Alex Mercer
Department of Mathematics and Statistics, University of Maryland
STAT 200: Introduction to Statistics
Dr. Elizabeth Vance
July 2026
This report evaluates the relationship between weekly self-study hours and final examination scores using a sample of 100 undergraduate students enrolled in an introductory statistics course. Pearson correlation coefficient analysis was performed to determine the strength and direction of the linear association between the two variables. The independent predictor variable, weekly study hours, was correlated with the dependent variable, final statistics exam scores. The calculated Pearson correlation coefficient was r = 0.82, indicating a strong positive linear relationship. The coefficient of determination was computed as r-squared = 0.6724, demonstrating that approximately 67.2% of the variance in final exam scores is shared with weekly study hours. Hypothesis testing yielded a highly significant test statistic of t(98) = 14.18 (p < 0.001), leading to the rejection of the null hypothesis. While the association is strong, causation cannot be directly inferred due to potential confounding factors such as prior mathematical aptitude or student motivation. It is recommended to expand the analysis to multivariate regression modeling to control for these covariates.
Analyzing the linear association between academic effort and performance outcomes represents a common inquiry in education research. Bivariate correlation analysis provides the mathematical framework to measure the degree of linear relationship between two continuous variables without specifying a causal direction. The primary objective of this study is to analyze the correlation between weekly study hours and final statistics exam scores. The concept of measuring linear association was pioneered by Sir Francis Galton in 1888, who introduced the concept of co-relations to study biological traits (Galton, 1888). This concept was later formalized mathematically by Karl Pearson in 1895 as the product-moment correlation coefficient (Pearson, 1895).
To evaluate if a linear relationship exists between study hours and exam scores, a statistical hypothesis test was formulated. The null hypothesis (H₀) states that there is no linear relationship between weekly study hours and final exam scores in the population. The alternative hypothesis (Hₐ) states that a significant linear relationship exists. Mathematically, the hypotheses are stated as:
H0: ρ = 0
Ha: ρ ≠ 0
Where ρ (rho) represents the population Pearson correlation coefficient. The significance level for this analysis was set a priori at alpha = 0.05.
The Pearson correlation coefficient is computed as the ratio of the covariance of the two variables to the product of their standard deviations. Covariance measures how the variables vary together, while standard deviations measure individual variability. The mathematical formula for the sample correlation coefficient (r) is defined as:
r = Cov(X, Y) / (sX * sY) = Σ((Xi - X̄)(Yi - Ȳ)) / √[Σ(Xi - X̄)2 * Σ(Yi - Ȳ)2]
In this equation, Xi and Yi represent individual observations, X̄ and Ȳ are the sample means, and sX and sY are the sample standard deviations. The resulting value of r ranges from -1.0 to +1.0, representing perfect negative and perfect positive linear relationships, respectively, while a value of 0.0 indicates a complete absence of a linear association (Field, 2013).
The dataset analyzed in this study comprises a random sample of N = 100 undergraduate students enrolled in STAT 200 during the Spring 2026 semester. Data collection was performed via an end-of-term survey where students self-reported their average weekly study hours dedicated to statistics coursework outside of scheduled lectures. Final exam scores were retrieved from the course grade book with student consent. This study design utilizes a quantitative survey method, which provides continuous, interval-scale measurements suitable for parametric correlation testing (Field, 2013).
Two continuous variables were defined for this bivariate analysis: the independent variable, Weekly Study Hours (measured as a continuous value representing hours per week), and the dependent variable, Exam Score (measured as a percentage from 0% to 100%). Both variables are continuous and measured on an interval scale, satisfying the first basic requirement of Pearson correlation modeling. According to sampling logs, the median completion time for the survey was 4.2 minutes (SD = 0.8), ensuring high response engagement.
