Suppose the correlation between SAT Verbal scores and Math scores is 0.57 and that these scores are normally distributed. If a student's Verbal score places her at the 90th percentile, at what percentile would you predict her Math score to be?
You are evidently studying bivariate distributions, and I am not an expert on that subject. Convert the 90th percentile to the number of standerd deviations from the mean score. I get 1.28 sigma. With a correlation of r=0.57, the number 1-r^2 = 0.675 probably plays a role in the number of standard deviations from the mean Math score. That would be 0.864 sigma, or the 80th percentile.
To find the percentile at which you would predict a student's Math score to be, given her Verbal score, you need to use the correlation coefficient and the standard deviations of the two test scores. Here's how you can do that:
1. Start by determining the area to the left of the 90th percentile on the standard normal distribution. This can be done by using a Z-table or calculator. The area to the left of the 90th percentile is 0.90.
2. Since the correlation coefficient is 0.57, you can use it to determine the fraction of the total variation in Math scores that can be predicted by the Verbal scores. Square the correlation coefficient to get the coefficient of determination: R^2 = 0.57^2 = 0.3249. This means that 32.49% of the variability in Math scores can be explained by Verbal scores.
3. Next, calculate the standard deviation of the Math scores by multiplying the standard deviation of the Verbal scores by the square root of (1 - R^2). Let's say the standard deviation of the Verbal scores is "σV" and the standard deviation of the Math scores is "σM". Then, σM = σV * sqrt(1 - R^2).
4. Now that you have the standard deviation of the Math scores, you can find the Z-score corresponding to the 90th percentile using the standard normal distribution. This Z-score represents the number of standard deviations away from the mean a score is.
5. Use the Z-score to find the corresponding area to the left of the Z-score using a Z-table or calculator.
6. The area to the left of the Z-score represents the percentile at which you would predict the student's Math score to be. Multiply this area by 100 to get the percentile.
By following these steps, you can determine the percentile at which you would predict the student's Math score to be based on her Verbal score and the given correlation coefficient.
To find the percentile at which the student's Math score is predicted to be, we can use the information about the correlation between the two scores.
The given correlation coefficient between SAT Verbal scores and Math scores is 0.57. This indicates a positive correlation between the two variables.
To determine the percentile of the student's Math score based on her Verbal score, we can use the concept of conditional probability. We know that she scored at the 90th percentile in Verbal, so her Verbal score is greater than or equal to 90% of the other students' scores.
With a positive correlation of 0.57, we can predict that her Math score will also tend to be higher than the average. However, without knowing the specific standard deviation and mean for both Verbal and Math scores, we cannot determine the exact percentile at which her Math score would be predicted.