Question

Mopeds (small motorcycles with an engine capacity below 50cm3) are very popular in Europe because of their mobility, ease of operation, and low cost. An article 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?

Answer 1

a) 96.64% probability that maximum speed is at most 50 km/h

b) 24.67% probability that maximum speed is at least 48 km/h

c) 86.64% probability that maximum speed differs from the mean value by at most 1.5 standard deviations

Explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

`\mu = 46.8, \sigma = 1.75`

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

This is the pvalue of Z when X = 50. So

`Z = (X - \mu)/(\sigma)`

`Z = (50 - 46.8)/(1.75)`

`Z = 1.83`

`Z = 1.83` has a pvalue of 0.9664

96.64% probability that maximum speed is at most 50 km/h.

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

This is 1 subtracted by the pvalue of Z when X = 48.

`Z = (X - \mu)/(\sigma)`

`Z = (48 - 46.8)/(1.75)`

`Z = 0.685`

`Z = 0.685` has a pvalue of 0.7533

1 - 0.7533 = 0.2467

24.67% 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?

Z = 1.5 has a pvalue of 0.9332

Z = -1.5 has a pvalue of 0.0668

0.9332 - 0.0668 = 0.8664

86.64% probability that maximum speed differs from the mean value by at most 1.5 standard deviations