The SAT mathematics scores in the state of Florida for this year are approximately normally distributed with a mean of 500 and a
standard deviation of 100. Using the empirical rule, what is the probability that a randomly selected score lies between 500 and 700? Express your answer as a decimal.
<u></u><u>The correct answer is 47.5%, or 0.475.</u>
Explanation: The empirical rule states that in any normal distribution:
68% of data will fall within 1 standard deviation of the mean; 95% of data will fall within 2 standard deviations of the mean; and 99.7% of data will fall within 3 standard deviations of the mean.
The mean is 500 and the standard deviation is 100. This means that 700 is 2 standard deviations away from the mean:
(700-500)/100=200/100=2.
We know that 95% of data will fall within 2 standard deviations from the mean. However, included in the 95% is data less than the mean and greater than the mean. Since we are only concerned with the scores from 500 to 700, we only want the half that is greater than the mean:
