Normal distribution

Question

Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)

𝜇 = 51; 𝜎 = 14

P(40 ≤ x ≤ 47) = 

Answer 1

Hi,

We can solve this using the z-score table (you should have one, if not look up z-score table it's the top result) where the z-scores tell us the number of standard deviations away from the mean a particular value is (given the mean and standard deviation, as we are given in your post).

40 corresponds to 0.786 standard deviations to the left of the mean or -0.79 in the z-score table. The value at z = -0.79 is 0.2148 which is the total probability to the left of this z-score.

47 corresponds to 0.286 standard deviations to the left of the mean or 0.29 in the z-score table. The value at z = -0.29 is 0.3859 which is the total probability to the left of this z-score.

To determine the probability of values between 40 and 47 inclusive we subtract the respective probabilities. That is,

P(40 <= x <= 47) = P(x <= 47) - P(x <= 40)

P(40 <= x <= 47) = 0.3859 - 0.2148 = 0.1711 which is equivalent to a 17.11% probability that we get values between 40 and 47 inclusive.