(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?

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Description: (a) The average daily travel time to work for people living in a particular city isnormally distributed. It is known that the lower quartile is 61 minutes, theinterquartile range is 62 minutes, and the standard deviation is 16 minutes.i) Show that t

A First Course in Probability (10th Edition)

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ISBN:9780134753119

Author:Sheldon Ross

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Chapter1: Combinatorial Analysis

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Question

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(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?

Expert Solution

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,

$P (X > 120) = 1 - P (X < 120) = 1 - P (\frac{X - μ}{σ} < \frac{120 - 92}{16}) = 1 - P (Z < 1.75) = 1 - 0.9599 (f r o m n o r m a l t a b l e) = 0.0401$

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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