If the mean of a data set is 80 and the standard deviation is 5, about what percent of the data falls between 65 and 95

Question

If the mean of a data set is 80 and the standard deviation is 5, about what percent of the data falls between 65 and 95

Answer 1

If x is data we want to find probability that x is between 65 and 95: P(65 < x < 95) where x has mean 80 and standard deviation 5.

If we can assume x is normally distribution we can transform to standard normal distribution (mean 0 standard deviation 1), using the z-score:

z = (x - mean)/standard deviation

so P(65 < x < 95) = P( (65-80)/5 < z < (95-80)/5) = P(-3 < z < 3) = P(z < 3) - P(z < -3)

using standard normal probability table:

P(z < 3) - P(z < -3) = 0.9987 - 0.0013 = 0.9974