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Chapter 5: Problem 69

The following data on \(x=\) score on a measure of test anxiety and \(y=\) exam score for a sample of \(n=9\) students are consistent with summary quantities given in the paper "Effects of Humor on Test Anxiety and Performance"' (Psychological Reports [1999]: \(1203-1212\) ): $$ \begin{array}{llllrlllll} x & 23 & 14 & 14 & 0 & 17 & 20 & 20 & 15 & 21 \\ y & 43 & 59 & 48 & 77 & 50 & 52 & 46 & 51 & 51 \end{array} $$ Higher values for \(x\) indicate higher levels of anxiety. a. Construct a scatterplot, and comment on the features of the plot. b. Does there appear to be a linear relationship between the two variables? How would you characterize the relationship? c. Compute the value of the correlation coefficient. Is the value of \(r\) consistent with your answer to Part (b)? d. Is it reasonable to conclude that test anxiety caused poor exam performance? Explain.

Short Answer

Expert verified
a. The scatterplot will show the distribution of scores and anxiety levels. b. If the data points show a downward slope, it might suggest a negative linear relationship. c. The correlation coefficient can be calculated using the given formula, this will offer a numerical measure of the direction and strength of the potential relation. d. Even though correlation might suggest a relation, causality cannot be established based solely on this data.

Step by step solution

01

Scatterplot and Relationship

a. Plot the data on a graph, with test anxiety levels on the x-axis and the corresponding exam scores on the y-axis. The resulting scatterplot will offer visual insight on the relationship between test anxiety and exam scores. The direction, strength, and closeness of the data points can help determine the nature of this relationship. b. If the data points are closely concentrated in a downward slant, there may be a negative linear relationship. If this is the case, a student experiencing higher test anxiety generally scores lower on their exam and vice versa.
02

Correlation Coefficient

c. To quantify this perceived relation, the correlation coefficient, denoted by \(r\), is calculated. It's a numerical measure ranging from -1 to +1. A negative value indicates a negative linear relation and positive value indicates a positive relation. A value near -1 indicates a strong negative linear relation, a value near +1 indicates a strong positive relation, and a value near 0 suggests there is no linear relation at all. To find \(r\), use the scores provided and the formula \(r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2] [n\Sigma y^2 - (\Sigma y)^2]}}\) where \(危xy\) is the sum of the product of corresponding \(x\) and \(y\) values, \(危x\), \(危y\), \(危x^2\), and \(危y^2\) are the sums of \(x\), \(y\), \(x^2\), and \(y^2\) values respectively, and \(n\) is the number of data points.
03

Causation

d. Based on the information you have, it is not possible to conclusively state that test anxiety causes poor exam performance. Although the scatterplot and correlation coefficient could suggest a potential relationship between the two variables, this is not sufficient to draw conclusions about cause and effect since correlation does not imply causation.

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Key Concepts

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

Scatterplot Analysis
When examining the relationship between test anxiety and exam performance, a scatterplot provides a visual representation that makes interpreting data more manageable. Imagine plotting dots on a graph where one axis represents a student's level of test anxiety and the other represents their exam score. As you place each pair of scores on the graph, a pattern begins to emerge.

These plotted points can show trends, such as a cluster of students with low anxiety and high exam scores or vice versa. If you notice that as the test anxiety score increases (moving rightwards on the scatterplot), the exam scores tend to decrease (moving downwards), this suggests a negative relationship. In simple terms, more anxiety corresponds to lower scores. However, a scatterplot doesn't tell us everything; it's a snapshot that prompts further analysis. It won't reveal, for example, whether test anxiety causes poor performance, only that the two may be connected.
Correlation Coefficient
Diving deeper into the numbers requires calculating the correlation coefficient. This single number, denoted as \(r\), quantifies the direction and strength of a linear relationship between two variables. Values of \(r\) range from -1 to +1, with extremes indicating stronger relationships. A negative \(r\) suggests a negative association 鈥� in our case, that higher test anxiety might be linked with lower exam scores.

Calculating \(r\) involves a formula that uses sums and squares of your data. If \(r\) is close to -1, it suggests a strong negative relationship, supporting the trend you might've spotted on your scatterplot. Still, even a strong correlation does not mean one variable causes the other to change. It simply implies a consistent and predictable relationship.
Causation vs Correlation
It's crucial to note the difference between correlation and causation. Just because two variables move in sync doesn't mean one causes the other to change. For instance, while our data might show that higher test anxiety often goes hand in hand with lower exam performance, there could be other factors at play 鈥� perhaps study habits, understanding of the material, or external stressors.

Causation implies a direct relationship where one effect is due to another. To prove causation, controlled experiments and further statistical tests are necessary. In the case of test anxiety and exam performance, other research methods would be needed to support a claim of causation beyond the correlation observed in the data.

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Most popular questions from this chapter

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