Question

The article "Impacts of On-Campus and OffCampus Work on First-Year Cognitive Outcomes" (Journal of College Student Development [1994]: \(364-\) 370) reported on a study in which \(y=\) spring math comprehension score was regressed against \(x_{1}=\) previous fall test score, \(x_{2}=\) previous fall academic motivation, \(x_{3}=\) age, \(x_{4}=\) number of credit hours, \(x_{5}=\) residence \(\left(1\right.\) if on campus, 0 otherwise), \(x_{6}=\) hours worked on campus, and \(x_{7}=\) hours worked off campus. The sample size was \(n=210\), and \(R^{2}=.543\). Test to see whether there is a useful linear relationship between \(y\) and at least one of the predictors.

Short answer

Based on the calculated F statistic and corresponding P-value, if the P-value is less than 0.05, we can conclude that there is a useful linear relationship between y (spring math comprehension score) and at least one of the predictors. If not, we fail to reject the null hypothesis.

Step-by-step solution

Formulate the Hypotheses

The null hypothesis (\(H_{0}\)) is that none of the predictors is linearly related to the dependent variable. The alternative hypothesis (\(H_{a}\)) is that at least one predictor is linearly related to the dependent variable. Mathematically, \(H_{0}: β_{1} = β_{2} =···= β_{k} = 0\) and \(H_{a}\): at least one β is not 0.

Set the Significance Level

Assuming that a common significance level of 0.05 is used, if the p-value of the F statistic is less than this level, we would reject the null hypothesis.

Compute the F Statistic and Corresponding P-Value

The F statistic for testing the overall significance is given by the formula \(F = \frac{(R^2/k)}{(1-R^2)/(n-k-1)}\) where R^2 is the coefficient of determination, n is the sample size, and k is the number of predictors. In this case, \(R^{2} = .543\), \(n=210\), and \(k=7\) (as there are seven predictors). The F statistic can be computed using these values and the p-value can be determined using a F Distribution Table.

Conclusion

After calculating the F statistic and the P-value, if the P-value is less than the significance level (0.05), then we would reject the null hypothesis and conclude that there is a useful linear relationship between y (spring math comprehension score) and at least one of the predictors.

Key concepts

Hypothesis Testing

In the context of linear regression, hypothesis testing helps us determine whether there is a significant relationship between the predictor variables and the outcome variable. It usually involves two hypotheses:

• The null hypothesis ( H_0): suggests that there is no linear relationship between the predictor variables and the dependent variable. Essentially, it assumes all the coefficients are equal to zero.
• The alternative hypothesis ( H_a): indicates that at least one predictor variable is significantly related to the outcome, meaning at least one of the coefficients is different from zero.

When we perform hypothesis testing, our ultimate goal is to evaluate whether we should reject the null hypothesis. This decision depends largely on the p-value derived from statistical tests, which measure the strength of evidence against the null hypothesis. A common benchmark is a significance level of 0.05.

Predictor Variables

Predictor variables, also known as independent variables, are inputs or factors that we use to predict an outcome in regression analysis. In our study, the predictor variables include previous fall test score, previous fall academic motivation, age, number of credit hours, residence status, hours worked on campus, and hours worked off campus.
Each of these variables may have different levels of influence on the dependent variable, which in this case is the spring math comprehension score.

• Previous fall test score and academic motivation: These are likely direct indicators of a student's current academic performance.
• Age and number of credit hours: These variables could reflect a student's maturity and workload, potentially affecting their success.
• Residence status: Living on-campus may provide more study resources, influencing comprehension scores.
• Hours worked: Time spent working might detract from study time, impacting test scores negatively.

F Statistic

The F statistic is a critical value derived from an ANOVA (Analysis of Variance) test in linear regression. It helps to determine the overall significance of the model by testing if at least one predictor has a non-zero coefficient. The formula to calculate the F statistic is given by: \[F = \frac{(R^2/k)}{(1-R^2)/(n-k-1)}\]Here:

• \(R^2\) is the coefficient of determination, showing how well data points fit the statistical model.
• \(k\) is the number of predictors.
• \(n\) is the sample size.

A higher F value typically points to a more significant relationship between predictors and the dependent variable. However, we need to check the corresponding p-value to make any conclusions.

P-Value

The p-value allows us to quantify the probability of observing the data, or something more extreme, if the null hypothesis is true. It helps us determine the strength of the evidence against the null hypothesis in the context of hypothesis testing. A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, leading us to reject it. In practice, if the p-value derived from the F statistic in our regression model is less than 0.05, we would conclude that the relationship between the dependent variable and at least one predictor is statistically significant. However, if the p-value is greater, the evidence is not strong enough to reject the null hypothesis, implying no significant linear relationship.

Coefficient of Determination

The coefficient of determination, denoted as \(R^2\), represents the proportion of variance in the dependent variable that is predictable from the independent variables. An \(R^2\) value ranges from 0 to 1, where a value closer to 1 indicates a better fit of the model to the data.In our example, with an \(R^2\) of 0.543, it suggests that 54.3% of the variation in spring math comprehension scores can be explained by the predictor variables. Such a result implies a moderately strong relationship between the data points and the model. Remember, a high \(R^2\) does not imply causation or the utility of each individual predictor—hypothesis testing and analysis of individual coefficients complement this understanding.