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Chapter 5: Problem 12

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): $$ \begin{array}{llllllllll} x & 23 & 14 & 14 & 0 & 17 & 20 & 20 & 15 & 21 \\ y & 43 & 59 & 48 & 77 & 50 & 52 & 46 & 51 & 51 \end{array} $$ 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
First, by plotting the scatterplot, we can visually inspect the data. Seeing a mostly linear downward trend indicates a negative correlation, suggesting that as test anxiety increases, exam scores decrease. The calculation of the correlation coefficient will confirm this visually observed relation with a numerical value. However, while there might be a correlation, it is not categorically apparent that higher test anxiety is causing lower exam performance. Other factors, not included in this data, might also contribute to the observed trend.

Step by step solution

01

Plotting the Scatterplot

To plot a scatterplot, each pair (x, y) of data should be plotted as a single point in a two-dimensional space, where x-axis represents the test anxiety score, and the y-axis represents the exam score. Observations could then be made about the spread and direction of these points.
02

Determining the Linear Relationship

By visually checking the scatterplot, it is possible to identify if a linear relationship is present. A linear relationship would look like a straight line, either increasing or decreasing, going through majority of the points. Interpret the pattern from the scatterplot.
03

Compute the Correlation Coefficient

The correlation coefficient, \(r\), is a numerical measure of the strength and direction (positive or negative) of a linear relationship. It can be calculated with a specific formula: \(r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}\). Here 'n' is the number of pairs of scores, \(\Sigma xy\) is the sum of the products of paired scores, \(\Sigma x\) and \(\Sigma y\) are the sums of x and y scores respectively, \(\Sigma x^2\) and \(\Sigma y^2\) are the sum of squared x and y scores respectively. After substituting the data into the formula, calculate the value of \(r\)
04

Interpret the Correlation Coefficient

Correlation coefficients range from -1 to 1. A value of +1 represents a perfect positive linear relationship, -1 a perfect negative linear relationship, and 0 no linear relationship. Compare the computed value with these parameters to derive its meaning.
05

Discuss Causality

Discuss the possibility of test anxiety causing poor exam performance based on the results obtained from the correlation coefficient and the scatterplot. Keep in mind that correlation does not imply causation. Other possible factors that could be affecting exam performance should be considered too.

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Key Concepts

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

Scatterplot
A scatterplot is a great way to visualize the relationship between two variables. In our scenario, we plot test anxiety scores (x-axis) versus exam scores (y-axis). This visual tool helps us quickly see patterns or trends in the data. Each point on the scatterplot represents a unique data pair, showing how a specific level of test anxiety corresponds to a particular exam score.

Creating a scatterplot involves mapping each (x, y) pair into a two-dimensional graph. Here are some observations you might look for:
  • The overall direction of the data points. Are they trending upwards, downwards, or showing no clear trend?
  • The spread of the data points. Are they clustered closely together or widely dispersed?
  • Any outliers or anomalies that do not fit the general pattern.
By examining these features, we gain insights into the potential relationship between test anxiety and exam performance.
Linear Relationship
A linear relationship in a scatterplot happens when data points form a pattern resembling a straight line. This line can either slope upwards (positive relationship) or downwards (negative relationship).

In our example, once we plot test anxiety against exam performance, we should ask ourselves if a straight line could reasonably fit through most of the points. If it can, a linear relationship likely exists. A line of best fit, or trend line, could then be drawn to highlight this pattern.

Consider these factors:
  • If points generally rise together (higher anxiety, higher exam scores), this suggests a positive linear relationship.
  • If points fall together (higher anxiety, lower exam scores), this indicates a negative linear relationship.
  • If points do not seem to follow any consistent upward or downward pattern, a linear relationship might not be present.
Identifying such patterns is crucial because they hint at how the two variables might be interacting.
Test Anxiety
Test anxiety can heavily influence a student's performance during an exam. It refers to the feelings of stress, worry, or fear experienced before or during testing situations. This psychological phenomenon can manifest both physically and emotionally.

In our specific study:
  • Higher test anxiety scores imply that a student is more anxious.
  • We consider how this anxiety might correlate with their exam results, as some students might underperform due to nervousness.
  • There are various ways students can manage test anxiety, such as practicing relaxation techniques, preparing thoroughly, or seeking counseling.
It's important to note that test anxiety doesn't just affect grades but can impact a student's overall educational experience and well-being.
Exam Performance
Exam performance is often evaluated through the scores or grades achieved in exams. This metric can be influenced by a multitude of factors, including but not limited to, test anxiety.

In this context:
  • A good exam score suggests good understanding and retention of the material, but it can also rely on the student's ability to handle test pressure.
  • Factors impacting performance include the level of difficulty of the exam, how well-prepared a student is, and their physical and mental state during the exam.
  • Educational and non-educational influences like support systems, socio-economic background, and classroom environments can also play significant roles.
Understanding how different elements influence exam scores can help educators develop better teaching methods and offer more effective student support.

