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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. $$ \begin{aligned} &\begin{array}{r|rrrrrrr} x & 0 & 10 & 20 & 30 & 45 & 50 & 70 \\ \hline y & 500 & 590 & 410 & 470 & 450 & 480 & 510 \end{array}\\\ &\begin{array}{l|rrrrrrr} x & 80 & 100 & 120 & 140 & 160 & 170 & 190 \\ \hline y & 450 & 360 & 400 & 300 & 410 & 280 & 350 \end{array} \end{aligned} $$ 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 the Data

To begin, we'll prepare the data points for plotting. The data consists of two variables: distance upslope \(x\) and grain sorghum yield \(y\). We'll pair these values from the given data where \(x = (0, 10, 20, ..., 190)\) and \(y = (500, 590, 410, ..., 350)\).

02

Create a Scatter Plot

Plot the data points on a graph with \(x\) values on the horizontal axis and \(y\) values on the vertical axis. Each point corresponds to a pair \((x, y)\) from the table. Visually inspect the plot to assess if a linear pattern is apparent. The points should ideally show a linear trend if a simple linear regression model is plausible.

03

Assess Linear Model Plausibility

After creating the scatter plot, observe if there is a visible linear trend. If the data points roughly form a straight line, a simple linear regression model may be appropriate.

04

Test Model Utility (Fit the Model)

Fit a simple linear regression model to the data using the least squares method. Calculate the slope and intercept of the line and the correlation coefficient \(r\). Perform a hypothesis test for the slope: \(H_0: \beta = 0\) vs \(H_a: \beta eq 0\). If the p-value is below the significance level (e.g., \(\alpha = 0.05\)), reject the null hypothesis, indicating the model is useful.

05

Calculate Regression Parameters

Using statistical software or a calculator, find the slope \(b\) and intercept \(a\) of the best-fit line from the formulae: \[\hat{y} = a + bx\]Evaluate the goodness of fit using \(R^2\). A higher \(R^2\) value suggests a better fit of the model to the data.

06

Estimate Yield at Upslope Distance 75

Substitute \(x = 75\) into the regression equation obtained from the model fitting to predict the yield at this upslope distance. Calculate the confidence interval for the predicted yield using the standard error of the estimate and t-distribution.

07

Use Regression Equation for Prediction

Using the regression equation, estimate \(\hat{y}\) when \(x = 75\). For the confidence interval, calculate:\[CI: \hat{y} \pm t_{\alpha/2} \times SE\\]where \(t_{\alpha/2}\) is the critical value from the t-distribution and \(SE\) is the standard error of the prediction.

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Key Concepts

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

Scatter Plot

A scatter plot is an essential tool in simple linear regression. It provides a visual representation of the relationship between two quantitative variables. Here, we are analyzing the relationship between the distance upslope \((x)\) and grain sorghum yield \((y)\) in a sloped watershed.

To create a scatter plot, you plot each pair of values \(x, y\) as a point on the Cartesian plane, where the horizontal axis represents the distance upslope \(x\) and the vertical axis represents the yield \(y\).

This visual inspection is crucial because it allows us to identify whether there is an apparent linear trend in the data. If the points roughly form a straight line, then we can say there is a potential linear relationship, making the simple linear regression model plausible. If the points do not align linearly, a more complex model may be necessary. The scatter plot is the first step in validating our assumption of linearity in a simple linear regression analysis.

Hypothesis Testing

In simple linear regression, hypothesis testing is used to determine whether the relationship between the independent variable \(x\) and the dependent variable \(y\) is statistically significant.

The key component of hypothesis testing in this context involves testing the slope of the regression line. Specifically, we conduct a test for the null hypothesis \(H_0: \beta = 0\) versus the alternative hypothesis \(H_a: \beta e 0\).

What does this mean? \(\beta\) represents the slope of the regression line, which indicates the relationship's strength and direction. A slope of zero would signify no linear relationship between \(x\) and \(y\).

By performing a hypothesis test, we calculate a p-value. If this p-value is less than our chosen significance level (commonly \(\alpha = 0.05\)), we reject the null hypothesis, suggesting that there is indeed a significant linear relationship. This means that changes in \(x\) are associated with changes in \(y\), confirming model utility.

Confidence Interval

Confidence intervals provide a range of values within which we expect the true value of a parameter to fall with a certain level of confidence, usually 95%.

In the context of simple linear regression, after estimating the yield \(\hat{y}\) using the regression equation, we can construct a confidence interval around this estimate. This interval gives us a sense of the precision of our prediction.

We use the formula:
\[CI: \hat{y} \pm t_{\alpha/2} \times SE\]
Here, \(\hat{y}\) is the estimated yield from the regression equation, \(t_{\alpha/2}\) is the critical value from the t-distribution which depends on our confidence level, and \(SE\) is the standard error of the prediction.

This approach helps in understanding how much our estimate might vary due to randomness in the data. A narrower confidence interval indicates a more precise estimate, while a wider interval suggests more uncertainty. Creating confidence intervals can thus guide decision-making based on the estimation of yields at different upslope distances.

Least Squares Method

The least squares method is a fundamental technique in simple linear regression. It is used to find the best-fitting line through the data points in a scatter plot by minimizing the sum of the squares of the vertical deviations (errors) of each point from the line.

The goal of this method is to find the slope \(b\) and the intercept \(a\) of the line described by the equation \(\hat{y} = a + bx\).

Here's how it works:

• First, calculate the average of the x-values and y-values.
• Next, using these averages and each individual point, calculate the slope \(b\) using the formula:
  \[b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}\]
• Then, find the intercept \(a\) using:
  \[a = \bar{y} - b\bar{x}\]

By applying the least squares method, we ensure that the line of best fit minimizes the discrepancies between the observed data points and the line itself, enabling more accurate predictions and insights.