91Ó°ÊÓ

The decline of water supplies in certain areas of the United States has created the need for increased understanding of relationships between economic factors such as crop yield and hydrologic and soil factors. The article "Variability of Soil Water Properties and Crop Yield in a Sloped Watershed" (Water 91Ó°ÊÓ Bull., 1988: 281-288) gives data on grain sorghum yield \((y\), in \(\mathrm{g} / \mathrm{m}\)-row \()\) and distance upslope \((x\), in \(\mathrm{m})\) on a sloping watershed. Selected observations are given in the accompanying table. $$ \begin{array}{l|rrrrrrr} x & 0 & 10 & 20 & 30 & 45 & 50 & 70 \\ \hline y & 500 & 590 & 410 & 470 & 450 & 480 & 510 \\ x & 80 & 100 & 120 & 140 & 160 & 170 & 190 \\ \hline y & 450 & 360 & 400 & 300 & 410 & 280 & 350 \end{array} $$ a. Construct a scatter plot. Does the simple linear regression model appear to be plausible? b. Carry out a test of model utility. c. Estimate true average yield when distance upslope is 75 by giving an interval of plausible values.

Step by step solution

01

Organize Data for Plotting

First, we will organize the given data points for the scatter plot. The data consists of pairs \((x, y)\) where \(x\) is the distance upslope and \(y\) is the grain sorghum yield. We have the following observations: \((0, 500), (10, 590), (20, 410), (30, 470), (45, 450), (50, 480), (70, 510), (80, 450), (100, 360), (120, 400), (140, 300), (160, 410), (170, 280), (190, 350)\).

02

Create Scatter Plot

Plot the data using \(x\) values on the horizontal axis and \(y\) values on the vertical axis. Look for any patterns or trends. This will help assess whether a linear relationship or trend is visible.

03

Analyze Scatter Plot Pattern

After plotting, observe the distribution of points. A simple linear regression is plausible if points appear to have a linear pattern, even if not perfectly aligned. If the points show a clear upward or downward trend, a linear model might be reasonable to consider.

04

Calculate Regression Parameters

To make the test of model utility, it's crucial to compute the slope \(b\) and intercept \(a\) of the regression line using formulas: \(b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\) and \(a = \bar{y} - b\bar{x}\). This requires calculating the means \(\bar{x}\) and \(\bar{y}\).

05

Perform Hypothesis Test for Regression Utility

Perform a hypothesis test at the significance level \(\alpha\) (usually 0.05) to check the regression's utility. Use the t-test, testing \(H_0: \beta = 0\) against \(H_a: \beta eq 0\). If the p-value is lower than \(\alpha\), the regression model is useful.

06

Calculate Prediction Interval

Estimate the true average yield for \(x = 75\). Use the regression model to calculate \(\hat{y} = a + b \cdot 75\). Compute the prediction interval, using the standard error of the estimate and the t-distribution.

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Key Concepts

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

Understanding Scatter Plots in Linear Regression

A scatter plot is a graphical representation that helps visualize the relationship between two variables. For the given problem, we plot the distance upslope \(x\) on the horizontal axis and the grain sorghum yield \(y\) on the vertical axis. Each point on the plot represents an observation from the dataset, helping us ascertain patterns or trends.

If the points show a cloud with a linear alignment, it indicates a potential linear relationship between the variables. A well-aligned trend suggests that changes in one variable may be associated with changes in the other. For students analyzing the scatter plot, keep in mind that while perfect alignment is rare, a general direction (either upward or downward) can indicate a significant relationship worth further investigation with a linear regression model.

Understanding Regression Utility

Once you have plotted the scatter plot, the next task is to calculate the regression line. This involves calculating both the slope \(b\) and the intercept \(a\) of the line using given formulas. This line represents the best summary of the relationship between the distance upslope \(x\) and yield \(y\).

Regression utility tests evaluate how well the linear regression model predicts the dependent variable, in this case, the grain sorghum yield. It involves hypothesis testing to establish the usefulness of the regression model, asking whether the slope \(b\) of the line is significantly different from zero. A slope very close to zero indicates that \(x\) doesn't explain much variance in \(y\), hence questioning the model's utility.

The Role of Hypothesis Testing in Regression Analysis

To decide if a linear regression model is worthwhile, we conduct hypothesis testing. In our context, the null hypothesis \(H_0: \beta = 0\) suggests no linear relationship exists between the upslope distance and yield. Conversely, the alternative hypothesis \(H_a: \beta eq 0\) posits that a significant relationship does exist.

A t-test allows us to evaluate these hypotheses using a chosen significance level (like 0.05). If the p-value is less than this level, you reject the null hypothesis, indicating that the regression model is useful. Understanding whether your findings could have occurred by chance is crucial in hypothesis testing, as it strengthens the credibility of the conclusions drawn from your data.

Exploring Prediction Intervals in Regression

Prediction intervals offer a way to estimate the range in which future data points are expected to fall, considering the model and its inherent variability. In the exercise, we estimate the sorghum yield for a distance of 75 meters upslope using the regression equation.

Here, the predicted yield \(\hat{y}\) is calculated using the equation \(\hat{y} = a + b \cdot 75\). To construct a prediction interval, one must compute the standard error, which accounts for variability around the regression line, and apply the t-distribution to find interval boundaries.

Prediction intervals are wider than confidence intervals for the average response because they account for more uncertainty, emphasizing the variability you might encounter when predicting single data points. This tool is crucial for making informed decisions based on regression analyses.