Mopeds (small motorcycles with an engine capacity below 50 cm3) 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 nor mal 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?

a)

And we can find this probability using the normal standard table or excel and we got:

b)

And we can find this probability using the complement rule, the normal standard table or excel and we got:

c)

And we can find this probability with this difference:

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the velocities of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the normal standard table or excel and we got:

Part b

And we can find this probability using the complement rule, the normal standard table or excel and we got:

Part c

And we can find this probability with this difference:

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

d will be the best answer

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