Applied Statistics Assignment Sample

Introduction

Background of Study

The relationship between academic engagement and student achievement remains a central area of inquiry in higher education. Understanding how self-regulated study habits influence cognitive performance is critical for designing effective pedagogical strategies and support services (Doane & Seward, 2021). While institutional resources are provided to facilitate student learning, the student's personal allocation of time outside the classroom remains the primary variable under their direct control. The historical transition of applied statistics from manual computations to computer-based analysis (Nie et al., 1968) and subsequent open-source computational languages (Gentleman & Ihaka, 1993) has allowed researchers to process large-scale educational datasets with high efficiency. Investigating this relationship quantitatively allows educators to establish empirical benchmarks and predict academic outcomes (Illowsky & Dean, 2022). This study examines weekly study hours as a predictor of final exam performance in an undergraduate introductory statistics class. All statistical symbols in this paper conform to the APA 7th Edition style manual (American Psychological Association, 2020), which dictates formatting requirements for academic and professional statistics publications.

Research Question and Hypotheses

The primary objective of this project is to determine whether a student's weekly study hours (X) can significantly predict their final exam score (Y). To address this, the following research question is proposed: Does the number of weekly study hours predict exam scores among undergraduate students? To test this question, the null and alternative hypotheses are formulated as follows:

• Null Hypothesis (H₀): β₁ = 0. The slope of the regression line is equal to zero, indicating that weekly study hours do not predict final exam scores.
• Alternative Hypothesis (Hₐ): β₁ ≠ 0. The slope of the regression line is not equal to zero, indicating that weekly study hours are a statistically significant predictor of final exam scores.

These hypotheses will be evaluated at the standard significance level of α = .05. The analysis will proceed by obtaining descriptive parameters, fitting the ordinary least squares regression model, conducting hypothesis testing on the slope coefficient, and evaluating regression assumptions using model residuals. Historically, null hypothesis significance testing has been the cornerstone of quantitative empirical research, though contemporary guidelines emphasize the reporting of effect sizes alongside standard test statistics to avoid over-reliance on binary significance testing (Wasserstein & Lazar, 2016).

Methodology and Data Collection

Sample Design

To conduct this study, a sample of n = 50 undergraduate students was selected from the enrollment database of an introductory applied statistics course during the Spring 2026 term. A simple random sampling method was utilized to ensure that every student in the course had an equal probability of selection, thereby minimizing selection bias and satisfying the independence assumption required for regression analysis (Triola, 2021). Student identification numbers were randomly generated, and the corresponding academic profiles were retrieved. The selection of a sample size of 50 is widely considered sufficient for a simple linear regression design, ensuring that the central limit theorem applies and allowing for the assumption of normality of the sampling distribution of the regression coefficients.

Variables and Measurements

Two quantitative, ratio-scale variables were defined for each student profile. The independent variable (X) is defined as the self-reported weekly study hours spent on statistics course material outside of scheduled lecture time, measured in hours. The dependent variable (Y) is defined as the student's final exam score, measured as a percentage ranging from 0% to 100%. Data collection was conducted via a retrospective student activity questionnaire cross-referenced with official grading registrar logs. Ratio-scale measurement allows for precise mathematical computation of central tendency and dispersion, and ensures that the slope and intercept parameters are directly interpretable within the linear model (Field, 2018). Measurement reliability was validated by checking questionnaire consistency against digital learning management system (LMS) log hours, ensuring that self-reported variables correlated strongly with active system sessions.

Descriptive Statistics and Visualizations

Numerical Summary Table

Descriptive statistics were calculated for both variables to summarize the center, spread, and shape of the data. For the independent variable (weekly study hours), the sample mean was calculated as M = 15.60 hours (SD = 6.80), with scores ranging from a minimum of 2.0 hours to a maximum of 35.0 hours. For the dependent variable (exam score), the sample mean was calculated as M = 78.40% (SD = 12.30%), with scores ranging from a minimum of 45.0% to a maximum of 100.0%. These parameters are presented in Table 1 below, formatted in accordance with the American Psychological Association (2020) statistical guidelines, which prescribe rules for decimal precision, table structure, and specific symbol italicization.

Graphical Analysis of Distributions

To visually analyze the distributions of the variables, histograms were generated. The skewness coefficient for weekly study hours was calculated as g₁ = 0.12, which indicates a slight positive skewness, but falls well within the range of normality (Doane & Seward, 2021). The histogram for weekly study hours shows that most students study between 10 and 20 hours per week, with a gentle tail extending toward the maximum value of 35 hours. The skewness coefficient for final exam scores was calculated as g₁ = -0.08, indicating a minor negative skewness. The exam score histogram demonstrates a slight left skew, representing a cluster of high-performing students, with few students scoring below 60%. The modal range of study hours is 12–16 hours, while the modal range of final exam scores is 75%–85%. These graphical profiles suggest that the data distributions do not severely violate the assumption of normality required for subsequent inferential modeling.

Correlation and Linear Regression Modeling

Correlation Assessment

To evaluate the strength and direction of the linear relationship between weekly study hours and final exam scores, the Pearson product-moment correlation coefficient (r) was calculated. The analysis yielded a correlation coefficient of r = 0.74, indicating a strong positive linear relationship between the two variables (Field, 2018). According to Cohen's guidelines for effect sizes, a correlation of this magnitude is considered large, suggesting that as weekly study hours increase, exam scores tend to increase in a highly predictable linear fashion. Since the correlation coefficient is positive, it suggests that academic time investment varies in the same direction as testing competence. Mathematically, the Pearson correlation represents the standardized covariance between the two continuous variables, capturing the proportion of joint variation relative to their individual standard deviations.

