Question

A random sample of \(n=347\) students was selected, and each one was asked to complete several questionnaires, from which a Coping Humor Scale value \(x\) and a Depression Scale value \(y\) were determined ("Depression and Sense of Humor" (Psychological Reports [1994]: \(1473-1474\) ). The resulting value of the sample correlation coefficient was \(-.18\). a. The investigators reported that \(P\) -value \(<.05\). Do you agree? b. Is the sign of \(r\) consistent with your intuition? Explain. (Higher scale values correspond to more developed sense of humor and greater extent of depression.) c. Would the simple linear regression model give accurate predictions? Why or why not?

Short answer

a. Yes, a P-value <.05 signifies a statistically significant correlation given the sample size. However, the weak correlation coefficient of -.18 suggests this must be interpreted carefully.\nb. Yes, the negative sign of 'r' indicates an inverse relationship between humor and depression which aligns with intuition, implying people with more humor sense have less depression.\nc. No, a simple linear regression model with such weak correlation isn't likely to give accurate predictions due to possible nonlinearity and multifactorial nature of human behavior.

Step-by-step solution

Understand the context and variables

Based on the exercise, we see that two values - the Coping Humor Scale (x) and the Depression Scale (y) have been determined for 347 students. These represent two key variables we're exploring for a correlation. The sample correlation coefficient is given as -0.18 and we'd need to interpret this value.

Interpret sample correlation coefficient

The sample correlation coefficient, given as -0.18, represents a slight negative correlation between the two variables - humor and depression. Meaning, as the humor scale increases, the depression scale slightly decreases, and vice versa. Generally, a correlation coefficient of -0.18 is considered very weak.

Analyzing the P-value

The P-value is reported to be <0.05. In this context, the P-value is the probability that we would obtain the observed sample correlation coefficient or a more extreme value if the population correlation coefficient were actually zero (if there was no correlation in the population). A P-value of <0.05 would typically be considered statistically significant, meaning there is likely a correlation between the two variables (humor and depression) in the population. However, considering the weak correlation coefficient (-.18), the conclusion of a statistically significant correlation must be taken with some caution.

Interpreting the direction of correlation

The negative sign in correlation coefficient '-.18' signifies an inverse relationship between the two variables, humor scale and depression scale. This observation is intuitive as it corresponds to widely held belief that individuals with a more developed sense of humor tend to have a lower degree of depression.

Applicability of simple linear regression model

The simple linear regression model might not give accurate predictions, especially considering the weak correlation (-.18) between the two variables. Linearity assumption in simple linear regression model might not hold true for such weak correlations. The model won't likely capture all the nuances of these variables considering human behavior is influenced by a myriad of interrelated factors, not merely constrained to humor and depression.

Key concepts

Statistical Significance

Statistical significance is a measure used in hypothesis testing that helps determine whether the relationship observed in a dataset is due to chance or a specific factor. It is a crucial concept for students to understand, as it forms the basis for deciding whether to reject the null hypothesis. In simple terms, if a result is statistically significant, it means that the findings are not likely due to random variation but rather indicate a real effect or association.

In the context of the given exercise, the investigators reported a P-value < 0.05 for the correlation between humor and depression among students. This implies that the negative correlation coefficient of -0.18 is statistically significant, even though it's weak. This means that the observed relation between humor and depression is unlikely to be due to chance. However, given the weak correlation, one should proceed with caution when interpreting the results. Statistical significance does not imply a strong or practically important relationship; it merely suggests that the finding is not random.

Understanding statistical significance can prevent students from drawing incorrect conclusions from data analyses. It distinguishes between effects that are likely to be genuine and those that might just be flukes of the sample taken.

P-value Analysis

P-value analysis is a tool used to quantify the evidence against a null hypothesis. The P-value measures the probability of observing a test statistic as extreme, or more extreme, than the one produced by the sample data, assuming that the null hypothesis is true. A small P-value indicates that the observed data is unlikely under the null hypothesis, and thus provides evidence against it.

In our exercise, the reported P-value is less than 0.05, which is commonly used as a threshold for statistical significance. This number means that there is less than a 5% chance that the sample correlation coefficient of -0.18 could have occurred if there really was no correlation (null hypothesis) in the population. Therefore, the investigators have sufficient evidence to suggest that the negative correlation is statistically significant.

P-value analysis is a cornerstone of hypothesis testing and statistical inference. Students should be aware that the P-value itself does not measure the size or importance of an effect. In fact, a small P-value can be associated with minor effects if the sample size is large, as the P-value is also sensitive to the number of observations.

Simple Linear Regression

Simple linear regression is a technique used to model the relationship between a single independent variable (predictor) and a dependent variable (response) by fitting a linear equation to observed data. The goal is to predict the value of the dependent variable based on the value of the independent variable.

In the exercise we're discussing, applying a simple linear regression model to predict depression based on humor scale values could potentially be challenging. A simple linear regression would assume that there is a direct and consistent relationship between these two variables, which is represented by our correlation coefficient of -0.18. This value implicates a slight negative relationship; as the humor level increases, the symptoms of depression decrease.

However, the weak correlation suggests that humor alone may not be a strong predictor of depression, and there may be other factors at play. Simple linear regression may not produce accurate predictions in this scenario, as it does not capture the complexity of human emotions and the vast number of variables that can influence depression levels. For precise model predictions, additional variables and possibly more complex models would likely be needed.