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Chapter 11: Problem 49

Compute and interpret the correlation coefficient for the following grades of 6 students selected at random: $$ \begin{array}{c|cccccc} \text { Mathematics grade } & 70 & 92 & 80 & 74 & 65 & 83 \\ \hline \text { English grade } & 74 & 84 & 63 & 87 & 78 & 90 \end{array} $$

Short Answer

Expert verified
The correlation coefficient calculated from the sum values as given would yield the strength and direction of relationship between English and Mathematics grades. From here, interpret the calculated value in context of educational scores: a strong positive relationship would mean as Mathematics scores increase, so do English scores; a strong negative relationship would mean that as Mathematics scores increase, English scores decrease; and an absence of relationship would mean the two sets of scores are not related.

Step by step solution

01

Tabulate the Scores

Place the Mathematics and English grades in two columns. Next, for each set of scores, calculate the squared values, and the product of each set of scores. The table will look something as follows:\n\[\begin{array}{c|cccccc} \text { Mathematics grade (X) } & 70 & 92 & 80 & 74 & 65 & 83 \\text { English grade (Y) } & 74 & 84 & 63 & 87 & 78 & 90 \ \text { X^2 } & 4900 & 8464 & 6400 & 5476 & 4225 & 6889 \\text { Y^2 } & 5476 & 7056 & 3969 & 7569 & 6084 & 8100 \\text { XY } & 5180 & 7728 & 5040 & 6438 & 5070 & 7470 \\end{array} \]
02

Calculate the Sums

Calculate the sum of X (Mathematics grade), Y (English grade), X^2, Y^2 and XY.\nFor our table, the sums are as follows: \(\sum X\) = 464, \(\sum Y\) = 476, \(\sum X^2\) = 36354, \(\sum Y^2\) = 38254, \(\sum XY\) = 36926.
03

Input Sums into Correlation Coefficient Formula

Substitute the sums calculated in step 2 into the correlation coefficient formula and solve for r. \[r = \frac{36926 - \frac{(464)(476)}{6}}{\sqrt{[(36354) - \frac{(464)^2}{6}] [( 38254) - \frac{(476)^2}{6}]}}\]
04

Simplify the Equation

Upon substituting, simplify the equation to get a simplified form of the equation.
05

Interpret the Correlation Coefficient

If r is near +1, it means there's a strong positive relationship between Mathematics and English grades. If it is near -1, there's a strong negative relationship. If it is near 0, there's no, or only a weak, relationship. Furthermore, remember the boundary conditions |r| > 0.8 is generally described as a strong correlation, 0.5 < |r| < 0.8 a moderate correlation, and |r| < 0.5 a weak correlation.

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Key Concepts

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

Statistics and the Correlation Coefficient
The correlation coefficient is a key concept in statistics that tells you how two variables are related. It's a measure that indicates the strength and direction of a linear relationship between two variables on a scatterplot. Its value ranges from -1 to +1.
  • A value of +1 implies a perfect positive linear relationship.
  • -1 indicates a perfect negative linear relationship.
  • 0 means that there is no linear relationship.
In the context of our exercise, we're analyzing the relationship between two students' grades in Mathematics and English. We used a formula to calculate this, which often involves statistical concepts like sums of products and squares. These calculations help derive how each variable behaves in relation to the other, giving insight into the strength and nature of their connection.
Interpreting this value is crucial. If you find the correlation coefficient close to +1 or -1, it indicates a strong relationship. Meanwhile, values further away from these extremities suggest a weaker relationship.
Mathematics Education and Understanding Relationships
Understanding the concept of a correlation coefficient is a vital part of mathematics education. It helps students make connections between different mathematical concepts and real-world phenomena. In a classroom setting, learning how to compute and interpret the correlation coefficient enhances critical thinking skills.
For example, when given a dataset like the grades of students, calculating the correlation can help students to see concrete examples of how mathematical theories are applied in everyday life. They can engage with real numbers and make observations about how changes in one variable might affect another.
This not only reinforces their mathematical knowledge but also builds confidence in interpreting data critically. By practicing these calculations, students learn the importance of precision in mathematics, understanding both the procedures and the potential implications of their results.
Data Analysis Using Correlation Coefficients
In data analysis, the correlation coefficient serves as a fundamental tool for interpreting relationships between datasets. It enables analysts to quickly and effectively assess the degree to which changes in one variable might predict changes in another. Data analysis often begins with calculating summary statistics, like means and standard deviations, as well as measures of association such as the correlation coefficient. This foundational analysis sets the stage for deeper insights into complex datasets.
  • By looking at correlation coefficients, analysts can identify potential trends.
  • This helps in constructing predictive models or researching causative factors.
  • It's used in various fields such as economics, sociology, and the natural sciences.
Being adept at calculating and interpreting this coefficient ensures that one can handle real-world data challenges efficiently. It plays a significant role in data-driven decision-making processes, offering clarity and direction in both practical and theoretical pursuits.

