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Chapter 5: Problem 13

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

Expert verified
After calculating the given formula, the correlation coefficient (\(r\)) is approximately 0.1673. This value suggests a mild positive correlation between high school GPA and first-year college GPA. It means students with higher GPAs in high school tend to have higher GPAs in their first year of college, but the correlation is not very strong.

Step by step solution

01

Understand the Formula

We are given a formula to find the correlation coefficient \(r\), which shows the linear relationship between two data sets, in this case high school GPA (\(x\)) and first-year college GPA (\(y\)). The formula is \[ r = \frac{ \sum xy - \frac{ (\sum x)(\sum y)}{n}}{\sqrt{\sum x^{2} - \frac{ (\sum x)^{2}}{n}} \sqrt{\sum y^{2} - \frac{ (\sum y)^{2}}{n}}} \]
02

Substitute the given values

Now, we substitute the given values into the formula: \(n=2600\), \(\sum x = 9620\), \(\sum y = 7436\), \(\sum xy = 27,918\), \(\sum x^{2} = 36,168\), \(\sum y^{2} = 23,145\). The substituted formula becomes: \[ r = \frac{27,918 - \frac{(9620)(7436)}{2600}}{\sqrt{36,168 - \frac{(9620)^{2}}{2600}} \sqrt{23,145 - \frac{(7436)^{2}}{2600}}} \]
03

Evaluate the Fractions

First we calculate the numerator and the two denominators separately. We take the sum of products of x and y and subtract the product of the sum of x and sum of y divided by total number of elements from it. Similarly, we calculate the two denominators separately. After the calculations, we get: \[ r = \frac{32.463}{14.8405*13.0776} \]
04

Calculate the Correlation Coefficient

We divide the numerator by the product of the two denominators to find r. \[ r = \frac{32.463}{193.866} \]
05

Interpret the result

After calculation, the final value of r falls between -1 and 1. Value close to 1 indicates positive strong correlation, a value close to -1 indicates a strong negative correlation, and value close to 0 indicates no correlation. Whatever the final value of r obtained, it gives information about the correlation between x and y.

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Key Concepts

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

High School GPA
In the academic journey, a student's High School GPA serves as a cumulative measure of their academic performance over the years spent in high school. GPA, which stands for Grade Point Average, is calculated using grades received in courses, weighted by the number of credits each course carries. The GPA is often on a 4.0 scale, where an 'A' equals 4.0, 'B' equals 3.0, and so on.

High School GPA is critical because it is a key consideration for university admissions officers when evaluating prospective students. It reflects a student's dedication, work ethic, and ability to handle academic workload. A high GPA can open doors to scholarships and acceptance letters from preferred colleges. However, it is only one part of the application process; colleges also consider extracurricular activities, personal statements, and recommendation letters.

In the context of the correlation coefficient exercise, high school GPA ( ox) is one of the variables used to calculate the relationship strength with first-year college GPA. Knowing the high school GPA provides a foundational understanding of a student's past academic achievements and is essential for assessing their future academic performance.
College GPA
As students transition to college, their College GPA becomes an important indicator of their academic performance. The College GPA functions similarly to a high school GPA but can represent a broader, more challenging curriculum. Students encounter a diverse range of subjects and academic expectations, potentially affecting their GPA differently than in high school.

A strong College GPA can lead to academic honors, involvement in research opportunities, and improved job prospects post-graduation. It requires consistent effort and time management. The college experience is marked by independence, making self-discipline critical in maintaining a healthy GPA.

In our exercise, the College GPA is denoted by oy and represents another data set for evaluating the correlation with high school GPA. Recognizing these two variables helps understand how high school academic performance can predict first-year college success. A positive correlation between high school and college GPA suggests that skills and habits developed in high school are positively carried over to college studies.
Linear Relationship
A linear relationship in statistics describes how two variables, say X and Y, consistently change in relation to each other. In a perfect linear relationship, an increase in one variable results in a consistent increase or decrease in the other. This connection can be graphically represented with a straight line, indicating constant proportionality. The strength and direction of this relationship are often quantified using the correlation coefficient, denoted by r.

In our case, the exercise investigates the linear relationship between high school GPA and first-year college GPA through this correlation coefficient. The value of r, calculated using the formula, can range from -1 to 1:

  • A value close to 1 indicates a strong positive linear relationship, where high school academic success is mirrored in college.
  • A value close to -1 suggests a strong negative linear relationship, which is unusual in our context as better past academics would usually mean better future performance.
  • A value near 0 implies no linear relationship, meaning high school performance does not predict college performance.

Understanding this linear relationship is crucial for educators and students aiming to improve academic outcomes. By identifying the strength of this correlation, stakeholders can devise strategies to support students transitioning from high school to college.

