(a) The average daily travel time to work for people living in a particular city is normally distributed. It is known that the lower quartile is 61 minutes, the interquartile range is 62 minutes, and the standard deviation is 16 minutes. i) Show that the mean is 92 minutes. ii) What percentage of the city's workforce has an average travel time that exceeds 2 hours? iii) If a randomly chosen person starts work at 9 am, at what time do they need to leave home in order to have a 90% chance of arriving on time? iv) This person is running late and leaves home at 8 : 17 am. What is the probability that they arrive on time anyway?
(a) The average daily travel time to work for people living in a particular city is normally distributed. It is known that the lower quartile is 61 minutes, the interquartile range is 62 minutes, and the standard deviation is 16 minutes. i) Show that the mean is 92 minutes. ii) What percentage of the city's workforce has an average travel time that exceeds 2 hours? iii) If a randomly chosen person starts work at 9 am, at what time do they need to leave home in order to have a 90% chance of arriving on time? iv) This person is running late and leaves home at 8 : 17 am. What is the probability that they arrive on time anyway?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Step 1
Given:
Q1 = 61 minutes
IQR = 62 minutes
Standard deviation = 16 minutes.
Part 1:
For normally distributed data we have,
Q1 = Mean – 0.5 × IQR
Thus, mean = Q1 + 0.5 × IQR = 61 + 0.5 × 62 = 92.
Therefore, the mean is 92 minutes.
Step 2
Part 2:
Let X denote the average daily travel time to work for people in a particular city.
Thus, X~N(92, 162)
The probability that the city’s workforce has an average travel time that exceeds 2 hours (120 minutes) is computed as,
Thus, the percentage of the city’s workforce has an average travel time that exceeds 2 hours (120 minutes) is 4.01%
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