Question

Legumes, such as peas and beans, are important crops whose production is greatly affected by pests. The article "Influence of Wind Speed on Residence Time of Uroleucon ambrosiae alatae on Bean Plants" (Environmental Entomology [1991]: \(1375-1380\) ) reported on a study in which aphids were placed on a bean plant, and the elapsed time until half of the aphids had departed was observed. Data on \(x=\) wind speed \((\mathrm{m} / \mathrm{sec})\) and \(y=\) residence half time were given and used to produce the following information. $$ \begin{array}{ll} a=0.0119 \quad b=3.4307 \quad n=13 \\ \text { SSTo }=73.937 \quad \text { SSResid }=27.890 \end{array} $$ a. What percentage of observed variation in residence half time can be attributed to the simple linear regression model? b. Give a point estimate of \(\sigma\) and interpret the estimate. c. Estimate the mean change in residence half time associated with a \(1-\mathrm{m} / \mathrm{sec}\) increase in wind speed. d. Calculate a point estimate of true average residence half time when wind speed is \(1 \mathrm{~m} / \mathrm{sec}\).

Short answer

a. The calculated \(R^2\)-value will express the percentage variation in residence half time attributable to the model. b. The estimated \(\sigma\) suggests the standard deviation of the error term in the regression model. c. The regression slope 'b' gives the average change in residence half time per 1-m/sec wind speed increase. d. When the wind speed is 1 m/sec, the approximated average residence half time can be calculated using the regression equation.

Step-by-step solution

Compute Coefficient of determination \(R^2\)

The coefficient of determination, denoted \(R^2\), is obtained by finding the proportion of the total variance (SSTo) that is explained by the regression (SSReg=SSTo-SSResid). It is computed by the formula: \[R^2=1-\frac{SSResid}{SSTo}\]This will help in answering question a, which asks for the percentage of observed variation in residence half time attributed to the model.

Estimate the standard error \(\sigma\)

The standard error of the regression \(\sigma\) can be estimated using the residual sum of squares (SSResid) and the sample size (\(n\)) via the formula:\[\hat{\sigma}=\sqrt{\frac{SSResid}{n-2}}\]This estimate reflects the standard deviation of the error terms in our model and is needed to answer question b.

Interpret the slope 'b'

The coefficient of the predictor variable 'b' in our model represents how much the dependent variable (residence half time) increases for every one unit increase in the independent variable (wind speed). This answers question c.

Use the regression equation to forecast

Using the regression equation 'y = a + bx', the dependent variable can be estimated for a given value of the independent variable. Here, using a wind speed of 1 m/sec ('x=1'), we can calculate the predicted half-time residence:\(y=0.0119+(3.4307*1)\)This answers question d.

Key concepts

Coefficient of Determination

The coefficient of determination, often notated as \(R^2\), plays a critical role in statistical regression analysis. It measures the proportion of the variance in the dependent variable that can be predicted from the independent variable. In simpler terms, \(R^2\) represents how well the regression model fits the observed data.

When evaluating \(R^2\), values range from 0 to 1, where 0 indicates that the model does not explain any of the variability of the response data around its mean, and 1 indicates that the model explains all the variability. For the study on aphids and wind speed, calculating \(R^2\) using \[R^2 = 1 - \frac{SSResid}{SSTo}\] reveals how much of the variance in aphids’ residence half-time is accounted for by the variations in wind speed. This statistical measure helps to understand the strength of the relationship between wind speed and the residence half time of aphids on bean plants.

Standard Error of Regression

The standard error of regression, \(\hat{\sigma}\), is a measure of how much observed data points deviate from the regression line; in other words, it indicates the typical error in the predictions made by the linear regression model. To estimate the standard error for the aphid study, we use the formula \[\hat{\sigma} = \sqrt{\frac{SSResid}{n - 2}}\]

By plugging in the residual sum of squares (SSResid) and the sample size \((n)\), we can calculate the standard error, which tells us the precision of our regression estimate. A smaller standard error would suggest a more precise estimation from the regression model. Interpreting \(\hat{\sigma}\) gives us an understanding of how tightly grouped the observed values are around the predicted values, thus indicating the model's accuracy in predicting residence half time given varying wind speeds.

Interpretation of Slope

The regression model for the aphid study includes a slope, denoted as \(b\), which represents the rate of change in the dependent variable (residence half time) for each one-unit increase in the independent variable (wind speed). In the computed regression equation, \(b = 3.4307\) means that each additional meter per second of wind speed is, on average, associated with an increase of 3.4307 units in the residence half time of the aphids.

Understanding the slope is crucial as it provides concrete information about the nature of the relationship between wind speed and aphid residence time. It means that if we were to increase the wind speed, the aphids would likely take longer to depart from the bean plants. Interpreting the slope in the context of the study shows how significant wind speed is as a predictor of aphid behavior.

Regression Equation Forecasting

Forecasting with a regression equation involves using the identified model to predict future values of the dependent variable based on new values of the independent variable. In our case, the regression equation \(y = a + bx\) is used to predict the residence half time (\(y\)) for a given wind speed (\(x\)).

For instance, to predict the average residence half time when wind speed is 1 m/sec, we use the estimated parameters \(a = 0.0119\) and \(b = 3.4307\) to calculate \[y = 0.0119 + (3.4307 \times 1)\]. This prediction can assist researchers and farmers in understanding how changes in wind speed may potentially affect the length of time that aphids will remain on bean plants, thus aiding in the management of these pests.