Name
Purdue University Globle
MM207 Statistics
Prof. Name
Date
A Pearson correlation coefficient of r = 0.774 shows a strong positive linear relationship between study hours and test scores. In other words, students who spend more time studying tend to earn higher test scores. The scatter plot supports this finding by showing an overall upward pattern in the data. Although correlation does not establish that studying more directly causes higher grades, the results indicate that study time and test performance are strongly associated in this dataset.
The scatter plot provides a visual representation of the relationship between the amount of time students spend studying and their test performance. The data points generally move upward from left to right, meaning that higher study hours are associated with higher test scores.
The Pearson correlation coefficient, r = 0.774084839, provides statistical support for this pattern. Because the coefficient is positive and relatively close to +1, the data indicate a strong positive linear association between the two variables.
This relationship suggests that students who dedicate more hours to studying generally perform better on tests. However, test scores can also be influenced by other factors, such as prior knowledge, learning strategies, attendance, motivation, sleep, and test-taking skills. Therefore, the correlation should be interpreted as an association rather than proof of a cause-and-effect relationship.
The analysis examines two quantitative variables: study hours and test scores. These variables can be displayed on a scatter plot to determine whether a linear relationship exists between them.
The independent variable is study hours, which is represented on the X-axis. It measures the amount of time a student spends studying.
The dependent variable is test score, which is represented on the Y-axis. It represents the student’s academic performance as measured by the test result.
Using these variables makes it possible to evaluate whether differences in study time are associated with differences in test performance.
The scatter plot contains individual observations representing students or data points in the dataset. Study hours are displayed horizontally, while test scores are displayed vertically. The test score values may also be displayed above the corresponding points to make individual observations easier to interpret.
The most important feature of the graph is its overall upward direction. As study hours increase, test scores generally increase as well. The points do not fall perfectly along a straight line, which indicates that students with similar study times can still receive different scores.
This variation is important because a correlation of 0.774 does not mean that every additional hour of studying will produce exactly the same increase in test performance. Instead, it indicates that the two variables tend to move together in a positive linear pattern.
A Pearson correlation coefficient describes the direction and strength of a linear relationship between two quantitative variables. The coefficient ranges from −1 to +1.
A positive value means that the variables tend to increase together, while a negative value means that one variable tends to increase as the other decreases. A value close to zero indicates little or no linear association.
For the study-hours dataset, the correlation is:
r = 0.774084839
Because the value is positive, students who report more study hours tend to have higher test scores. Because the absolute value is relatively large, the association is considered strong.
A Pearson correlation of 0.774 indicates a strong positive linear relationship between study hours and test scores. The result means that higher amounts of study time tend to be associated with higher test scores within the dataset.
A commonly used descriptive interpretation of correlation strength is:
0.00–0.19: Very weak
0.20–0.39: Weak
0.40–0.59: Moderate
0.60–0.79: Strong
0.80–1.00: Very strong
Under this interpretation, r = 0.774 falls within the strong positive correlation range.
It is also useful to consider the coefficient of determination. Squaring the correlation coefficient gives approximately 0.599, or about 59.9%. This means that approximately 59.9% of the variation in test scores is associated with the linear relationship with study hours in this dataset. This should not be interpreted as showing that studying causes 59.9% of a student’s test performance.
The reported Pearson correlation can also be evaluated against critical values for statistical significance. In this analysis, the comparison values are:
Significance level α = 0.05: Critical value = 0.514
Significance level α = 0.01: Critical value = 0.641
Calculated Pearson correlation: r = 0.774
Because the absolute value of the calculated correlation, 0.774, is greater than both reported critical values, the correlation meets the stated significance criteria at the 0.05 and 0.01 levels, assuming the appropriate sample size and test conditions for those critical values.
A statistically significant result provides evidence against the null hypothesis of no linear correlation. It does not, however, establish causation. Statistical significance and practical significance should also be considered separately when interpreting research findings.
The scatter plot and Pearson correlation complement one another. The graph provides a visual understanding of the data, while the correlation coefficient summarizes the direction and strength of the linear relationship numerically.
In this case, both forms of evidence point toward the same conclusion: study hours and test scores have a strong positive linear association.
The scatter plot is particularly useful because it can reveal patterns that a single correlation coefficient cannot fully describe. For example, a graph may help identify possible outliers, clusters, or nonlinear patterns. Therefore, researchers should generally examine the scatter plot along with the correlation statistic rather than relying on the coefficient alone.
The analysis of study hours and test scores produces several important findings. The data show a clear positive trend, with higher study hours generally corresponding to higher test scores. The Pearson correlation of r = 0.774 indicates that this association is strong and positive.
The reported statistical comparison also indicates that the correlation is significant at both the 0.05 and 0.01 levels. Overall, the evidence supports the conclusion that study time is strongly associated with academic performance in this dataset.
At the same time, the results should be interpreted carefully. Students’ test performance can be affected by many variables, and correlation alone cannot demonstrate that increasing study time will necessarily cause a particular improvement in test scores.
The MM207 Unit 5 analysis demonstrates a strong positive relationship between study hours and test scores. The Pearson correlation coefficient of 0.774084839 indicates that students who spend more time studying tend to achieve higher test scores. The upward pattern in the scatter plot provides visual evidence supporting this statistical result.
The findings suggest that study time is an important factor associated with academic performance in the dataset. However, other academic and personal factors may also contribute to differences in test scores. Therefore, the most appropriate interpretation is that study hours and test scores are strongly positively correlated, not that study hours alone determine test performance.
The scatter plot shows a strong positive linear relationship between study hours and test scores. The data generally trend upward, indicating that students who study more tend to achieve higher test scores.
The Pearson correlation coefficient reported for the dataset is r = 0.774084839, which indicates a strong positive linear association between the two variables.
The independent variable is study hours. It is placed on the X-axis of the scatter plot and represents the amount of time students spend studying.
The dependent variable is test score. It is placed on the Y-axis and represents academic performance measured by test results.
Yes. A Pearson correlation of 0.774 is generally interpreted as a strong positive correlation. It indicates that higher study hours tend to be associated with higher test scores.
No. Correlation does not prove causation. The result shows that study hours and test scores are strongly associated, but other variables may also influence academic performance.
A positive correlation means that two variables tend to increase together. In this case, as study hours increase, test scores also tend to increase.
A scatter plot allows researchers to visually examine the relationship between two quantitative variables. It can help identify trends, clusters, unusual observations, and whether a linear relationship appears reasonable.
Based on the reported critical values, r = 0.774 exceeds the critical values of 0.514 at α = 0.05 and 0.641 at α = 0.01. Therefore, the correlation is statistically significant under the stated criteria.
Squaring r = 0.774084839 gives approximately 0.599, meaning about 59.9% of the variation in test scores is associated with the linear relationship with study hours in this dataset. This does not mean that studying causes 59.9% of test performance.
American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://apastyle.apa.org/products/publication-manual-7th-edition
Lane, D. M. (n.d.). Correlation. In Online Statistics Education: An Interactive Multimedia Course of Study. Rice University. https://onlinestatbook.com/2/describing_bivariate_data/correlation.html
National Institute of Standards and Technology. (n.d.). Measures of association and correlation. NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/
OpenStax. (2023). Introductory statistics 2e. Rice University. https://openstax.org/details/books/introductory-statistics-2e
