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

Answer:

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

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{50 - 46.8}{1.75}$

$Z = 1.83$

$Z = 1.83$ 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.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{48 - 46.8}{1.75}$

$Z = 0.685$

$Z = 0.685$ 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