Module 9 Homework Assignment: Paired Data Below Consists

Module 9 Homework Assignmentthe Paired Data Below Consists Of Test Sco

The paired data below consists of test scores and hours of preparation for 5 randomly selected students. x represents hours of preparation, and y represents test scores. Use the given data and MS Excel to find the correlation coefficient r, the regression equation, and construct a scatter plot. Additionally, perform the following analyses and interpretations based on the regression and data:

1. Evaluate whether the linear correlation coefficient r indicates a good model at a significance level of 0.05. Explain your reasoning.
2. Calculate the best predicted test score for a student who spends 7 hours preparing for the test.
3. Find the standard error of estimate using the appropriate formula or MS Excel regression tools.
4. Determine the 99% prediction interval for the test score of a person who prepared for 7 hours, given that the standard error E equals 34.677. Interpret this interval.
5. Compute the explained variation in the test scores.
6. Calculate the unexplained variation in the test scores.
7. Find the total variation in the data.
8. Calculate the coefficient of determination (r²) and interpret its meaning.
9. Assess the impact of adding the data point (3, 100) to the dataset on the regression analysis. Determine whether this point is an outlier, influential point, or both, and explain your reasoning.

Paper For Above instruction

The analysis of the relationship between students' hours of preparation and their corresponding test scores provides valuable insights into learning behaviors and test performance. By employing statistical tools such as correlation, regression, and variation measures, educators and researchers can better understand the predictive power of preparation time and identify significant data points that might skew results.

To initiate this investigation, the primary step involves calculating the correlation coefficient (r) between hours of preparation (x) and test scores (y). The Pearson correlation coefficient quantifies the strength and direction of a linear relationship between these variables. Using MS Excel, this calculation is straightforward, leveraging the function =CORREL(array_x, array_y). Suppose the computed r is close to +1, indicating a strong positive linear relationship, suggesting that increased preparation correlates with higher test scores.

Following this, the regression equation takes shape, typically expressed as ŷ = a + bx, where b is the slope indicating the average change in test score with each additional hour of preparation, and a is the intercept. MS Excel's regression analysis tool can generate these parameters, along with the correlation coefficient, coefficients of determination, and residual analysis. The strength of the model can be evaluated by examining the significance of the regression coefficients and the value of r. An r value statistically significant at the 0.05 level supports the model's validity.

Next, predicting a test score for a student who studies for 7 hours involves substituting x=7 into the regression equation. For example, if the regression equation is ŷ = 30 + 10x, then ŷ = 30 + 10(7) = 100. This predicted score provides practical predictive insight and helps in assessing expected performance levels.

The standard error of the estimate (SE) measures the typical deviation of observed values from the predicted values. It can be obtained through Excel regression outputs or calculated manually using the residuals. The standard error enables assessment of the model's accuracy; smaller values imply better predictive performance.

Constructing a 99% prediction interval provides a range within which an individual test score for a student preparing for 7 hours is likely to fall with 99% confidence. The interval accounts for the variability inherent in the data, with the formula being:

Prediction interval = ŷ ± t* × SE_pred

where t* is the critical t-value for 99% confidence, and SE_pred is the standard error of the prediction, which incorporates both the standard error of estimate and the leverage of the specific x value. Given E = 34.677, the interval offers a range enclosing the likely test score, emphasizing the uncertainty around the prediction.

Decomposing the total variation into explained and unexplained parts reveals the model's explanatory power. The explained variation quantifies how much of the variance in test scores is accounted for by the regression, computed as r² multiplied by the total variation. Conversely, the unexplained variation reflects the residual variability not captured by the model. These measures are essential in evaluating model fit and predicting accuracy.

Adding a new data point, especially an outlier like (3, 100), impacts the regression results by potentially altering the slope, intercept, and correlation. An influential point is one that significantly affects the regression line's position. If the point (3, 100) deviates markedly from the pattern of existing data, it might be an outlier and influence the regression analysis, skewing results and reducing the model's reliability. Visual inspection through scatter plots and influence diagnostics such as Cook's distance can determine this.

In conclusion, thorough statistical evaluation combining correlation, regression, and variation analyses provides comprehensive insight into the relationship between preparation time and test performance. Recognizing influential data points ensures the robustness of the model, guiding educators in understanding and improving student outcomes.

References

• Albright, S. C., Winston, W. L., & Zappe, C. (2016). Data Analysis & Decision Making. Cengage Learning.
• Everitt, B. S., & Skrondal, A. (2010). The Cambridge Dictionary of Statistics. Cambridge University Press.
• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
• Glass, G. V., & Hopkins, K. D. (1996). Statistical Methods in Education and Psychology. Prentice Hall.
• Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to Linear Regression Analysis. Wiley.
• Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.
• Wooldridge, J. M. (2015). Introductory Econometrics: A Modern Approach. Cengage Learning.
• Yusuf, A. A., et al. (2014). Regression Analysis and Its Application. Journal of Statistical Methods and Applications, 3(2), 123-134.
• Zellner, A. (2009). An Introduction to Bayesian Regression Analysis. Springer.