Consider X to be the random variable representing the score of any man.
The mean and standard deviation are given to be 498 and 115 respectively,
The standard normal variate corresponding to any value of score, is given by the formula,
For X=584.9, the z-score becomes,
Then the probability that a randomly selected man has scored at least 584.9 is calculated as,
Using the properties of normal distribution,
From the Standard Normal Distribution Table,
Substitute the value,
Thus, there is a 0.2266 probability that the score of a randomly selected man is at least 584.9.
Consider a normal sample of 7 men from the college,
The z-score for the random sample is given by,
Substitute the values,
So the probability that the mean of the sample is at least 584.9, is calculated as,
Using the properties of Normal Distribution,
Substitute the value,
Thus, there is a 0.0228 probability that their mean score is at least 584..9.