If SAT math scores are bell-shaped with mean 400 and range from 0 to 800, estimate the standard deviation. Show any work supporting your answer.
To estimate the standard deviation, we need to consider the information given about the SAT math scores.
First, we know that the scores are bell-shaped, which suggests that they follow a normal distribution. In a normal distribution, approximately 68% of the scores fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations.
Given that the mean of the SAT math scores is 400, we can use this information to estimate the standard deviation.
Using the 68-95-99.7 rule, we know that the range of scores within one standard deviation of the mean is 400 - 200 (mean minus one standard deviation) to 400 + 200 (mean plus one standard deviation). This gives us a range of 200 to 600.
However, we also know that the range of scores for the SAT math is from 0 to 800. This means that the range of scores within one standard deviation must be half of the total range, which is 800 - 0 = 800. Therefore, the range within one standard deviation is 800 / 2 = 400.
Since we already have the range within one standard deviation, we can estimate the standard deviation as half of this range. Therefore, the estimated standard deviation is 400 / 2 = 200.
So, based on the given information and the assumption of a normal distribution, we estimate that the standard deviation of SAT math scores is 200.