The scatter diagram of midterm scores and final scores in a large class is football shaped. For about 80% of the students, the regression estimate of final score based on midterm score is correct to within 15 points.
For about 95% of the students, the regression estimate of final score based on midterm score is correct to within ___________ points
To determine the range of points within which the regression estimate of final score based on midterm score is correct for about 95% of the students, we can use the concept of the standard deviation.
The percentage given in the question corresponds to a certain number of standard deviations from the mean. In statistics, this is often referred to as the z-score. For a normal distribution, we can use a table or a calculator to find the z-score associated with a given percentage.
For example, for the 80% mentioned in the question, we know that it corresponds to a z-score of approximately 1.28. This means that approximately 80% of the students fall within 1.28 standard deviations of the mean.
To find the z-score corresponding to the 95% range, we can use the inverse normal distribution function or a table. The z-score for a 95% confidence interval is approximately 1.96.
Now, we need to find the number of points within which the regression estimate of the final score is correct for about 95% of the students. This can be calculated by multiplying the z-score by the standard deviation of the final scores. However, since we don't have the exact standard deviation, we can make an assumption that the distribution of final scores is symmetrical and has a football-shaped scatter diagram.
Using this assumption, we can estimate the range by multiplying the z-score (1.96) by the average distance between the regression estimate and the actual final score for the 80% range. If the regression estimate is correct to within 15 points for 80% of the students, the average distance is half of this, which is 7.5 points.
Calculating the range:
Range = z-score * average distance
Range = 1.96 * 7.5
Range ≈ 14.7
Therefore, the regression estimate of the final score based on midterm score is correct to within approximately 14.7 points for about 95% of the students.