In Professor Smith's statistics course, the correlation between students' total scores before the final exam and their final exam scores is r = 0.67. The pre-exam totals for all students in the course have a mean of 275 and a standard deviation of 26. The final exam scores have a mean of 71 and a standard deviation of 6. Professor Smith has lost Jane's final exam, so decides to predict her final exam score based on her pre-exam course total, which is 293. Use least-squares regression to predict Jane's final exam score.
Predicted Final Exam Score =
Please explain to me how to do this. My teacher does not teach this kind of stuff in class and he expects us to know this.
Pre-exam: mean = 275, sd = 26
Final exam: mean = 71, sd = 6
Correlation: r = 0.67
Regression equation is in this format:
predicted y = a + bx
...where a = intercept and b = slope.
To find the equation, you need to substitute the information given in the problem into a workable formula:
predicted y = (rSy/Sx)X - (rSy/Sx)xbar + ybar
...where r = correlation, Sy = sd of y, Sx = sd of x, and X is the variable in 'a + bx' equation.
Note: xbar = mean of x; ybar = mean of y.
I'll let you take it from here. (Once you calculate the predicted y formula, substitute 293 for x in the formula to predict Jane's final exam score.)
what is a correlation coefficient between two variables
To predict Jane's final exam score based on her pre-exam course total using least-squares regression, you can follow these steps:
Step 1: Calculate the slope of the regression line.
The slope of the regression line can be calculated using the formula:
slope = r * (std_dev_final_exam / std_dev_pre_exam_total)
Here, r is the correlation coefficient (0.67), std_dev_final_exam is the standard deviation of the final exam scores (6), and std_dev_pre_exam_total is the standard deviation of the pre-exam course totals (26).
slope = 0.67 * (6 / 26)
Step 2: Calculate the intercept of the regression line.
The intercept of the regression line can be calculated using the formula:
intercept = mean_final_exam - (slope * mean_pre_exam_total)
Here, mean_final_exam is the mean of the final exam scores (71) and mean_pre_exam_total is the mean of the pre-exam course totals (275).
intercept = 71 - (slope * 275)
Step 3: Substitute the given pre-exam course total for Jane (293) into the regression equation to get the predicted final exam score.
predicted_final_exam_score = intercept + (slope * Jane's pre-exam total)
predicted_final_exam_score = intercept + (slope * 293)
By substituting the calculated values for slope, intercept, and Jane's pre-exam total into the equation, you can find the predicted final exam score for Jane.
To predict Jane's final exam score using least-squares regression, you will need to follow a few steps:
Step 1: Understand the given information
In this problem, you know the correlation between the pre-exam totals and the final exam scores (r = 0.67), as well as the means and standard deviations of both variables.
Step 2: Calculate the slope (b) of the regression line
The slope of the regression line can be calculated using the formula:
b = r * (standard deviation of Y / standard deviation of X)
Here, Y represents the final exam scores and X represents the pre-exam totals. So, plug in the values:
b = 0.67 * (6 / 26) = 0.1546
Step 3: Calculate the y-intercept (a) of the regression line
The y-intercept of the regression line can be calculated using the formula:
a = mean of Y - (slope * mean of X)
Here, the mean of Y represents the mean final exam score (71) and the mean of X represents the mean pre-exam total (275). Substitute these values:
a = 71 - (0.1546 * 275) = 30.675
Step 4: Use the regression equation to predict Jane's final exam score
The regression equation is given by:
y = a + (b * x)
Here, x represents Jane's pre-exam total (293). Substitute the values to find the predicted final exam score:
y = 30.675 + (0.1546 * 293) = 75.4238
Therefore, the predicted final exam score for Jane is approximately 75.42.
Note: The least-squares regression is a statistical method used to find the best-fit line that minimizes the sum of squared differences between the observed and predicted values. In this case, it is used to predict a student's final exam score using their pre-exam total.