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Most popular questions from this chapter

An experiment was designed for the Department of Materials Engineering at Virginia Polytechnic Institute and State University to study hydrogen embrittlement properties based on electrolytic hydrogen pressure measurements. The solution used was \(0.1 \mathrm{~N}\) \(\mathrm{NaOH},\) the material being a certain type of stainless steel. The cathodic charging current density was controlled and varied at four levels. The effective hydrogen pressure was observed as the response. The data follow. $$ \begin{array}{ccc} & \text { Charging Current } & \text { Effective } \\ & \text { Density, } x & \text { Hydrogen } \\ \text { Run } & \left(\mathrm{mA} / \mathrm{cm}^{2}\right) & \text { Pressure, } \boldsymbol{y} \text { (atm) } \\ \hline 1 & 0.5 & 86.1 \\ 2 & 0.5 & 92.1 \\ 3 & 0.5 & 64.7 \\ 4 & 0.5 & 74.7 \\ 5 & 1.5 & 223.6 \\ 6 & 1.5 & 202.1 \\ 7 & 1.5 & 132.9 \\ 8 & 2.5 & 413.5 \\ 9 & 2.5 & 231.5 \\ 10 & 2.5 & 466.7 \\ 11 & 2.5 & 365.3 \\ 12 & 3.5 & 493.7 \\ 13 & 3.5 & 382.3 \\ 14 & 3.5 & 447.2 \\ \mathrm{~L} 5 & 3.5 & 563.8 \end{array} $$ (a) Run a simple linear regression of \(\boldsymbol{y}\) against \(x\). (b) Compute the pure error sum of squares and make a test for lack of fit. (c) Does the information in part (b) indicate a need for a model in \(x\) beyond a first-order regression? Explain.

The grades of a class of 9 students on a midterm report \((x)\) and on the final examination \((y)\) are as follows: $$ \begin{array}{c|ccccccccc} \mathbf{X} & 77 & 50 & 71 & 72 & 81 & 94 & 96 & 99 & 67 \\ \hline \boldsymbol{y} & 82 & 66 & 78 & 34 & 47 & 85 & 99 & 99 & 68 \end{array} $$ (a) Estimate the linear regression line. (b) Estimate the final examination grade of a student who received a grade of 85 on the midterm report.

The following data were collected to determine the relationship between pressure and the corresponding scale reading for the purpose of calibration. $$ \begin{array}{cc} \text { Pressure, } x \text { (lb/sqin.) } & \text { Scale Reading, } y \\ \hline 10 & 13 \\ 10 & 18 \\ \text { to } & 16 \\ 10 & 15 \\ 10 & 20 \\ 50 & 86 \\ 50 & 90 \\ 50 & 88 \\ 50 & 88 \\ 50 & 92 \end{array} $$ (a) Find the equation of the regression line. (b) The purpose of calibration in this application is to estimate pressure from an observed scale reading-Estimate the pressure for a scale reading of 54 using \(\dot{x}=(54-a) / b\)-

A study of the amount of rainfall and the quantity of air pollution removed produced the following data: $$ \begin{array}{cc} \text { Daily Rainfall, } & \text { Particulate Removed, } \\ x(0.01 \mathrm{~cm}) & y\left(\mu \mathrm{g} / \mathrm{m}^{3}\right) \\ \hline 4.3 & 126 \\ 4.5 & 121 \\ 5.9 & 116 \\ 5.6 & 118 \\ 6.1 & 114 \\ 5.2 & 118 \\ 3.8 & 132 \\ 2.5 & 141 \\ 7.5 & 108 \end{array} $$ (a) Find the equation of the regression line to predict the particulate removed from the amount of daily rainfall. (b) Estimate the amount of particulate rernoved when the daily rainfall is \(x=4.8\) units.

The following data represent the chemistry grades for a random sample of 12 freshmen at a certain college along with their scores on an intelligence test administered while they were still seniors in high school: $$ \begin{array}{ccc} & \text { Test } & \text { Chemistry } \\ \text { Student } & \text { Score, } \boldsymbol{x} & \text { Grade, } \boldsymbol{y} \\ \hline \mathbf{1} & 65 & 85 \\ \mathbf{2} & 50 & 74 \\ \mathbf{3} & 55 & 76 \\ 4 & 65 & 90 \\ 5 & 55 & 85 \\ 6 & 70 & 87 \\ 7 & 65 & 94 \\ 8 & 70 & 98 \\ 9 & 55 & 81 \\ 10 & 70 & 91 \\ 11 & 50 & 76 \\ 12 & 55 & 74 \end{array} $$ (a) Compute and interpret the sample correlation coefficient. (b) State necessary assumptions on random variables. (c) Test the hypothesis that \(p=0.5\) against the alternative that \(p>0.5 .\) Use a P-value in the conclusion.
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