91Ó°ÊÓ

A scatter plot is a graphical representation that shows the relationship between two numerical variables. In this exercise, the scatter plot portrays the association between the upslope distance \(x\) (in meters) and the grain sorghum yield \(y\) (in grams per meter) on a sloping watershed. You plot each pair of \(x\) and \(y\) as a point on a graph where the x-axis represents the distance upslope, and the y-axis depicts the yield.

Drawing a scatter plot helps visually identify if there’s a trend or pattern. You can begin by locating each point based on the given data. For example, for \(x = 0\) and \(y = 500\), you place a point at the intersection of these values on the graph. Continue this for all data pairs.

In this case, as you dissect the scatter plot, observe the overall direction of the points. You are looking for either a positive or negative trend. If the points loosely form a line that either slopes upward or downward, a simple linear regression model might be plausible, indicating a possible linear relationship between the variables.

The regression model utility test helps determine if the linear relationship observed in the scatter plot is statistically significant. This test will scrutinize if the slope \(b\) of the linear regression line is significantly different from zero. A slope of zero implies no linear relationship.

The procedure begins with formulating two hypotheses:

• The null hypothesis \(H_0: b = 0\) asserts that there's no linear relationship, meaning the slope is zero.
• The alternative hypothesis \(H_a: b eq 0\) suggests that a linear relationship exists, meaning the slope is not zero.

You then calculate the test statistic \(t\), which involves the obtained slope \(b\) and its standard error \(SE_b\). Using the formula:

\[ t = \frac{b}{SE_b} \]

You compute this statistic to check how many standard deviations the slope is away from zero.

Next, compare the \(t\) value against the critical value from the t-distribution, which depends on the chosen significance level (e.g., \(\alpha = 0.05\)) and degrees of freedom \(n-2\). This comparison will guide your decision on whether to reject the null hypothesis. If you reject \(H_0\), the model is deemed statistically significant and useful.

Confidence interval estimation provides a range in which we believe a certain parameter, like the average yield at a given distance upslope, is likely to fall. When using a linear regression model, the confidence interval helps gauge the reliability of the estimated value for a specific input value, such as when x = 75 meters.

To calculate this interval, first use the regression equation to obtain the estimated yield \(\hat{y}\) at \(x = 75\). Substitute the value into the equation:

For example:

• Regression Equation: \(\hat{y} = a + b(75)\)

Next, calculate the margin of error using a critical \(t\)-value for your chosen confidence level (e.g., 95%) and the standard error of the prediction \(SE_{\hat{y}}\):

\[ \hat{y} \pm t^* \cdot SE_{\hat{y}} \]

The output gives you an interval of plausible values predicting where the true average yield for \(x = 75\) meters might lie. Essentially, this method allows you to assert with reasonable confidence that the true mean yield falls within this range.