# Standardized Normal Distribution Problem

Mopeds (small motorcycles with an engine capacity below $50~cm^3$) are very popular in Europe because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections” (J. of Automobile Engr.,2008: 1615–1623) described a rolling bench test for determining maximum vehicle speed.A normal distribution with mean value 46.8 km/h and standard deviation 1.75 km/h is postulated. Consider randomly selecting a single such moped.

a.What is the probability that maximum speed is at most 50 km/h?

b.What is the probability that maximum speed is at least 48 km/h?

c. What is the probability that maximum speed differs from the mean value by at most 1.5 standard deviations?

I am working on part c), at the moment. How I calculated it was in terms of the adjusted mean and deviation:

$P( \mu - \sigma \le Z \le \mu + \sigma) = P(-\sigma \le Z \le \sigma) = P(-1.5 \le Z \le 1.5) = \phi(1.5) - \phi(-1.5)$

This doesn't lead to the correct answer. What have I done wrong?

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I'm terribly sorry. This was a question I was going to ask, but decided not to; and stackexchange saved the title. – Mack Mar 9 '13 at 18:21
I get about $0.8664$. You did not indicate what numerical answer you got. – André Nicolas Mar 9 '13 at 18:39
Yes, that is what I get too. The answer keys says that 0.1336 is the answer. – Mack Mar 9 '13 at 19:18
That's the probability we differ from the mean by at least $1.5$ standard deviation units. If you reported the wording of the problem correctly, the answer key is wrong. – André Nicolas Mar 9 '13 at 19:22