Question

A radar unit is used to measure speeds of cars on a motorway. The speeds are normally distributed with a mean of 90 km/hr and a standard deviation of 10 km/hr. What is the probability that a car picked at random is travelling at more than 100 km/hr?

Round your answer to the nearest thousandth.

Answer 1

To find the probability that a car is traveling at more than 100 km/hr, we need to calculate the z-score for 100 km/hr using the provided mean and standard deviation.

The formula for the z-score is:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) is the value we are examining (100 km/hr),
- \(\mu\) is the mean (90 km/hr), and
- \(\sigma\) is the standard deviation (10 km/hr).

Substituting the values in:
\[
z = \frac{(100 - 90)}{10} = \frac{10}{10} = 1
\]

Next, we look up the z-score of 1 in the standard normal distribution table to find the probability of a car traveling less than 100 km/hr. The table shows:

\[
P(Z < 1) \approx 0.8413
\]

To find the probability of a car traveling more than 100 km/hr, we need to subtract this from 1:

\[
P(X > 100) = 1 - P(Z < 1) = 1 - 0.8413 = 0.1587
\]

Therefore, the probability that a car picked at random is traveling at more than 100 km/hr is approximately:

\[
\boxed{0.159}
\]

when rounded to the nearest thousandth.