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Most popular questions from this chapter

Cost-to-charge ratio (the percentage of the amount billed that represents the actual cost) for inpatient and outpatient services at 11 Oregon hospitals is shown in the following table (Oregon Department of Health Services, 2002). A scatterplot of the data is also shown. $$ \begin{array}{ccc} \hline \text { Hospital } & \begin{array}{l} \text { Outpatient } \\ \text { Care } \end{array} & \begin{array}{l} \text { Inpatient } \\ \text { Care } \end{array} \\ \hline 1 & 62 & 80 \\ 2 & 66 & 76 \\ 3 & 63 & 75 \\ 4 & 51 & 62 \\ 5 & 75 & 100 \\ 6 & 65 & 88 \\ 7 & 56 & 64 \\ 8 & 45 & 50 \\ 9 & 48 & 54 \\ 10 & 71 & 83 \\ 11 & 54 & 100 \\ \hline \end{array} $$ The least-squares regression line with \(y=\) inpatient costto-charge ratio and \(x=\) outpatient cost-to-charge ratio is \(\hat{y}=-1.1+1.29 x\). a. Is the observation for Hospital 11 an influential observation? Justify your answer. b. Is the observation for Hospital 11 an outlier? Explain. c. Is the observation for Hospital 5 an influential observation? Justify your answer. d. Is the observation for Hospital 5 an outlier? Explain.

Cost-to-charge ratios (the percentage of the amount billed that represents the actual cost) for 11 Oregon hospitals of similar size were reported separately for inpatient and outpatient services. The data are $$ \begin{array}{lcc} \text { Hospital } & \text { Inpatient } & \text { Outpatient } \\ \hline \text { Blue Mountain } & 80 & 62 \\ \text { Curry General } & 76 & 66 \\ \text { Good Shepherd } & 75 & 63 \\ \text { Grande Ronde } & 62 & 51 \\ \text { Harney District } & 100 & 54 \\ \text { Lake District } & 100 & 75 \\ \text { Pioneer } & 88 & 65 \\ \text { St. Anthony } & 64 & 56 \\ \text { St. Elizabeth } & 50 & 45 \\ \text { Tillamook } & 54 & 48 \\ \text { Wallowa Memorial } & 83 & 71 \\ \hline \end{array} $$ a. Does there appear to be a strong linear relationship between the cost-to- charge ratio for inpatient and outpatient services? Justify your answer based on the value of the correlation coefficient and examination of a scatterplot of the data. b. Are any unusual features of the data evident in the scatterplot? c. Suppose that the observation for Harney District was removed from the data set. Would the correlation coefficient for the new data set be greater than or less than the one computed in Part (a)? Explain.

In the study of textiles and fabrics, the strength of a fabric is a very important consideration. Suppose that a significant number of swatches of a certain fabric are subjected to different "loads" or forces applied to the fabric. The data from such an experiment might look as follows: $$ \begin{aligned} &\text { Hypothetical Data on Fabric Strength }\\\ &\begin{array}{lccccccc} \hline \begin{array}{l} \text { Load } \\ \text { (lb/sq in.) } \end{array} & \mathbf{5} & \mathbf{1 5} & \mathbf{3 5} & \mathbf{5 0} & \mathbf{7 0} & \mathbf{8 0} & \mathbf{9 0} \\ \hline \begin{array}{l} \text { Proportion } \\ \text { failing } \end{array} & 0.02 & 0.04 & 0.20 & 0.23 & 0.32 & 0.34 & 0.43 \\ \hline \end{array} \end{aligned} $$ a. Make a scatterplot of the proportion failing versus the load on the fabric. b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the loads and fit the line \(y^{\prime}=a+b\) (Load). What is the significance of a positive slope for this line? c. What proportion of the time would you estimate this fabric would fail if a load of \(60 \mathrm{lb} / \mathrm{sq}\) in. were applied? d. In order to avoid a "wardrobe malfunction," one would like to use fabric that has less than a \(5 \%\) chance of failing. Suppose that this fabric is our choice for a new shirt. To have less than a \(5 \%\) chance of failing, what would you estimate to be the maximum "safe" load in \(\mathrm{lb} / \mathrm{sq}\) in.?

Both \(r^{2}\) and \(s_{e}\) are used to assess the fit of a line. a. Is it possible that both \(r^{2}\) and \(s_{e}\) could be large for a bivariate data set? Explain. (A picture might be helpful.) b. Is it possible that a bivariate data set could yield values of \(r^{2}\) and \(s_{e}\) that are both small? Explain. (Again, a picture might be helpful.) c. Explain why it is desirable to have \(r^{2}\) large and \(s_{e}\) small if the relationship between two variables \(x\) and \(y\) is to be described using a straight line.

The relationship between hospital patient-to-nurse ratio and various characteristics of job satisfaction and patient care has been the focus of a number of research studies. Suppose \(x=\) patient-to-nurse ratio is the predictor variable. For each of the following potential dependent variables, indicate whether you expect the slope of the least-squares line to be positive or negative and give a brief explanation for your choice. a. \(y=\) a measure of nurse's job satisfaction (higher values indicate higher satisfaction) b. \(y=\) a measure of patient satisfaction with hospital care (higher values indicate higher satisfaction) c. \(y=\) a measure of patient quality of care.
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