Question

According to the article "First-Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students" \((\) Journal of College Student Development \([1999]: 599-\) 605), there is a mild correlation between high school GPA \((x)\) and first-year college GPA \((y)\). The data can be summarized as follows: $$ \begin{array}{clc} n=2600 & \sum x=9620 & \sum y=7436 \\ \sum x y=27,918 & \sum x^{2}=36,168 & \sum y^{2}=23,145 \end{array} $$ An alternative formula for computing the correlation coefficient that is based on raw data and is algebraically equivalent to the one given in the text is $$ r=\frac{\sum x y-\frac{\left(\sum x\right)\left(\sum y\right)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to compute the value of the correlation coefficient, and interpret this value.

Short answer

The correlation coefficient value lies between -1 and 1 and its magnitude indicates the strength of the correlation, either positive or negative, between a student's high school and first-year college GPA. Given its magnitude and direction, it can be concluded about the impact of high school GPA on first-year college GPA. The step-by-step instructions provided above provide a detailed guide to calculate and interpret the correlation coefficient accurately.

Step-by-step solution

Identify Information in the Problem

Identify the important information from the problem: Total number of students, n = 2600, sum of high school GPA, \(\sum x = 9620\), sum of first-year college GPA, \(\sum y = 7436\), sum of product of high school GPA and first-year college GPA, \(\sum xy = 27918\), sum of squares of high school GPA, \(\sum x^2 = 36168\) and sum of squares of first-year college GPA, \(\sum y^2 = 23145\). This data will be substituted into the given formula for the correlation coefficient.

Substitute the Identified Values into the Formula

Substitute the identified values into the formula for the correlation coefficient. In the denominator of the formula, note the subtractive nature. This means that the subtractive term must be calculated first before the square root operation is applied to them.

Perform the Calculations

After values substitution, perform the calculations accurately. Note that arithmetic operations must be performed in the order specified by the BODMAS rule (brackets, orders, division and multiplication, addition and subtraction). Using a square root function on the denominators, one needs to watch out for negative results.

Interpret the Correlation Coefficient

After calculating the correlation coefficient, interpret its value. If it's close to 1 or -1, there's a strong correlation, either positive or negative respectively. If it's close to 0, there's less or no correlation. The signs, positive or negative, indicate the direction of the relationship between variables.

Key concepts

Understanding High School GPA

High school Grade Point Average (GPA) is a measure of a student's academic achievement at a high school level, typically on a scale of 0 to 4.0. It represents the average value of the accumulated final grades earned in courses over time. More than just a number, a high school GPA can indicate a student's work ethic, capabilities, and future potential.

• Calculation: The GPA is calculated by taking the number of grade points a student earned in a given period of time and dividing that amount by the total number of credits taken.
• Significance: Universities often look at high school GPA when considering admission because it can reflect a student's ability to perform academically in a rigorous environment.
• Correlation with College GPA: The correlation between high school GPA and first-year college GPA is frequently studied to understand how academic performance evolves as students transition from high school to college.

It is this transition and its predictability that the given exercise explores through statistical analysis.

Transition to First-year College GPA

First-year college GPA continues from the foundation set by high school GPA as an indicator of a student's ability to succeed in higher education. It plays a crucial role in determining a student's college trajectory, affecting academic standing, financial aid, and even future employment opportunities.

• Adaptation: The shift from high school to college can be considerable, with changes in academic expectations, autonomy, and learning environment.
• Impact: A strong correlation between high school GPA and first-year college GPA may suggest that the skills and behaviors effective in high school continue to be relevant in college.
• Usage: Colleges may use the correlation data to identify students who might need additional support to achieve their academic goals.

In the context of the exercise, the relationship between these two GPAs offers a glimpse into the consistency of academic performance across different educational systems.

The Role of Statistical Analysis

Statistical analysis is the process of collecting, examining, interpreting, and presenting data to discover underlying patterns and trends. In education, it is used to make informed decisions based on empirical information.

• Understanding Correlation: Through statistical analysis, educators and administrators can understand the relationship between different variables, such as high school GPA and first-year college GPA.
• Decision Making: The insights gained guide policy development, academic support programs, and tailored interventions for students at risk.
• Research: Educational research relies on statistical analysis to validate theories and practices in real-world settings.

The correlation coefficient derived in the exercise is a product of statistical analysis, highlighting its importance in interpreting academic data effectively.

Applying the BODMAS Rule

The BODMAS rule is an acronym representing the correct sequence of operations in mathematics to solve equations accurately: Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

• Correct Order: Following BODMAS ensures that arithmetic calculations are performed in the intended order, preventing errors in complex equations.
• Common Missteps: A typical mistake in mathematics is to perform calculations out of this order, leading to incorrect results, especially relevant in statistical formulas like the correlation coefficient.
• Practical Use: In our exercise, applying BODMAS is essential to accurately calculate correlation, especially when working through the formula's denominator.

Understanding and adhering to the BODMAS rule is pivotal in many mathematical operations, including those required to solve statistical problems such as determining the correlation coefficient between two sets of data.