Pre study scores versus post-study scores for a class of 120 college freshman english students were considerated. The residual plot for the least squares regression line showed no pattern. The least square regression line was y = 0.2 + 0.9x with a correlation coefficient r = 0.76. What percent of the variation of post- study scores can be explained by the variation in pre-study scores? Based on the value of coefficient of determinartiton,we have ? Since the residual plot has no pattern we should? Suppose that one student had a pre-study score of 20 and a residual of 2.2?
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To find the percentage of the variation of post-study scores that can be explained by the variation in pre-study scores, we need to square the correlation coefficient.
In this case, the correlation coefficient is given as r = 0.76. By squaring it, we get r^2 = 0.76^2 = 0.5776.
So, approximately 57.76% of the variation in post-study scores can be explained by the variation in pre-study scores.
The coefficient of determination (denoted as R^2) represents the proportion of the variance in the dependent variable (post-study scores) that can be explained by the independent variable (pre-study scores). In this case, the coefficient of determination is equal to r^2, which is 0.5776 or 57.76%.
Since the residual plot for the least squares regression line shows no pattern, it indicates that the assumptions of linearity, independence, and equal variance are likely satisfied for the data. This suggests that the linear regression model is appropriate for predicting the post-study scores based on the pre-study scores.
If one student had a pre-study score of 20 and a residual of 2.2, it means that the predicted post-study score for that student would be 0.2 + 0.9(20) = 18.2. The residual of 2.2 indicates that the actual post-study score for that student deviates from the predicted score by 2.2 units.