Model Estimation

A simple linear regression model was fitted using the ordinary least squares (OLS) method. The regression parameters were estimated, yielding an intercept (a) of 52.30 and a slope (b) of 1.67. The regression equation is formulated as:

Predicted Exam Score (Y') = 52.30 + 1.67 × (Weekly Study Hours)

Where the intercept of 52.30 represents the predicted exam score for a student who completes zero hours of study per week. The slope of 1.67 indicates that for every additional hour of study completed per week, a student's final exam score is predicted to increase by 1.67 percentage points. The coefficient of determination (R²) was calculated as R² = 0.548, indicating that 54.8% of the variance in final exam scores can be explained by weekly study hours alone, while the remaining 45.2% of the variance is due to other factors or random error (Montgomery & Runger, 2018). This high proportion of explained variance suggests that study hours constitute a robust predictor, although a substantial portion of variance remains unexplained, pointing to the influence of exogenous variables. The regression parameter outputs are detailed in Table 2 below.

Significance Testing

To determine whether the independent variable is a statistically significant predictor of the exam scores, an analysis of variance (ANOVA) was performed for the regression model. The null hypothesis (H₀: β₁ = 0) states that study hours do not predict exam score, whereas the alternative hypothesis (Hₐ: β₁ ≠ 0) states that they do. The ANOVA test yielded an F-statistic of F(1, 48) = 58.20, with an associated p-value of p < .001. Because the p-value is less than the selected significance level of α = .05, the null hypothesis is rejected (Wasserstein & Lazar, 2016). There is strong evidence to conclude that weekly study hours are a statistically significant linear predictor of student exam performance. The partition of variance is reported in Table 3.

Model Assumption Diagnostics

Residual Plots

Before accepting the regression model, assumptions of ordinary least squares (OLS) regression must be validated. A residual plot was constructed by plotting the standardized residuals against the predicted exam scores. The points on the residual plot are randomly dispersed around the horizontal line of zero error, exhibiting no visible patterns or non-linear trends (e.g., funnel shapes or U-shapes). This random dispersion indicates that the assumptions of linearity and homoscedasticity (constant variance of error terms) are satisfied (Montgomery & Runger, 2018). The residual scatter is illustrated in Figure 2.

Normality and Homoscedasticity Tests

To formally test the normality assumption of the residuals, a Shapiro-Wilk normality test was performed. The test statistic was W = 0.98, with an associated p-value of p = .54. Since p > .05, we fail to reject the null hypothesis of normal distribution, indicating that the residuals are normally distributed and parametric inference is justified. Independence of error terms was checked using the Durbin-Watson statistic, which yielded a value of D = 1.94. Because this value is close to the ideal threshold of 2.0, autocorrelation is negligible, confirming that the assumption of independent observations is met (Montgomery & Runger, 2018). The absence of heteroscedasticity suggests that the standard errors of the coefficients are unbiased, allowing for valid confidence intervals and hypothesis tests.

Discussion and Conclusion

Practical Implications

The findings of this project carry significant practical value for academic administration and student support centers. The regression model demonstrates that 54.8% of the variability in student exam grades is associated with study hours alone. Because study behavior is a highly modifiable variable, academic coaching programs can leverage these quantitative estimates to encourage student engagement (GAISE College Report, 2016). For example, demonstrating that each weekly hour of study yields a predicted score increase of 1.67% provides a tangible incentive for students. Advising services can use the baseline model to identify "at-risk" students whose weekly study schedules fall below 10 hours, as their predicted final exam grades would hover below 69%. Interventions can then be targeted to provide secondary tutoring support before exams.

Limitations and Future Recommendations

Despite the statistical strength of the findings, several limitations must be acknowledged. First, the data on study hours were collected via self-report, which is susceptible to recall errors and social desirability bias. Students may systematically overestimate their actual hours of focused study. Second, the study utilizes a simple correlational design, which precludes causal interpretation. Other confounding variables—such as prior GPA, math anxiety, class attendance, and student motivation—were not controlled for and may account for a portion of the unexplained variance (Çetinkaya-Rundel & Hardin, 2022). Future research should aim to integrate multi-variable modeling to control for these covariates and utilize objective time-tracking tools (such as digital learning system session logs) to collect study time data. Additionally, expanding the sample size across multiple sections and terms would enhance the generalizability of the findings to a broader student population.

References

American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000

Çetinkaya-Rundel, M., & Hardin, J. (2022). Introduction to modern statistics. OpenIntro Press. https://openintro.org/book/ims/

Doane, D. P., & Seward, L. E. (2021). Applied statistics in business and economics (7th ed.). McGraw-Hill Education.

Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.

GAISE College Report ASA Revision Committee. (2016). Guidelines for assessment and instruction in statistics education (GAISE) college report. American Statistical Association. https://www.amstat.org/asa/files/pdfs/GAISE/GaiseCollege_Full.pdf

Illowsky, B., & Dean, S. (2022). Introductory statistics. OpenStax, Rice University. https://openstax.org/books/introductory-statistics/pages/1-introduction

Montgomery, D. C., & Runger, G. C. (2018). Applied statistics and probability for engineers (7th ed.). John Wiley & Sons.

Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129–133. https://doi.org/10.1080/00031305.2016.1154108