Question

Assume a random variable x is normally distributed with mean = 90 and standard deviation = 5. Find the indicated probability.

P(x<85)

Answer 1

16%

Explanation:

The indicated probability is actually the area under the standard normal curve to the left of the mean. I used the function normalcdf( on my TI-83 Plus calculator to find this quantity:

normalcdf(-1000,85,90,5) = 0.1587.

Note #1: This quantity (area / probability) is the area to the left of 85.

Note #2: by the Empirical Rule, 68% of data lies within 1 s. d. of the mean, so the area between the mean (90) and the score (85) is half of 68%, or 34%. Subtracting this from 50% (the area to the left of the mean), we get 16%, which is roughly equivalent to the 0.1587 we got earlier.