Final answer:
The fraction of random numbers within standard deviations of the mean can be determined using the empirical rule, which states that approximately 68% of values lie within one standard deviation of the mean, 95% of values lie within two standard deviations of the mean, and 99.7% of values lie within three standard deviations of the mean.
Step-by-step explanation:
The question is asking for the fraction of random numbers that are within standard deviations of the mean. To determine this, we need to use the concept of the empirical rule, which states that approximately 68% of values lie within one standard deviation of the mean, 95% of values lie within two standard deviations of the mean, and 99.7% of values lie within three standard deviations of the mean.
So, for this specific scenario where the mean is 180 and the standard deviation is 20, we would expect approximately 68% of the random numbers to be within the range of 160 to 200, 95% of the random numbers to be within the range of 140 to 220, and 99.7% of the random numbers to be within the range of 120 to 240.