Prior to calculating the Pearson correlation coefficient, several parametric assumptions were verified. First, the assumption of interval scale measurement was met as both study hours and exam scores are continuous scale measurements. Second, the assumption of normality was assessed. Shapiro-Wilk tests were performed on both variables, yielding W = 0.985 (p = 0.312) for study hours and W = 0.988 (p = 0.451) for exam scores, indicating no significant deviation from normal distributions. Third, the assumption of homoscedasticity was verified by plotting the standardized residuals of the relationship, which showed an even distribution of variance across all values. Finally, the assumption of linearity was verified using visual inspections of scatter plots to confirm that the bivariate distribution does not exhibit curvilinear trends (Field, 2013).
Exploratory data analysis was conducted to summarize the univariate distributions of the sample variables. Descriptive statistics, including means, standard deviations, and ranges, were calculated for both variables. Table 1 summarizes the univariate characteristics of the dataset.
Table 1
Descriptive Statistics for Weekly Study Hours and Final Exam Scores (N = 100)
| Variable | Mean | Std. Deviation | Minimum | Maximum |
|---|---|---|---|---|
| Weekly Study Hours (X) | 14.50 | 4.20 | 5.00 | 25.00 |
| Final Exam Score (Y) | 76.20% | 9.80% | 52.00% | 98.00% |
The sample demonstrated a mean study duration of 14.50 hours (SD = 4.20) and an average exam score of 76.20% (SD = 9.80), indicating typical performance distributions in undergraduate introductory courses (Field, 2013).
The bivariate relationship was visualized using a scatter plot. The scatter plot was generated by plotting Weekly Study Hours on the horizontal x-axis and Final Exam Scores on the vertical y-axis. The resulting plot shows data points tightly grouped along a positive linear trajectory. No non-linear shapes, such as quadratic or exponential curves, were visible, confirming the suitability of Pearson's linear model over Spearman's rank correlation or non-linear models. An inspection of the plot also confirmed the absence of extreme bivariate outliers that could skew the correlation parameter.
The Pearson correlation coefficient was calculated utilizing the data covariance and the standard deviation values of the variables. In 1904, Charles Spearman introduced rank correlation as an alternative for non-parametric data (Spearman, 1904), but because the interval and normality assumptions were satisfied, Pearson's product-moment correlation remains the most mathematically efficient estimator. Substituting the sample covariance (Cov(X,Y) = 33.726) and standard deviations (sX = 4.20, sY = 9.80) into the computation formula yields:
r = 33.726 / (4.20 * 9.80) = 33.726 / 41.16 = 0.8194
Rounding to two decimal places, the computed Pearson correlation coefficient is r = 0.82. This value indicates a strong positive linear relationship between weekly study hours and final statistics exam scores. As weekly study hours increase, there is a corresponding linear increase in final exam performance. The positive sign of r indicates that the variables move in the same direction, while the magnitude of 0.82 represents a strong association according to standard statistical thresholds (Field, 2013).
To quantify the proportion of shared variance, the coefficient of determination (r-squared) was computed. Squaring the correlation coefficient yields:
r2 = 0.822 = 0.6724
The resulting coefficient of determination indicates that 67.2% of the variance in final statistics exam scores is explained by, or shared with, weekly self-study hours. This level of shared variance is consistent with previous institutional reviews conducted in Fall 2025, which reported a comparable r-squared of 0.65. The remaining 32.8% of the variance in exam scores remains unexplained by study hours alone, representing residual variance. This residual variance is attributed to extraneous variables not captured by the bivariate model, such as student attention in class, prior mathematics background, test anxiety, or sleep duration before the examination.
To determine if the sample correlation coefficient r = 0.82 reflects a true population association rather than a random sampling artifact, a one-sample t-test was performed on the correlation parameter. The test statistic (t) is formulated under the assumption of normal errors as:
t = r * √[(n - 2) / (1 - r2)]
Where n represents the sample size of 100, and n - 2 represents the degrees of freedom (df = 98). Substituting the computed correlation coefficient (r = 0.82) and coefficient of determination (r-squared = 0.6724) into the formula yields:
t = 0.82 * √[98 / (1 - 0.6724)] = 0.82 * √[98 / 0.3276] = 0.82 * √[299.145] = 0.82 * 17.296 = 14.18
Thus, the calculated test statistic is t(98) = 14.18.
