Suppose that you are given the following information about the freshman year GPAs and the SAT scores of students entering a certain college: Mean SAT= 1125, Standard Deviation SAT= 150, Mean GPA= 2.68, St. Dev. GPA= 0.82 and r=0.65.
a)Use the above information to compute the regression line you would use to predict freshman GPAs from SAT scores.
b) Interpret the slope and the y-intercept of the regression line you obtained.
c) Predict the freshman GPA of a student who had a 1300 SAT score.
d) What is r^2(r-squared) and what does it tell you about the quality of the predictions of freshman GPA based on SAT score?
I am really lost, but have worked some of the problems out, but I'm not sure if they are correct. For part a, I got y-hat= -1.317499963 + 0.0035533333x
For part c, I got y-hat= 3.301833327
For part d, I got (0.65)^2= 0.4225 X 100= 42.45%. Therefore, quality of predictions is moderate due to moderate correlation
Any help would be greatly appreciated!
To solve this problem step-by-step, we'll go through each part one by one:
a) To compute the regression line, we'll use the formula:
y-hat = b0 + b1x
where y-hat is the predicted GPA, b0 is the y-intercept, b1 is the slope, and x is the SAT score.
To find the slope (b1), we'll use the formula:
b1 = r * (St. Dev. GPA / St. Dev. SAT)
Plugging in the values:
b1 = 0.65 * (0.82 / 150) = 0.0035533333 (rounded to 10 decimal places)
To find the y-intercept (b0), we'll use the formula:
b0 = Mean GPA - (b1 * Mean SAT)
Plugging in the values:
b0 = 2.68 - (0.0035533333 * 1125) = -1.317499963 (rounded to 10 decimal places)
Therefore, the regression line is:
y-hat = -1.317499963 + 0.0035533333x
b) The slope (0.0035533333) tells us that for every increase of 1 SAT score unit, the predicted GPA will increase by approximately 0.0036 units. This means higher SAT scores are associated with higher GPAs.
The y-intercept (-1.317499963) tells us the predicted GPA when the SAT score is zero. However, this interpretation is not meaningful because SAT scores cannot be zero in reality.
c) To predict the freshman GPA for a student with a 1300 SAT score, we'll use the regression line:
y-hat = -1.317499963 + 0.0035533333 * 1300 = 3.301833327
Therefore, the predicted freshman GPA for a student with a 1300 SAT score is approximately 3.3018.
d) The coefficient of determination (r-squared, denoted as r^2) measures the proportion of variation in the dependent variable (GPA) that can be explained by the independent variable (SAT score). It ranges from 0 to 1.
To calculate r^2, we square the correlation coefficient (r):
r^2 = (0.65)^2 = 0.4225
This means that approximately 42.25% of the variation in freshman GPA can be explained by the variation in SAT scores. It implies that SAT scores have a moderate explanatory power in predicting freshman GPA.
Please note that the calculations have been rounded to 10 decimal places for simplicity.
Sure, I can help you with that!
a) To compute the regression line, you need to use the formula:
y-hat = b0 + b1 * x
where y-hat is the predicted value of the dependent variable (freshman GPA), b0 is the y-intercept, b1 is the slope, and x is the independent variable (SAT score).
To find the slope (b1), you can use the formula:
b1 = r * (St. Dev. GPA / St. Dev. SAT)
Plugging in the given values, we have:
b1 = 0.65 * (0.82 / 150) = 0.0035533333 (rounded)
Next, to find the y-intercept (b0), you can use the formula:
b0 = Mean GPA - b1 * Mean SAT
Plugging in the given values, we have:
b0 = 2.68 - 0.0035533333 * 1125 = -1.317499963 (rounded)
Therefore, the regression line you would use to predict freshman GPAs from SAT scores is:
y-hat = -1.317499963 + 0.0035533333 * x
b) The slope (0.0035533333) represents the change in the predicted freshman GPA for every one-unit increase in the SAT score. In this case, it means that for every increase of 1 point in the SAT score, the predicted freshman GPA will increase by approximately 0.0036.
The y-intercept (-1.317499963) represents the predicted freshman GPA when the SAT score is zero. However, in the context of this problem, it may not have a practical interpretation since it is unlikely for a student to have a zero SAT score.
c) To predict the freshman GPA of a student with a 1300 SAT score, you can plug the value of x into the regression line formula:
y-hat = -1.317499963 + 0.0035533333 * 1300
y-hat ≈ 3.3018333267
Therefore, the predicted freshman GPA for a student with a 1300 SAT score is approximately 3.30.
d) The coefficient of determination (r^2) measures the proportion of the variance in the dependent variable (freshman GPA) that can be explained by the independent variable (SAT score). In other words, it tells you how well the regression line fits the data.
To calculate r^2, you square the correlation coefficient (r). In this case, r^2 = (0.65)^2 ≈ 0.4225.
This means that approximately 42.25% of the variation in freshman GPA can be explained by the variation in SAT scores. The remaining 57.75% of the variation is due to other factors not included in the regression model.
Based on this value, you can conclude that the predictions of freshman GPA based on SAT score are somewhat moderate, as 42.25% is a moderate amount of variance explained.