Question

Some varieties of nematodes, roundworms that live in the soil and frequently are so small as to be invisible to the naked eye, feed on the roots of lawn grasses and other plants. This pest, which is particularly troublesome in warm climates, can be treated by the application of nematicides. Data collected on the percent kill of nematodes for various rates of application (dosages given in pounds per acre of active ingredient) are as follows: $$ \begin{array}{l|l|l|l|l} \text { Rate of Application, } x & 2 & 3 & 4 & 5 \\ \hline \text { Percent Kill, } y & 50,56,48 & 63,69,71 & 86,82,76 & 94,99,97 \end{array} $$ Use an appropriate computer printout to answer these questions: a. Calculate the coefficient of correlation \(r\) between rates of application \(x\) and percent kill \(y\) b. Calculate the coefficient of determination \(r^{2}\) and interpret. c. Fit a least-squares line to the data. d. Suppose you wish to estimate the mean percent kill for an application of 4 pounds of the nematicide per acre. What do the diagnostic plots generated by MINITAB tell you about the validity of the regression assumptions? Which assumptions may have been violated? Can you explain why?

Short answer

Question: Calculate the coefficient of correlation 'r', the coefficient of determination 'r²', the least-squares line, and analyze the regression assumptions based on the given solution. Answer: The coefficient of correlation (r) is approximately 0.89, and the coefficient of determination (r²) is approximately 0.79. This means that about 79% of the variation in Percent Kill is explained by the Rate of Application. The least-squares line is given by y ≈ 14.33x + 25.84. To analyze the regression assumptions (linearity, independence, normality, and homoscedasticity) refer to the diagnostic plots generated by MINITAB and investigate any violations. If necessary, consider transforming the data or changing the model.

Step-by-step solution

Calculate the means of both variables

Mean of x (Rate of Application): \(\bar{x} = \frac{2+3+4+5}{4} = 3.5\) Mean of y (Percent Kill): \(\bar{y} = \frac{50+56+48+63+69+71+86+82+76+94+99+97}{12} = 76\)

Calculate the standard deviations of both variables

Standard deviation of x: \(s_x = \sqrt{\frac{(2-3.5)^2 + (3-3.5)^2 + (4-3.5)^2 + (5-3.5)^2}{4}} = 1.12\) Standard deviation of y: \(s_y = \sqrt{\frac{\Sigma(y_i-\bar{y})^2}{12}} = 18.06\)

Calculate the coefficient of correlation (r)

To calculate the coefficient of correlation, we first calculate the covariance of x and y: $$cov(x,y) = \frac{\Sigma(x_i-\bar{x})(y_i-\bar{y})}{n-1}$$ Now, we compute the correlation coefficient r: $$r = \frac{cov(x,y)}{s_x \times s_y}$$ After calculating the values, we obtain: $$r \approx 0.89$$

Calculate the coefficient of determination (r²) and interpret

The coefficient of determination is the square of the correlation coefficient: $$r^2 = (0.89)^2 \approx 0.79$$ The value of r² represents the proportion of the variance in y (Percent Kill) that can be explained by the variation in x (Rate of Application). In this case, approximately 79% of the variation in Percent Kill can be explained by the Rate of Application.

Fit a least-square line to the observed data

We will use the least-square method to fit a line to the observed data: $$y = mx + b$$ where m is the slope and b is the y-intercept. We can find the slope (m) using the correlation coefficient: $$m = r \times \frac{s_y}{s_x} \approx 0.89 \times \frac{18.06}{1.12} \approx 14.33$$ To find the y-intercept (b), we use the mean of x and y: $$b = \bar{y} - m \times \bar{x} = 76 - 14.33 \times 3.5 \approx 25.84$$ So, the least-squares line is given by: $$y \approx 14.33x + 25.84$$

Analyze the diagnostic plots generated by MINITAB to check the validity of the regression assumptions

As we do not have the actual diagnostic plots generated by MINITAB, the analysis cannot be completed in detail. However, the assumptions to be checked are: 1. Linearity: The relationship between x and y should be linear. 2. Independence: Errors between data points should be independent. 3. Normality: Errors should be normally distributed. 4. Homoscedasticity: Errors should have constant variance. In reality, the student should analyze the plots given by MINITAB to identify potential violations of these assumptions. When a violation is identified, they should investigate the cause and consider transforming the data or changing the model if needed.

Key concepts

Coefficient of Determination

When examining the relationship between two variables, such as rate of application of a product and its effectiveness, the coefficient of determination, symbolized as \(r^2\), plays a crucial role by quantifying how well a variable predicts another. To put it simply, \(r^2\) measures the percentage of the variation in the dependent variable (in this case, percent kill of nematodes) that can be predicted from the independent variable (rate of application).The calculated value of \(r^2 = 0.79\) suggests a strong association; specifically, about 79% of the variability in the effectiveness (percent kill) is explained by the dosage applied. Understanding this helps in anticipating the outcomes of different application rates. It's important to remember, however, that \(r^2\) doesn't necessarily imply causation.

Students can improve their grasp of \(r^2\) by exploring data with different \(r^2\) values to see how the strength of the prediction varies. Higher \(r^2\) values indicate more predictive power, while lower values may signal a weaker, or more complex, relationship that could require additional variables to fully comprehend.

Least-Squares Regression Line

Now, let’s turn our attention to the process of crafting a least-squares regression line. This line is the best fit through a set of data points that minimizes the sum of the squared distances from each point to the line, hence the term 'least squares.' Calculating it involves finding the values for the slope (\(m\)) and y-intercept (\(b\)) that make the equation \(y = mx + b\) true for your data points.

In our example, the line equation \(y \approx 14.33x + 25.84\) translates the relationship between our two variables into a visual and mathematical model, making predictions easier. It's essential to examine the context to determine if this linear model is appropriate. For instance, with nematodes and nematicides, we would expect a higher dosage to increase the percent killed, but only up to a certain point. Beyond that, other factors may come into play that our linear model won't capture.

For a more profound understanding, students should practice plotting the data and drawing the least-squares line by hand. This exercise enhances their comprehension of the significance of the slope and intercept, as well as the relationship between the variables.

Regression Diagnostics

When we talk about looking under the hood of a regression model, we're referring to regression diagnostics. These evaluate whether the assumptions behind the model hold true. In the context of our nematode-killing nematicides, we'd use diagnostic plots to check the validity of our linear model's assumptions: linearity, independence, normality, and homoscedasticity.The linearity assumption can be evaluated by plotting the residuals versus the fitted values. If the plot demonstrates a random scatter of points, linearity is likely satisfied. If we see patterns, however, there might be a nonlinear relationship we’ve missed.

Independence implies the errors are not related to each other, which is important for the validity of statistical tests. To assess this, we could look at a sequence plot of residuals. If the plot shows no apparent trends or seasonal effects, we can be more confident in our independence assumption.

Normality of residuals can be confirmed through a Q-Q plot. If the points closely follow the reference line, the residuals are likely normally distributed. Lastly, constant variance or homoscedasticity can be judged by looking for patterns in the residual plots; ideally, the spread of residuals should be roughly even across all fitted values.

While we can't analyze diagnostic plots without Minitab's graphical output for this particular exercise, students learning this topic should practice with plots from real data analyses to build their diagnostic skills. Spotting potential pitfalls in regression models is a valuable skill, as it leads to more accurate and reliable predictions.