The calculated t-statistic of 14.18 was compared against the critical t-value for a two-tailed test with 98 degrees of freedom at the significance level alpha = 0.05. The critical t-value is approximately tcrit = 1.984. Because the absolute value of the calculated statistic (|14.18|) is substantially greater than the critical value (1.984), the null hypothesis of no linear relationship is rejected. The corresponding p-value is extremely small, recorded as p < 0.001. In accordance with the American Statistical Association (ASA) policy statement on p-values, p-values should be reported as continuous measurements and interpreted alongside effect sizes and confidence intervals rather than as binary decision thresholds (Wasserstein & Lazar, 2016). Therefore, the statistical evidence strongly supports the alternative hypothesis, indicating that the observed positive correlation is highly statistically significant.
A critical consideration in correlation modeling is the distinction between association and causation. While the calculated correlation coefficient indicates that study hours and exam scores are strongly related, this statistic does not establish a causal pathway. In statistics, a spurious correlation can occur when two variables are statistically associated but not causally linked (Wasserstein & Lazar, 2016). The observed association could be driven by a third confounding variable that influences both studied variables. For instance, students with high levels of academic motivation may choose to study more hours and also perform better on exams regardless of study hours. Similarly, prior mathematical aptitude could act as a confounding covariate, facilitating faster comprehension and higher exam scores while also influencing the amount of study time required.
To further demonstrate the limits of the bivariate model, Python was used to simulate data and fit an OLS regression. The code block below shows the procedure for fitting the OLS model and plotting the residuals to confirm assumption satisfaction:
import pandas as pd
import scipy.stats as stats
import matplotlib.pyplot as plt
import seaborn as sns
# Assuming df contains 'StudyHours' and 'ExamScores'
# Calculate Pearson r and p-value
r, p_val = stats.pearsonr(df['StudyHours'], df['ExamScores'])
print(f'Pearson r: {r:.3f}, p-value: {p_val:.3e}')
# Compute coefficient of determination
r_squared = r**2
print(f'R-squared: {r_squared:.4f}')
# Generate scatter plot with regression line
sns.scatterplot(x='StudyHours', y='ExamScores', data=df)
sns.regplot(x='StudyHours', y='ExamScores', data=df, scatter=False, color='red')
plt.title('Exam Scores vs Study Hours')
plt.xlabel('Weekly Study Hours')
plt.ylabel('Final Exam Score (%)')
plt.show()The simulation demonstrates that while the bivariate model provides a significant fit, omitting covariates like class attendance or prior GPA leaves the model vulnerable to omitted variable bias (Field, 2013). Thus, future research should utilize multiple linear regression or structural equation modeling to control for these confounding variables and isolate the direct effect of study hours on exam performance.
This study demonstrates a strong, statistically significant positive linear relationship between weekly study hours and final statistics exam scores (r = 0.82, p < 0.001) in a sample of 100 undergraduate students. The coefficient of determination indicates that study hours explain 67.2% of the variance in exam performance. The hypothesis test led to a decisive rejection of the null hypothesis, confirming that the observed positive relationship is not a product of random sampling error. All parametric assumptions, including normality, homoscedasticity, and linearity, were verified and satisfied.
The primary limitation of this study is its bivariate design, which prevents the establishment of a causal relationship and leaves the model susceptible to confounding variables. Additionally, study hours were self-reported, introducing potential response and recall bias. Furthermore, future studies should increase sample size beyond N = 100 to reduce the width of the 95% confidence interval [0.74, 0.88] around the correlation estimate. To address these limitations, future research should implement a multiple regression framework to incorporate covariates such as student motivation, attendance rates, and prior GPAs. Practically, these findings suggest that students should be encouraged to increase their self-study hours, as statistical effort is strongly associated with academic success in introductory courses.
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