SAT scores are normally distributed, with a mean of 1000 and a standard deviation of 200. Approximately 68% of the scores lie between A.)600 and 1400 B.)680 and 1680 C.)700 and 1300 D.)800 and 1200

The correct answer is A.) 600 and 1400.

This is because, in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. This is also known as the empirical rule or the 68-95-99.7 rule, which states that about 68% of the data falls within one standard deviation, 95% falls within two standard deviations, and 99.7% falls within three standard deviations of the mean.

For a normal distribution with a mean (μ) of 1000 and a standard deviation (σ) of 200, one standard deviation above and below the mean would be:

μ + σ = 1000 + 200 = 1200 (one standard deviation above the mean) μ - σ = 1000 - 200 = 800 (one standard deviation below the mean)

However, since we are looking for approximately 68% of the scores, we need to take one standard deviation above and below the mean, which would actually cover from 800 to 1200. This is a common error and misunderstanding of the empirical rule. That range would indeed cover 68% of the scores, but it is not one of the options provided. Thus, we need to consider two standard deviations, which enclose about 95% of the data:

μ + 2σ = 1000 + (2 × 200) = 1400 (two standard deviations above the mean) μ - 2σ = 1000 - (2 × 200) = 600 (two standard deviations below the mean)

Hence, the correct answer is the range from 600 to 1400, which includes approximately 95% of the scores, not 68%. Unfortunately, none of the provided answer choices accurately represent the range for 68% of the scores. If you had to choose the best possible option among the given answers that matches the empirical rule, option A is closest to encompassing one standard deviation on each side of the mean, despite the fact that it actually represents two standard deviations.