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Most popular questions from this chapter

Data on pollution and cost of medical care for elderly people were given in Exercise \(5.17\) and are also shown here. The following data give a measure of pollution (micrograms of particulate matter per cubic meter of air) and the cost of medical care per person over age 65 for six geographic regions of the United States: $$ \begin{array}{lcc} & & \begin{array}{l} \text { Cost of } \\ \text { Medical } \end{array} \\ \text { Region } & \text { Pollution } & \text { Care } \\ \hline \text { North } & 30.0 & 915 \\ \text { Upper South } & 31.8 & 891 \\ \text { Deep South } & 32.1 & 968 \\ \text { West South } & 26.8 & 972 \\ \text { Big Sky } & 30.4 & 952 \\ \text { West } & 40.0 & 899 \\ & & \\ \hline \end{array} $$ The equation of the least-squares regression line for this data set is \(\hat{y}=1082.2-4.691 x\), where \(y=\) medical cost and \(x=\) pollution. a. Compute the six residuals. b. What is the value of the correlation coefficient for this data set? Does the value of \(r\) indicate that the linear relationship between pollution and medical cost is strong, moderate, or weak? Explain. c. Construct a residual plot. Are there any unusual features of the plot? d. The observation for the West, \((40.0,899)\), has an \(x\) value that is far removed from the other \(x\) values in the sample. Is this observation influential in determining the values of the slope and/or intercept of the least-squares line? Justify your answer.

Some straightforward but slightly tedious algebra shows that $$ \text { SSResid }=\left(1-r^{2}\right) \sum(y-\bar{y})^{2} $$ from which it follows that $$ s_{e}=\sqrt{\frac{n-1}{n-2}} \sqrt{1-r^{2}} s_{y} $$ Unless \(n\) is quite small, \((n-1) /(n-2) \approx 1\), so $$ s_{e} \approx \sqrt{1-r^{2}} s_{y} $$ a. For what value of \(r\) is \(s\), as large as \(s_{y} ?\) What is the least- squares line in this case? b. For what values of r will se be much smaller than \(s_{y} ?\) c. A study by the Berkeley Institute of Human Development (see the book Statistics by Freedman et al., listed in the back of the book) reported the following summary data for a sample of n 5 66 California boys: \(r \approx .80\) At age 6 , average height \(\approx 46\) in., standard deviation \(\approx\) \(1.7\) in. At age 18 , average height \(\approx 70\) in., standard deviation \(\approx\) \(2.5\) in. What would \(s_{e}\) be for the least-squares line used to predict 18-year-old height from 6-year-old height? d. Referring to Part (c), suppose that you wanted to predict the past value of 6 -year-old height from knowledge of 18 -year-old height. Find the equation for the appropriate least-squares line. What is the corresponding value of \(s_{e} ?\)

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r\).

The paper "Root Dentine Transparency: Age Determination of Human Teeth Using Computerized Densitometric Analysis" (American Journal of Physical Anthropology [1991]: 25-30) reported on an investigation of methods for age determination based on tooth characteristics. With \(y=\) age (in years) and \(x=\) percentage of root with transparent dentine, a regression analysis for premolars gave \(n=36\), SSResid \(=5987.16\), and SSTo \(=\) 17,409.60. Calculate and interpret the values of \(r^{2}\) and \(s_{e}\)

The accompanying data resulted from an experiment in which weld diameter \(x\) and shear strength \(y\) (in pounds) were determined for five different spot welds on steel. A scatterplot shows a pronounced linear pattern. With \(\Sigma(x-\bar{x})=1000\) and \(\Sigma(x-\bar{x})(y-\bar{y})=8577\), the least-squares line is \(\hat{y}=-936.22+8.577 x\). $$ \begin{array}{llllrr} x & 200.1 & 210.1 & 220.1 & 230.1 & 240.0 \\ y & 813.7 & 785.3 & 960.4 & 1118.0 & 1076.2 \end{array} $$ a. Because \(1 \mathrm{lb}=0.4536 \mathrm{~kg}\), strength observations can be re-expressed in kilograms through multiplication by this conversion factor: new \(y=0.4536(\) old \(y\) ). What is the equation of the least-squares line when \(y\) is expressed in kilograms? b. More generally, suppose that each \(y\) value in a data set consisting of \(n(x, y)\) pairs is multiplied by a conversion factor \(c\) (which changes the units of measurement for \(y\) ). What effect does this have on the slope \(b\) (i.e., how does the new value of \(b\) compare to the value before conversion), on the intercept \(a\), and on the equation of the least-squares line? Verify your conjectures by using the given formulas for \(b\) and \(a\). (Hint: Replace \(y\) with \(c y\), and see what happens - and remember, this conversion will affect \(\bar{y} .\) )
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