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STAT 312 Spring 2024 Investigation 2: Exam score regression (assigned on Mon April 15, due on Tues April 23) You may work with in a group of as many as three students on this assignment, provided that you all contribute to the work. Type your answers to these questions into a file. (You do not need to re-type the questions.) Then save that file as a pdf file, and submit the pdf file in Canvas. Only one member of the group should submit a pdf file, making sure that all group member names appear at the top of the document. The file examscores , which is available in different formats in Canvas, contains score on the first exam and scores on the final exam for 100 students in a recent class of mine. You will use an applet called Least squares regression , also available from a link in Canvas, to analyze these data. After you open the applet: 1. Click on the “Clear” button below the box containing the (default) dataset. 2. Copy/paste the exam data from the file into the empty data box. 3. Click on the “Use Data” button. 4. Make sure that 100 appears as the sample size. 5. Make sure that final exam score is the response variable (on the vertical axis) in the scatterplot.. 6. Click the box to “Show Regression Line.” a) Include a screen capture of the scatterplot, with the regression line, in your report. b) Report the equation of the least-squares regression line for predicting final exam score from first exam score. finalexam ^ = 28.02 + 0.5790 x exam1 c) Write a sentence to interpret the slope coefficient. (Be sure to include the value of the slope coefficient.) - For every one point increase in exam 1 scores, the predicted final exam score increases by a factor of 0.5709.

d) Report the value of r 2 , and write a sentence to interpret this value in this context. - R^2 = 0.222. - 22.2% of the variability in the final exam scores is explained by the least-squares regression line with first exam scores. e) Use the least-squares regression line to predict the final exam score for a student who scores 70 on the first exam. (Show how you calculate this.) - I calculated this by drawing a vertical line from 70 on the x axis (first exam score) to the least squares regression line. Then I drew a horizontal line from the regression line to the y axis (final exam score). This is shown in blue. If a student scored 70 on the first exam, they would most likely score about 68 on the final exam. f) Use the least-squares regression line to predict the final exam score for a student who scores 90 on the first exam. (Show how you calculate this.) - I calculated this by drawing a vertical line from 90 on the x axis (first exam score) to the least squares regression line. Then I drew a horizontal line from the regression line to the y

axis (final exam score). This is shown in blue. If a student scored 90 on the first exam, they would most likely score about 89 on the final exam. g) Who is predicted to achieve a higher score on the final exam – a student who scores 70 on the first exam, or a student who scores 90 on the first exam? How many points does the line predict for the difference between the final exam scores for two such students? Show how you could calculate this from the slope coefficient without knowing the intercept coefficient. - A student who scored 90 on the first exam is predicted to score higher on the final exam as opposed to a student who scored 70 on the first exam. The least-squares regression line predicted that the student who scored 70 on the first exam will get a 68 on the final. The least-squares regression line predicted that the student who scored 90 on the first exam will get an 89 on the final exam. The difference between 89 and 68 is 21. This is calculated by just looking at the least squares regression line. h) Which of the 100 students over-performed their predicted final exam score by the largest amount? (Report the exam1 and final exam scores for this student.) Also calculate the predicted value and residual value for this student. (Show how you calculate these.) - The student who scored 72.86 on the first exam and a 92.5 on the final exam was the student who over performed their predicted final exam score by the largest amount. - The predicted final exam score value for this student is 28.02 + 0.579 (72.86) = 70.20594. - The residual value for this student was 92.5. - The difference between the residual value and the predicted value for this student is 22.29406. i) Which of the 100 students under-performed their predicted final exam score by the largest amount? (Report the exam1 and final exam scores for this student.) Also calculate the predicted value and residual value for this student. (Show how you calculate these.) - The student who scored 88.57 on exam 1 and 51.25 on the final exam underperformed their predicted final exam score by the largest amount. - The predicted final exam score for this student is 28.02 + 0.579(88.57) = 79.30203. - The residual value for this student was 51.25. - The difference between the residual value and the predicted value for this student is 28.05203. j) Now consider only the 12 students who scored below 70 on the first exam. How many, and what percentage, of them achieved a higher score on the final exam than they obtained on their first exam?

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