Question

ch3 - population sigma known (σ)
example 1: the average speed of a car is 55mph with a standard deviation of 4.5. assume that the speed of
cars is normally distributed. find the
a) probability that the speed of car is less than 52.5 mph.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 52.5\), \(\mu=55\), and \(\sigma = 4.5\).

$$z=\frac{52.5 - 55}{4.5}=\frac{- 2.5}{4.5}\approx - 0.56$$

Step2: Find the probability using the standard normal table

We want to find \(P(X\lt52.5)\), which is equivalent to \(P(Z\lt - 0.56)\).
Looking up the value of \(z=-0.56\) in the standard - normal table (the cumulative distribution function of the standard normal distribution \(\varPhi(z)\)), we get \(P(Z\lt - 0.56)=0.2877\)

Answer:

The probability that the speed of a car is less than \(52.5\) mph is approximately \(0.2877\)