Out of 2,000 random but normally distributed numbers with a mean of 45 and a standard deviation of x, approximately 1,360 numbers are found to be between 40 and 50. what is the value of x?

Let denote the random variable representing a given number in the total set of numbers. We're told that of the numbers fall within a given range, so we know



where is normally distributed with mean 45 and an unknown variance .

Let's make the transformation to a random variable with a standard normal distribution:



Since is symmetric, we have



The mean of is 0, and by symmetry we know that exactly half of the distribution falls to the left of , so . We're left with




This probability corresponds to a value of , which means