Question

Refer to Exercise 12.S.2. Consider the null hypothesis that sulfur dioxide concentration has no effect on yield. (a) Assuming that the linear model holds, formulate this as a hypothesis about the true regression line. (b) Write a directional alternative, in symbols, that says increasing sulfur dioxide tends to decrease yield. (c) How many degrees of freedom are there for the test that compares the hypotheses in (a) and (b)? (d) The sample slope, \(b_{1}\), is -2.262 and the standard error of the slope is 0.4421 . Compute the value of the test statistic. (e) The \(P\) -value for the test is \(0.0002 .\) If \(\alpha=0.05,\) state your conclusion regarding the null hypothesis in the context of this setting.

Short answer

The null hypothesis \(H_0: \beta_1 = 0\) is rejected since the test statistic is \(t = -5.116\) with a \(P\)-value of \(0.0002\), which is less than the significance level of \(0.05\), indicating a significant effect of sulfur dioxide on yield.

Step-by-step solution

Formulate the null hypothesis

The null hypothesis (\(H_0\)) states that the slope of the true regression line is zero, implying that sulfur dioxide concentration has no effect on yield. Symbolically, this is written as \(H_0: \beta_1 = 0\), where \(\beta_1\) represents the slope of the regression line relating sulfur dioxide concentration to yield.

Write the alternative hypothesis

The alternative hypothesis (\(H_a\) or \(H_1\)) should indicate that an increase in sulfur dioxide concentration tends to decrease yield, implying a negative relationship. This is expressed as \(H_a: \beta_1 < 0\).

Determine degrees of freedom

The degrees of freedom for a test of slope in a simple linear regression is given by \(n - 2\), where \(n\) is the total number of observations. This is because we estimate two parameters: the slope and the intercept.

Compute the test statistic

To find the test statistic, we use the formula \(t = \frac{b_1}{SE(b_1)}\), where \(b_1\) is the sample slope and \(SE(b_1)\) is the standard error of the slope. Substituting the given values, we get \(t = \frac{-2.262}{0.4421}\).

Calculate the value of test statistic

By plugging the values into the formula, we calculate the test statistic as \(t = -5.116\), which is the value that will be compared against a t-distribution with \(n - 2\) degrees of freedom to evaluate the significance of the slope.

Make a conclusion based on the P-value

Since the given \(P\)-value is \(0.0002\), which is less than the significance level \(\alpha = 0.05\), we reject the null hypothesis. This suggests that there is statistically significant evidence to conclude that increasing sulfur dioxide concentration tends to decrease yield.

Key concepts

Null Hypothesis

The null hypothesis in a statistical analysis represents a statement of no effect or no relationship between the variables studied. In the context of regression, it asserts that there is no association between the independent and dependent variables. Specifically, in linear regression analysis, the null hypothesis posits that the slope coefficient (\beta_1) is equal to zero (\beta_1 = 0). This means that changes in the independent variable (such as sulfur dioxide concentration) do not lead to any change in the dependent variable (yield). Establishing the null hypothesis is a crucial step because it provides a starting point for statistical testing and is the assumption that will be tested against the data collected.

In the given exercise, students are tasked with the null hypothesis that sulfur dioxide concentration has no effect on yield, which is formally stated as:
\[\begin{equation} H_0: \beta_1 = 0 \end{equation}\]. Clearly understanding the formulation of the null hypothesis is essential as it guides the type of statistical test to be performed and helps interpret the results thereof.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis suggests that there is indeed an effect or a non-zero relationship between the variables in question. It is what the researcher aims to support or prove. The alternative hypothesis is symbolized as (\beta_1 eq 0) for a two-tailed test or (\beta_1 < 0) or (\beta_1 > 0) for a one-tailed test, depending on the direction of the relationship hypothesized. It's important for students to carefully consider the expected direction of effect when formulating the alternative hypothesis.

For the exercise mentioned, students are instructed to create a one-tailed directional alternative hypothesis that posits an inverse relationship between sulfur dioxide concentration and yield:
\[\begin{equation} H_a: \beta_1 < 0 \end{equation}\]. By stating that the slope is less than zero, they are expressing that as sulfur dioxide levels increase, the yield is expected to decrease, which is essential for guiding the direction of the statistical testing.

Degrees of Freedom

The concept of degrees of freedom is important in hypothesis testing because it defines the number of independent values that can vary in an analysis without breaking any constraints. In regression analysis, degrees of freedom are associated with the number of observations and the number of parameters being estimated.

In the context of the exercise, degrees of freedom for the test would be calculated as the total number of observations minus the number of estimated parameters (which, in simple linear regression, are the slope and intercept). The formula is given by:
\[\begin{equation} df = n - 2 \end{equation}\] where 'n' is the total number of observations. Understanding degrees of freedom is crucial for the proper interpretation of statistical tests, as it impacts the distribution used to determine the significance of the test statistic.

Test Statistic

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It's used to decide whether to support or reject the null hypothesis. In a regression analysis, the test statistic for evaluating the significance of the slope is typically a t-statistic calculated by dividing the estimated slope (b_1) by its standard error (SE(b_1)). The formula is as follows:
\[\begin{equation} t = \frac{b_1}{SE(b_1)} \end{equation}\].

This value is then compared to a theoretical t-distribution to determine statistical significance. The calculation of the test statistic is a critical step in hypothesis testing as it translates the sample data into a form that can be assessed using probability distributions.

P-value

The P-value is a crucial concept in hypothesis testing. It is the probability of obtaining the observed sample results, or more extreme results, under the assumption that the null hypothesis is true. A lower P-value indicates that the sample findings are unusual given the null hypothesis and hence provides evidence against the null hypothesis. In the exercise, the P-value of 0.0002 suggests strong evidence against the null hypothesis, especially when it is less than the commonly used significance level of \(\alpha = 0.05\), leading to its rejection. Teaching students about the implication of P-values in hypothesis testing helps them make informed decisions regarding the acceptance or rejection of hypotheses.

Linear Regression Analysis

Linear regression analysis is a powerful statistical tool used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data. This technique estimates the coefficients of the linear equation, involving one or more independent variables that best predict the outcome of the dependent variable.

In the exercise, the linear regression analysis is used to investigate the impact of sulfur dioxide on crop yield. By calculating the slope of the regression line, we can understand the rate at which yield changes for a unit change in sulfur dioxide concentration. The significance of this slope coefficient is then tested using the concepts of the null and alternative hypothesis, degree of freedom, test statistic, and P-value to determine if there is enough evidence to suggest a significant relationship between the variables in question.