: a 25 -year- old, white female with a college degree (16 years of education), who has a \(\$ 32,000\) -peryear job, is from the upper middle class, and considers herself left of center, but who is neither a materialist nor a postmaterialist. Predict her ecology score. b. If the woman described in Part (a) were Hispanic rather than white, how would the prediction change? c. Given that the other variables are the same, what is the estimated mean difference in ecology score for men and women? d. How would you interpret the coefficient of \(x_{2}\) ? e. Comment on the numerical coding of the ideology and social class variables. Can you suggest a better way of incorporating these two variables into the model?

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