Question

You’re waiting for Caltrain. Suppose that the waiting times are approximately Normal with a mean of 12 minutes and a standard deviation of 3 minutes. Use the Empirical Rule to estimate each of the following probabilities without using the normalcdf function of your calculator:

a) What is the probability that you’ll wait between 9 and 15 minutes for the train?

b) What is the probability that you’ll wait between 6 and 18 minutes for the train?

c) What is the probability that you’ll wait between 3 and 21 minutes for the train?

d) What is the probability that you’ll wait more than 12 minutes for the train?

e) What is the probability that you’ll wait between 12 and 18 minutes for the train?

f) What is the probability that you’ll wait between 3 and 18 minutes for the train?

g) What is the probability that you’ll wait more than 21 minutes for the train?

Answer 1

As per Empirical rule, for normal distribution, 68,95 and 99.7% of data lies within 1, 2 and 3 standard deviations of mean respectively.

Mean = 12 minutes

Standard deviation = 3 minutes

a) 12-3 = 9

12+3 = 15

9 and 15 is the interval of 1 standard deviation of mean

P(you’ll wait between 9 and 15 minutes for the train) = 0.68

b) 12 - 2x3 = 6

12 + 2x3 = 18

6 and 18 is the interval of 2 standard deviation of mean

P(you’ll wait between 6 and 18 minutes for the train) = 0.95

c) 3 is 3 standard deviations from mean

21 is 3 standard deviations from mean

P(you’ll wait between 3 and 21 minutes for the train) = 0.997

d) P(you’ll wait more than 12 minutes) = 0.5

e) P(you’ll wait between 12 and 18 minutes for the train) = P(you’ll wait between 6 and 18 minutes for the train)/2

= 0.95/2

= 0.475

f) P(you’ll wait between 3 and 18 minutes for the train) = P(you will wait between 3 and 12 minutes) + P(you will wait between 12 and 18 minutes)

= 0.997/2 + 0.95/2

= 0.9735

g) P(you will wait more than 21 minutes) = (1 - 0.997)/2

= 0.0015