Question

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. Suppose the maximum speed of a moped is normally distributed with mean value 46.8 km/h and standard deviation 1.75 km/h. Consider randomly selecting a single such moped. (a) What is the probability that maximum speed is at most 49 km/h

Answer 1

The probability that the maximum speed is at most 49 km/h is 0.8340.

Explanation:

Let the random variable X be defined as the maximum speed of a moped.

The random variable X is Normally distributed with mean, μ = 46.8 km/h and standard deviation, σ = 1.75 km/h.

To compute the probability of a Normally distributed random variable we first need to convert the raw score of the random variable to a standardized or z-score.

The formula to convert X into z-score is:

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

Compute the probability that the maximum speed is at most 49 km/h as follows:

Apply continuity correction:

P (X ≤ 49) = P (X < 49 - 0.50)

= P (X < 48.50)

`=P((X-\mu)/(\sigma)<(48.50-46.80)/(1.75))\\=P(Z<0.97)\\=0.83398\\\approx 0.8340`

*Use a z-table for the probability.

Thus, the probability that the maximum speed is at most 49 km/h is 0.8340.